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INTERNATIONALE SCHRIFTENREIHE ZUR NUMERISCHEN MATHEMATIK SERlE
INTERNATIONALE D'ANALYSE NUMERIQUE
Editors:
Ok. Blanc, Lausanne; A. Gkizzetti, Roma; A. Ostrowski, Montagnola;
J. Todd, Pasadena; H. Unger, Bonn; A. von Wijngaarden,
Amsterdam
VOL. 13
QUADRATURE FORMULAE
1970 SPRINGER BASEL AG
ISBN 978-3-0348-5837-3 ISBN 978-3-0348-5836-6 (eBook)
All Rights reserved No Part of this Book may be reproduced in any
form, by Photostat, Microfilm.
or any other means, without written permission from the Publishers
Library of Congress Catalog Card Number: 76 124441
© Springer Basel AG 1970 Originally published by Birkhiiuser Verlag
Basel, in 1970
Softcover reprint of the hardcover 1st edition 1970
DOI 10.1007/978-3-0348-5836-6
PREFACE
Quadrature formulae are generally obtained substituting the
integral to b b
evaluate J f(x) dx with the integral J <p(x) dx where <p(x)
is a function a a
approximating f(x), whose indefinite integral is expressible in
elementary functions. The choice of <p(x) is performed using
interpolation methods. Thus the problem of approximating the
integral of f(x) is reduced to the approximation of f(x) itself,
that is to a problem not equivalent to the first one. Therefore, it
is reasonable to think that it possible to obtain quadrature
formulae, without using interpolation methods. In a paper of 1913,
G. PEANO (see G. PEANO (1]) first made an attempt in this way and
succeeded in obtaining Cavalieri-Simpson's formula, with an
integral ex pression of the remainder, only by means of
integration by parts. This method was systematically employed by R.
VON MISES, who (in a paper of 1935; see R. VON MISES [1]) showed
how it is possible to get every qua drature formulae with the only
tool of integration by parts.
In 1935, also J. RADON (1] showed how the integral expression of
the remainder can be obtained by means of the Green-Lagrange
identity, relative to a linear differential operator and its
adjoint. However, these fundamental works remained nearly unknown
and still to-day the ele mentary method suggested by them is not
used in every book concerning to quadrature formulae. From 1954,
one of us (see A. GHIZZETTI [1], [2], [3], [4]) had the:
opportunity to reconsider and to probe the study of Radon's
procedure, pointing out that it allows to obtain not only the
expression of the remainder, but also of the part approximating the
integral. This result can be attained in more general conditions
than the usual ones, that is making use of Lebesgue integrals for
finite or infinite intervals. In this way a method, which allows to
obtain all the known quadrature for mulae and many others, is
carried out by a systematic application of some simple rules. It is
worthwhile to note that Radon's method includes that of von Mises,
since the integrations by parts used by von Mises, are already
performed in the Green-Lagrange identity, used by Radon.
In 1965, we have reconsidered the study of these methods and their
applications (see A. GHIZZETTI [5]; A. OSSICINI [1], [2], [3]; A.
GHIZZETTI and A. OSSICINI [1]), availing us of the valued help of
some collaborators of ours. In this way we succeeded in collecting
all necessary elements to draw up this book, in which we expound
the theory of quadrature formulae, only by means of the above
mentioned methods.
6 Preface
This book is divided into six chapters. Chapter 6 must be
considered independently; it contains exclusively the solutions to
all the problems, proposed to the reader in every preceding
chapter. Chapter 1 is devoted to the exposition of some preliminary
notions about differential linear operators which possess adjoint
operator. In Chapter 2, as application uf the preceding chapter,
the general rule for constructing all the quadrature formulae,
having a structure analogous to the classical ones, is given and
discussed. We call these formulae elementary to distinguish them
from the generalized ones, considered in Chapter 5. These formulae
are obtained dividing the integration interval in partial intervals
and applying an ele mentary formula to everyone of them. These
generalized formulae are just those, which we use for the numerical
computation of integrals; on this subject some questions arise
about convergence, which are considered in Chapter 5 itself.
Chapters 1, 2, 5 are the most important and this book could be
limited to them, since Chapters 3, 4 are exclusively devoted to
some applications of the general rule expounded in Chapter 2. They
could be considered just an collection of problems. However also
Chapters 3, 4 are important, since not only they explain, with
concrete examples, the power of the general theory, but in addition
they present some very difficult problems, whose solutions requires
the knowledge of notions and special techniques, useful in some
other cases. For this reason, in Chapter 3 we have collected, for
reader's convenience, some elements about certain special
functions, which OCCUr in the extensive collection of examples of
Chapter 4. These examples have been treated by the same method;
what varies from one example to another is the technique of
remainder estimate, by means of the integral represent ation
provided by the method.
Since in this book we make use of a unique method, which few
Authors have considered, the closely related Bibliography would be
not large; therefore we have added the list of the most recent
books on quadrature formulae; to these books we refer readers for
many practical questions (for example, numerical table about nodes
and coefficients of the classical for mulae). In some paragraphs,
we have put a bibliographic note to point out some papers on which
we have founded the draft of the paragraph itself. We are grateful
to "Birkhiiuser Verlag" which has received this books in one of its
valuable collections.
Rome, October, 1969 A. GHIZZETTI . A. OSSICINI
REMARK
Every chapter is divided into paragraphs, which are denoted by a
symbol as § 3.4 (to indicate the fourth paragraph of Chapter 3). In
every paragraph, formulas are indicated by three numbers into
brackets; for instance (3.4.7) means the 7th formula of § 3.4. For
theorems, an analogous notation is used, without brackets, and
having the 3th number Roman; for instance, theor. 3.4.II means the
second theorem of § 3.4. In the end of every chapter 1,2,3,4,5
there is a paragraph concerning problems proposed to readers; their
solutions are collected in Chapter 6. These problems are denoted as
§ 1.5., Problem 2, but of course the corresponding solution of
Chapter 6 must be considered.
TABLE OF CONTENTS
Chapter 1. Additional results on linear differential equations . .
11
1.1 Linear differential operators and their adjoints . 11 1.2 The
Green-Lagrange identity. . . . . . . . . 12 1.3 Relations between
the solutions of a linear differential
equation and those of the adjoint equation .... 13 1.4 Linear
differential equations considered in infinite
intervals 18 1.5 Pro blems . . . . 25
Chapter 2. Elementary quadrature formulae and a general procedure
for constructing them. . . . . . . . . . . . . . . . . . . 27
2.1 The elementary quadrature formulae on a finite interval 27 2.2
The elementary quadrature formulae on the interval
[a, +(0). . . . . . . . . . . . . . . . . . . . . 32 2.3 Elementary
quadrature formulae on the interval (- 00,
+00) . . . . . . . . . . . . . . . . . . 37 2.4 Some remarks on the
rule for obtaining quadrature
formulae . . . . 38 2.5 Gauss problem. . 41 2.6 Tchebychef problem
43 2.7 Problems . . . . . 45
Chapter 3. Special functions 47
3.1 Bernouilli polynomials and numbers. 47 3.2 Euler functions. . .
. . . . . . 50 3.3 Orthogonal polynomials connected to a given
weight
function. . . . . . . . . . . . 52 3.4 Jacobi polynomials. . . . .
. . 58 3.5 Particular cases of J aco bi polynomials. 64 3.6
Laguerre polynomials. . . . . . . 68 3.7 Hermite polynomials . . .
. . . . . 70 3.8 Some notions on divided differences. . 71 3.9 The
s-orthogonal polynomials connected to a given
weight . 74 3.10 Problems 78
10 Table of contents
Chapter 4. Various examples of elementary quadrature formulae
80
4.1 General remarks on the contents of this chapter 80 4.2
Deduction of three classical formulae . . . . . 80 4.3 The
Euler-MacLaurin formula . . . . . . . . 84 4.4 Other examples of
application of the generale rule. 86 4.5 The Gauss-Jacobi formulae
. . . . . . . . 92 4.6 Particular cases of the Gauss-Jacobi
formulae 97 4.7 The Bouzitat formulae of the first kind . 101 4.8
The Bouzitat formulae of the second kind 106 4.9 Gauss-Laguerre
formulae 110 4.10 Gauss-Hermite formulae. . . . . 114 4.11
Newton-Cotes formulae . . . . . 116 4.12 Examples of Tchebychef
problems 124 4.13 Quadrature formulae connected to s-orthogonal
poly-
nomials . . . . . . . . . . . . . . . . . . . . . 131 4.14 Some
examples of quadrature formulae exact for trigo
nometric polynomials . . . . . . . . . . . . . . . 139 4.15
Problems . . . . . . . . . . . . . . . . . . . . 145
Chapter 5. Generalized quadrature formulae and questions of conver-
gence . . . . . . . . . . . . . . .. .... 148
5.1 Generalized quadrature formulae in the case of the constant
weight . . . . . . . . . . . . . .. 148
5.2 Generalized quadrature formulae with any weight 153 5.3 An
example of generalized quadrature formula . 159 5.4 Problems . . .
163
Chapter 6. Solutions to problems 164
6.1 Solutions to problems of § 1.5 164 6.2 Solutions to problems of
§ 2.7 167 6.3 Solutions to problems of § 3.10 . 169 6.4 Solutions
to problems of § 4.15 174 6.5 Solutions to problems of § 5.4
183
Bibliography. 191
CHAPTER 1
Additional results on linear differential equations
1.1. Linear differential operators and their adjoints. In this
chapter we shall consider exclusively real-valued functions. If [a,
b] is a finite interval, we shall call L[a, b] the class of
functions f(x) Lebesgue-integrable (summable) in [a, b] and A Ck[a,
b] the class offunctions f(x) whose k-th derivative f(k)(x) is
absolu tely continuous in [a, b] (k = 0, 1,2, .. .).1) It is known
that if f(x) E A Ck[a, b] the derivative t(k+l)(X) exists almost
everywhere in [a, b] and that f(k+1)(x) E L[a, b].
Let us consider in [a, b] a linear differential operator of order n
(n = 1, 2,3, ... ):
n dn - k
with the following conditions on the coefficients ak(x)
(1.1.1)
ao(x) = 1; ak(x) E A Cn- k- 1 [a, b], (k = 1,2, ... , n - 1);
n,,(x) E L[a, b). (1.1.2)
The operator E can be applied to the functions u(x) E A cn-l[a, b],
obtaining the function (defined almost everywhere):
n E[u(x)] = E(u) = 2: ak(x) u(n-k)(x) E L[a, b] .
k=O (1.1.3)
In the following ~e shall have occasion to associate with the
operator E the reduced operators
, dr - k E, = L ak(x) dxr - k ' (r = 0,1, ... , n - 1) .2)
(1.1.4)
k=O
By virtue of the hypotheses (1.1.2) we can consider the so-called
adjoint operator E* of the operator E, defined by
n dn - k
n dn - k E* = '\' a*(x)--
.:::... k dxn - k k=O
(1.1.5)
(1.1.6)
1) For k = 0 we have the class A Cora, b] of absolutely continuous
functions; this class is also denoted by A C[a, b].
2) For T = n we obtain En = E.
12 1. Additional results on linear differential equations
and an easy computation shows that
at(x) = ~ (- l)n-i (; == ~) a~k-i)(x) , (k = 0, 1, ... , n).
(1.1.7)
From (1.1.2) and (1.1.7) easily follow
a~(x) = (- l)n; at(x) E A on-k-1 [a, b], (k = 1,2, ... , n - 1);
a!(x) E L[a, b],
(1.1.8)
analogous to (1.1.2), so that the operator E* can be applied to the
functions v(x) E A on-1[a, b] obtaining a function (defined almost
everywhere):
n E*[v(x)] = E*(v) = 2: (- l)n-k [ak(x) v(x)]<n-k) E L[a, b].
(1.1.9)
k~O
From (1.1.8) it follows that we can consider the operator E**,
adjoint to E*, but we do not obtain a new operator, since the
reader will easily prove that E** = E.
We shall also have the occasion to consider the adjoint operator E:
of the reduced operators E F , for which we obviously have
r
E:(v) = 2: (- 1),-k [ak(x) v(X)](,-k) , (r = 0, 1, ... , n - 1).
(1.1.10) k~O
Note also that there exists the recurrence formula
* d * E, (v) = - dx E,_dv) + a,(x) v(x) , (r = 1, 2, ... , n) .1)
(1.1.11)
1.2. The Green-Lagrange identity. We shall begin by proving that,
if u(x), v(x) E A Ok-1[a, b], (k = 1,2, 3, ... ) the relation
d k-1 v(x) U(k)(x) - (- l)k u(x) v(k)(x) = - .r: (- l)k-i-1 u(i)(x)
v(k-i-1)(X) (1.2.1)
dx i~O
holds almost everywhere in [a, b]. We prove it by induction,
observing first of all that for k = 1, (1.2.1)
is evident since it becomes v u' + u v' = ! (u v) •
To pass from k to k + 1, it is thus sufficient to write
v u(k+1L (- 1)k+1 u V(k+1) = ! [(- l)k U V(k)] + [v (:: t) -(- l)k
:: V(k)]
=~[(-I)kUV(k) + 1;1 (_I)k-i_l(dU)(i)v(k_i_1)] dx i~O dx
= ~ [( - l)k u V(k) + f (- l)k-i U(i) V(k-i)] = ~ f (- l)k-i u(i)
V(k-i). dx i~1 dx i~O
1) For r = n, remember that E: = E*.
1.3 Relations between the solutions 13
Now we can prove the Green-Lagrange identity, relative to a linear
dif ferential operator E of order n of the type (1.1.1) under the
hypotheses (1.1.2) and to its adjoint E* [see (1.1.5)]. The
Green-Lagrange identity states that for every pair of functions
u(x), v(x) E A on-I [a, b], the following equation bolds almost
everywhere in [a, b]:
d n-I
v(x) E[u(x)] - u(x) E*[v(x)] = dx k~ u(k)(X) E:_k_I[V(X)] ,
(1.2.2)
where the operators E:-k - I are defined by (1.1.10). We have n
n
vE(u) - uE*(v) = v J.: aj u(n-i) - u J.: (- l)n-; (at v)(n-i) ;=0
;=0
n-I = J.: [aj v . u(n-i) - (- l)n-i u . (at v)(n-i)] ,
i=O
and applying (1.2.1) to the functions u E A on-I [a, b], at v E A
on-i-I [a, b]
n-I d n-i-I
V E(u) - u E*(v) = I: - E (- l)n-i-k-I U(k) (at v)(n-i-k-I) i=O dx
k=O
d n-I n-k-I = - I: u(k) I: (- l)n-k-I-i (at v)(n-k-I-i) ;
dx k=O i=O
from which, using (1.1.10), (1.2.2) follows.
1.3. Relations between the solutions of a linear differential
equation and those of the adjoint equation. Let us use the symbols
E, E* with the meaning already given in (1.1.1), (1.1.2) and
(1.1.5). Let us consider in [a, b] the non-homogeneous linear
differential equation
E(u) =!(x) (1.3.1)
with !(x) E L[a, b]. Let [uI(x), u2(x), ... , un(x)], where Ut E A
On-Ira, b], be a fundamental system of solutions of the homogeneous
equation E(u) = 0, and let U(x) =f. 0 their Wronskian. It is known
that the general solution of (1.3.1) is expressed by
n '" u(x) = 2: lXi Ui(X) + J K(x, ;) !(;) d; ,
i=I a (1.3.2)
where lXI' lX2' ••• , lX,. are arbitrary constants and K(x,;) is
the so-called Oauchy resolvent kernel, which is (as function of x)
the particular solution of the homogeneous equation E(u) = 0 which
satisfies, at the point;, the following initial conditions:
[:~ K(x, ;) 1=; = dh, n-I , (h = 0, 1, ... , n - 1) .1)
(1.3.3)
1) ~rs is the Kronecker delta = 0 (if r =1= 8)
= 1 (if r = 8).
It may be written n K(x, ~) = L: u,(x) vd~) , (1.3.4)
i=1
where the functions v1(x), v2(x), ... , vn(x) are given by the
system of n linear equations n
L: U~h)(X) v,(x) = fih, n-l , (h = 0, I, ... , n - I) , (1.3.5)
i=1
whose determinant is U(X).l) Therefore (1.3.2) may also be
written
u(x) =ig u,(x) [(Xi + ! vM)f(~) d~] . (1.3.6)
As to the derivatives of u(x), it is well known that from (1.3.2)
and (1.3.6) follows
(1.3.7)
We observe another property of the functions Vi(X) introduced by
(1.3.5); we have almost everywhere
that is n L: u~n)(x) Vi(X) = - a1(x) .
;=1 (1.3.8)
The relations (1.3.5), (1.3.8) may be generalized in the way shown
by the following theorem:
Theorem 1.3.1. The function8 Vj(x) defined by the 8y8tem (1.3.5)
belong to the clas8 A on-1[a, b] and 8ati8fy the following
relations
n L: U~h)(X) E:[vi(x)] = fih, n-k-l , (h, k = 0, I, ... , n - I),
(1.3.9)
;=1
n L: u~n)(x) Et[Vi(X)] = - Uk+l(X) , (k = 0, I, ... , n - I) ,2)
(1.3.10)
;=1
where E: are the operator8 defined by (1.1.10).
Proof. Since Et(Vi) = Vi' for k = ° the equation (1.3.9) reduce to
(1.3.5) (which assures Vi E A G) and (1.3.10) reduces to (1.3.8).
We proceed by
1) From (1.3.5) and from the fact that Ui(X) E A Cn-l[a, b] it
follows immediately that v;(x) E A C[a, b]; but we shall prove
shortly that Vi(X) E A cn-l[a, b] (see theorem 1.3.1).
2) (1.3.10) holds almost everywhere.
1.3 Relations between the solutions 15
induction to assume that for a certain k (with 0 < k < n - I)
the equation (1.3.9) hold, that is Et(Vi) E A 0 or Vi E A Ok. We
prove that as a conse quence (1.3.10) holds, with the same k and
(1.3.9) with k + I instead of k; thus Vi E A Ok+!. We may prove
(1.3.10) immediately by observing that
n = - 2: a" I5n- h, n-k-l = - ak+l .
h=l
To prove the other assertion, we start from (1.3.9) and
differentiate both members; there results almost everywhere
n n d ~ uV'+ 1) E*(v ) + Y' U<.h) - E*(v ) = 0 L,-, ki ...J'dxkt
,
i=l ;=1 (h = 0, I, ... , n - I) . (1.3.11)
The value of the first term is 151t+1, n-k-1 if h = 0, 1, ... , n -
2 [from (1.3.9)] and - ak +! if h = n - I [from (1.3.10)]. For the
second term we have from (1.1.11)
vVith this, (1.3.11) yields
= - L u~h) Et+!(v;) + aU1 15k , n-1 . i=1
n
l51t+l, n-k-1 - 2: t4lt ) Et+!(v;) + 0 = 0 , i=1 '
(h = 0, I, ... , n - 2) , I n
- ak+l - 2: u~n-l) Et+l(Vi) + aU1 = 0 i=1
which may be gathered in
n
f (1.3.12)
2: u~h) Et+1(V;j = bh, n-k-2 , (h = 0, 1, ... , n - 1) , (1.3.13)
i=1
which is precisely (1.3.9) with k + 1 instead of k, q.e.d.
\Ve now prove another theorem:
Theorem 1.3.II. The functions [v1(.c), v2(x), ... ,vn(x)] defined
by the system (1.3.5) form a fundamental system of solutions of the
homogeneous adjoint equation
E*(v) = 0
1 V(:t;) = U(x) .
16 1. Additional results on linear differential equations
The general solution of the non-homogeneous adjoint equation
E *(v) = g(x) , (g(x) E L[a, b]) , (1.3.16) is given by
n x v(x) = ~ fJi Vi(X) + J K*(x,~) g(~) d~ , (1.3.17)
i=l a
where fJI> fJ2' .. , fJn are arbitrary constants, while for the
resolvent kernel K *(x, ~) we have n
K*(x,~) = - K(~, x) = - ~ Vi(X) Ui(~) . (1.3.18)
Therefore (1.3.17) may also be written
v(x) = i~ Vi(X) [fJi - J Ui(~) g(~) d~] . (1.3.19)
Proof. We begin by proving that E*(Vi) = 0 (i = 1,2, ... , n).
Starting from (1.3.9) written with k = n - 1, we may proceed in the
same fashion as in (1.3.11), (1.3.12), (1.3.13) (with k = n - 1, E!
= E*) thus obtaining
n ~ U~h) E*(Vi) = 0 , (h = 0, 1, ... , n - 1) ,
i=l
which, together with U(x) i= 0, give E*(Vi) = O. To prove (1.3.15)
we observe that from (1.1.10) it easily follows that the
derivative v(r)(x), (r = 0, 1, ... ,n - 1) of a v( ) E A en- 1[a,
b] may be expressed by means of E~(v), E~_l(V), ... ,Et(v) with a
formula of the type r
v(r)(x) = (- 1)' E~[v(x)] + L crs(x) E;_8[V(X)] . (1.3.20)
Thus we may write
1.4 Relations between the solutions 17
(
u~ . .. u\n_l») :: u; ... u~n-l) Un u~... U~n-l)
we get, by virtue of (1.3.9)
t5 t5 00 ... 0] t50,n-l t51,n-l ... ~n-1,n-l 00 ... 10
Vex) U(x) = (-I)(~) O,n-2 1,n-2'" n-l,n-2 = (-I)(~) . . . . . = 1
.
01. .. 00 10 ... 00
Finally we prove (1.3.18). Considering that in the first member of
the diffe rential equation (1.3.16) the coefficient of v(n) is not
equal to 1, but to (_I)n [see (1.1.8)], we may assert that the
resolvent kernel K*(x, ~) is given by
n 1: Vi(X) Wi(~)' where the functions w1(x), W2(x) , ... , wn(x)
are defined by the
i=1
system of linear equations n
1: v~h) W, = (_I)n t5h,n-1, (n = 0, 1, ... ,n - 1). i=l
By application of equation (1.3.20), this system may also be
written
ig [( _I)h Et(Vi) + 8~ Clis Et_8(Vi)] Wi = (_I)n ~h, n-1'
(n=~I, .. . ,n- 1),
or also, by subtracting from every equation an obvious linear
combination of the preceding ones,
n 1: Et(Vi) Wi = (_l)n-h bh n-l ,
i=l ' (n = 0, 1, ... , n - 1) .
Finally another transformation leads to n
1: Et(Vi) (- Wi) = ~h,n-l' (n = 0,1, ... ,11- 1). (1.3.21)
i=1
But from (1,3.9) we have n
1: Et(Vi) Ui = bo,n-h-l = bh,n-l, (n = 0,1, ... , n - 1), (1.3.22)
i=1
and since the determinant of Et(vj) is (-I)(~) Vex) =1= 0, the
comparison of (1.3.21) with (1.3.22) leads to Wi = - Uj. Equation
(1.3.18) follows imme diately.
2 Ghizzetti/Ossicini: Quadrature
18 1. Additional results on linear differential equations
To complete this theorem, we note that from (1.3.17) and (1.3.19)
we get n z
E:_h _ 1[V(X)] = }; fJI E:_h _ 1[Vi(X)] + J E:_h _ 1[K*(x, m g(~)
d~ 'i=l a
= ig E:_h_1[t'i(X)] [fJi - ! uM) g(~) d~] . (1.3.23)
A fundamental system (Ut, u2 , ••• ,un) of solutions of E(u) = °
and a funda mental system (VI' v2 , ••• ,vn) of solutions of E*(v)
= ° will be called associated if the latter can be obtained from
the first using (1.3.5).
The following theorem will also be useful:
Theorem 1.3.111. With the meaning previously given to Uf(X) and
Vj(x) the formula
holds.
n-l
(i, j = 1,2, ... , n) (1.3.24)
Proof. Let us denote by Ail the first member of (1.3.24). Holding
the index i fixed, for every h = 0,1, ... ,n - 1 we may write
n n n-l n-l n
}; U~h) Ail = }; U}h) }; U~k) E:_k _ 1(vl) = }; U~lc) }; u)hl E:-k
- 1(t'l) j=l j=l k=O lc=O j=l
and thus by (1.3.9) n n-l
}; U}hl Ail = }; 'U~k) ~h,lc = u~h) , (h = 0, 1, ... , n - 1) . j=l
k=O
This system of n linear equations in n unknowns Ail' Ai2' .•• ,Ain
in virtue of the fact that U(x) =F 0, has only the solution Ail =
~il which establishes (1.3.24), q.e.d.
1.4. Linear differential equations considered in infinite
intervals.1) The statements made in § 1,1, 1.2 and 1.3 pertain to
the case in which the inde pendent variable x varies in a finite
interval [a, b]. With slight modifications they hold also when x
varies in an infinite interval, for instance [a, +00).2) It
suffices to substitute for the class L[a, b] the class 40c [a, +
00) of locally 8ummable functions in [a, + 00) and for the class A
Glc[ a, b] the class A G~oc [a, + 00) of functions whose k-th
derivative is locally absolutely continuous in [a, + 00).3) With
only these modifications all the formulae and theorems of § 1.1,
1.2, 1.3 remain valid. We shall now give some other basic ideas
essentially connected with the infinite interval.
1) See A. GmzZETTI [2], [3], [4]. 2) Here we only refer to the case
of [a, + (0); similar considerations hold for the
interval (- 00, +(0), and we shall summarily state them at the end
of the paragraph. 3) A function f(x) is called locally 8ummable (or
locally ab80lutely continuoUB) in
[a, +(0) when it is summable (or absolutely continuous) in every
finite interval con tained in [a, +(0).
1.4 Linear differential equations considered in infinite intervals
19
Writing f(x) E I [a, + 00), that is, f(x) is integrable in [a, +
00), we shall mean thatf(x) is locally summable in [a, + 00) and
moreover that the integral +00
f f(x) dx is convergent'!) In a particular case it may happen that
this integral a
be absolutely convergent; we note that this is the same as saying
that f(x) is Lebesgue-integrable (summable) on the interval [a, +
00), that is f(x) E L [a, + 00). Let us now consider again the
differential equations (1.3.1) and (1.3.16), namely
E(u) = f(x)
under the new hypothesesf(x), g(x) E L\oc[a, + 00).
(1.4.1)
(1.4.2)
From § 1.3 we know that having introduced two associated systems,
(uv ... ,un), (vv ... ,vn) of solutions of the homogeneous
equations E(u) = 0, E*(v) = 0, we can derive from (1.4.1) and
(1.4.2) the equations [see (1.3.7) and (1.3.23)]:
u(h)(x) = i~ u<f')(x) [tXi + ! f(~) vM) d~], (h = 0, 1, ... , n
- 1) ,
E:_h_1[V(X)] =ig E:_h_1[Vi(X)] [Pi - J g(~) uM) d~], (h = 0, 1, ...
, n - 1) ,
where tXi, Pi are arbitrary constants.
( 1.4.3)
(1.4.4 )
Now let us assume that the term denoted by g(x) satisfies the
additional hypotheses
g(x) Ui(X) E I [a, + 00) , (i = 1,2, ... ,n) . (1.4.5)
We choose the arbitrary constants Pi so that lim [Pi - f g Ui d~] =
0; it is evident that we must put :1:->-+00 a
+00
Pi = f g(~) uM) d~ , (i = 1,2, ... , n) . (1.4.6) a
In this way we specify a certain particular solution of (1.4.2)
that we shall denote by V(x) and for which (1.4.4) becomes
n +00 E:_h_1[V(X)] = 1: E::-h- 1 [Vi(X)] f g(~) Ui(~) d~ ,
i~l z
(h = 0, 1, ... , n - 1) , (1.4.7)
z 1) This means that the lim f f(x) dx is finite. In the case
oftheinterval (- 00, + 00)
o z both the limits lim f f(x) dx and lim f f(x) dx must be
finite.
x-+-oo x X-J-+OO 0
2·
and in particular for h = n - 1
n +00 V(x) = ~ Vj(x) J g(;) '11M) d; (1.4.8)
i=1 x that is, recalling (1.3.18)
+00 V(x) = J K(;, x) gm d; . (1.4.9)
x
This particular solution V(x) will be denoted as that particular
solution of (1.4.2) which satisfies the initial conditions null at
the point + 00. This sentence is not to be taken literally; in fact
generally it is not true that
lim V(h)(X) = 0, (h = 0, 1, ... ,n - 1) . 1) (1.4.10)
X-++CX>
However, we shall use it for the following three reasons:
1°) if it is true that in general (1.4.10) do not hold, although n
linear in dependent combinations of V(h)(x) do exist (with
coefficient functions of x) which tend to zero for x -+ + 00, we
have in fact
n-l
~ U~h)(X) E!_h_l[V(X)] = [using (1.4.7)] h=O
n-l n +00 = ~ U~h)(X) ~ E!_h_l[Vt(X)] r g(;) uM) d;
h=O ;=1 i n +00 n-l
= ~ J g(;) uM) d; ~ U~h)(X) E!_h_l[Vt(X)] = [using (1.3.24)] ;=1 x
h=O
n +00 +00 = ~ J g(;) uM) d; . bjt = J g(;) uM) d;
;=1 x x
and therefore n-l
lim ~ U~h)(x) E!_h_l[V(X)] = 0, (i = 1,2, ... , n); 2) (1.4.11)
x-++oo h=O
1 1 1) For instance, having assumed n = 1, E(u) = u' + - u, E*(v) =
- v' + - v,
x x 1 1
g(x) = - in the interval [1, + 00) we have u1(x) = -, v1(x) = x.
The hypothesis x x
g u1 E 1 [1, + 00) is satisfied and it turns out that
+00 f 1 1 V(x) = x T . T dt; = 1 .
x
2) In the example of the preceding footnote, (1.4.11) becomes
1 u1(x) V(x) = -·1 ..... 0 (for x ..... + 00).
x
1.4 Linear differential equations considered in infinite intervals
21
x, 2°) the expression (1.4.9) of V(x) is similar to the expression
J K(~, x) g(~) d~
x of the solution of (1.4.2) which satisfies the initial conditions
null at the point xo; 1)
3°) having introduced the above mentioned sentence, in Chapter 2,
we shall have the advantage to be able to use the same language for
the quadrature formulae on finite intervals and for those on
infinite intervals.
In connection with this solution V(x) of (1.4.2) and with any
solution u(x) of (1.4.1.), we consider the following function
n-1
(1.4.12)
In the next chapter we shall see that, in the theory of quadrature
formulae on infinite intervals, it is interesting to discover under
what other conditions [besides (1.4.5)] we may assert that
lim w(x) = 0 . (1.4.13)
Thus we now give the following theorem:
Theorem 1.4.1. If the given function g(x) in equation (1.4.2)
verifies the following hypotheses [a particular case of
(1.4.5)]
g(x) Ui(X) E L [a, + (0), (i = 1,2, ... , n) (1.4.14)
and if the given function f(x) in equation (1.4.1) satisfies these
other hypotheses +00
f(x) Vi(X) J Ig(~) uM)1 d~ E L [a, + (0) , (i = 1,2, ... , n) ,
(1.4.15)
then, for every solution u(x) of (1.4.1) and for that particular
solution V(x) of (1.4.2) which satisfies the initial conditions
null at the point + 00, we have
g(x) u(x) E L [a, + (0) ,
f(x) V(x) E L [a, + (0) , n-l
lim w(x) = lim 2 U(h)(X) E!_h_l[V(X)] = 0 .
Proof. Equation (1.4.15) says that +00 +00 J If(x) Vi(X) I dx J
Ig(~) uM)1 d~ < + 00 ,
a x
(1.4.16)
(1.4.17)
(1.4.18)
whence, by a well-known theorem of Tonelli, the function f(x) Vi(X)
g(~) uM)
1) If we write (1.4.10) and (1.4.11) with lim , they are
equivalent; on the contrary X~Xo
they are not equivalent in the case of the point + 00.
2) This function is the same which occurs in the right member of
the Green-Lagrange identity (1.2.2).
22 1. Additional results on linear differential equations
of the two variables x and ~ is summable in the unbounded domain (~
> x ~ a) of the (x, ~) plane. Then.applying the Fubini theorem
we may write
+00 +00 +00 I;
J f(x) Vi(X) dx J g(~) uM) d~ = J g(~) Ui(~) d~ J f(x) Vi(X) dx a x
a a
and also, interchanging the variables x and ~ in the second
member:
+00 +00 +00 x
J f(x) Vi(X) dx J g(~) Ui(~) d~ = J g(x) Ut(X) dx J f(~) vM) d~.
(1.4.19) a x a a
Thus we may say that from a hypothesis given above we draw not only
the obvious conclusion
+00 f(x) Vt(X) J g(~) Ui(~) d~ E L [a, + 00 ) , (i = 1,2, ... ,n)
(1.4.20)
but also x
g(a..) Ui(X) J f(~) vM) d~ E L [a, + (0), (i = 1,2, ... , n)
(1.4.21) a
showing the validity of (1.4.19). Once that is done, we observe
that using (1.4.3), with h = 0, we obtain
g(x) u(x) = i~ [Xi g(x) Ui(X) + g(x) Ui(X) !f(~) vM) d~] , which,
together with (1.4.14) and (1.4.21) make (1.4.16) hold and
establishes the validity of the formula
From (1.4.8) we get n +00
f(x) V(x) = 2: f(x) vlv) J g(~) uM) d~ i=1 x
which, together with (1.4.20), proves (1.4.17) and the existence
of
+00 n +00 +00 J f(x) V(:t) dx = 2: J f(x} Vi(X) dx J g(~) uM) d~ .
a i=l a x
Now we observe that the Green-Lagrange identity (1.2.2) gives
d V E(u) - U E*(V) = dx w(x)
that is d
f(x) V(x) - g(x) u(x) = dx w(x) .
There follows x x J f(x) Vex) dx - J g(x) u(x) dx = w(x) -
w(a}
a a
1.4 Linear differential equations considered in infinite intervals
23
and therefore, with a passage to the limit for x --+ + 00, and
using (1.4.22) and (1.4.23):
n [+00 +00 ",.!.i:oo w(x) = w(a) +i~ ! f(x) v,(x) dx 1 g(~) uM)
d~
+00 +00 '" ] - ex, J g(x) Uj(X) dx - J g(x) u,(x) dx J f(~) vM) d~
.
a a a
In virtue of (1.4.19) this formula reduces to' n +00
lim w(x) = w(a) - J; ex, J g(x) Uj(X) dx "'-++ 00 i=l a
and to obtain (1.4.18) we need to prove that the value of this last
expression is O.
To this purpose note that (1.4.12) in connection with (1.4.3) and
(1.4.7) gives
w(x) = :#: i~ U~h)(X) [exi + /f(~) vM) d~ tR E!_h_l[vf(x)] JOOg(~)
uM) d~
= iR ig [ex i + J f(~) vM) d~] JOOg(~) uM) d~ :#olU}h)(X)
E!_h_l[v1(x)]
and then using (1.3.24)
w(x) = i~ it [exi + I f(~) vM) d~] T")g(~) uM) d~ i5"
= ig [ex i + J f(~) vM) d~] JOOg(~) uM) d~ . n +00
For x = a we obtain in particular w(a) = X exi J g(~) u .• m d~,
q.e.d. i=l a
In the case of the interval (- 00, + 00) the hypotheses analogous
to (1.4.5) is
g(x) u,(x) E 1(- 00, + 00), (i = 1,2, ... , n) . (1.4.24)
If it is satisfied, we may give in (1.4.4)1) to the arbitrary
constants {Ji the values
o +00
(Ji = - J g(~) uM) d~ or (J, = J g(~) uM) d~ -00 0
and therefore, with regard to the differential equation (1.4.2)
consider the particular solution
n '" '" V_(x) = - J; v,(x) J g(~) u,(~) d~ = - J K(~, x) g(~) d~
(1.4.25)
i=l -00 -00
1) In (1.4.3) and (1.4.4) the symbol a denotes now an arbitrary
fixed point; we shall assume for instance a = O.
24 1. Additional results on linear differential equations
which satisfies the initial conditions null at the point - 00, or
the particular solution
n +00 +00 V +(x) = l: Vi(X) f g(;) uM) d; = f K(;, x) g(;) d;
i=l x x (1.4.26)
mhich satisfies the initial conditions null at the point +
00.1)
Then, the theorem analogous to theorem 1.4.1, can be stated as
follows:
Theorem 1.4.II. If the given function g(x) of the equation (1.4.2)
satisfies the following hypotheses [particular case of
(1.4.24)]:
g(x) Ui(X) E L (- 00, + (0) , (i = 1, 2, ... , n) (1.4.27)
and if the givenfunctionf(x) of the equation (1.4.1) satisfies
these other hypotheses
'" f(x) Vi(X) f Ig(;) uM) I d; E L (- 00, 0] ,
(i d,2, ... , n) , } . -00 +00
'"
(1.4.28)
then for every solution u(x) of (1.4.1) and for the particular
solutions V _(x), V +(x) of (1.4.2) above introduced, we have
g(x) u(x) E L (- 00, + (0),
f(x) V _(x) E L (- 00, 0] , f(x) V +(x) E L [0, + (0) ,
n-l
lim w_(x) = lim l: u(h)(x) E:-h-1[V _(x)] = ° , X-'I>- -00 x-+
-00 It=o
n-l
lim w+(x) = lim l: U(h)(X) E:_h_1[V +(x)] = ° . "'-++00 "'-++00 h~O
I
(1.4.29)
(1.4.30)
(1.4.31)
The proof of this theorem is to be carried out reasoning separately
on the intervals (- 00, 0], [0, + (0) analogously to the proof of
theorem 1.4.I. It is only to be noted that the statement (1.4.29)
is a consequence of g(x) u(x) E L (- 00, 0], and g(x) u(x) E L [0,
+ (0) which are deduced separately. Note that in the last two
theorems, the n hypotheses (1.4.14) or (1.4.27) generally are not
mutually independent; some may be consequence of the others. The
same holds for (1.4.15) or (1.4.28).
1) The justification of these two sentences (which are not to be
taken literally) is analogous to the one previously given. Instead
of (1.4.11) we have now
n-l lim l.,' u~h)(x) E:_h_1[V-(x)] = 0,
"'-+-00 h~O n-l
lim }; u~h)(x) E!_h_l[V +(x)] = 0, "'-++00 h~O
(i = 1,2, ... , n)·
n E(u) = L: aTe u(n-k) = 0 ,
k=O
n aTe u(n-k) E(u) = E ( _ k)' -k- = 0, k=O n . x
(aTe constant) ,
(aTe constant) ,
25
(1)
(2)
(3)
and write their adjont equations E*(v) = O. In each one of the
three cases, devise pairs of associated systems [~(x), ... ,
un(x)], [vl(x), ... ,vn(x)] of solutions of the two equations E(u)
= 0, E*(v) = 0 (see § 1.3), In the cases (2) and (3) assume that
the roots .xl' £X2' ••• , £Xn of the respective characteristic
equations
be all simple.
i' aTe ( £x k) = 0 k=O n-
2. Prove that, if U(x) = [~(x), ... ,un(x)), V(x) = [vl(x), ...
,vn(x)] are two associated systems of solutions of the equations
E(u) = 0, E*(v) = 0 (see § 1.3), then, fixing arbitrarily a matrix
a = (CtTe), (i, k = 1, 2, ... , n), with constant elements and det
a =F 0, there result associated also the
- - systems U(x) = [ul(x), ... ,un(x)], V(x) = [vl(x), ... , vn(x)]
defined by
U(x) = V(x) aT , V(x) = V(X) 0-1 .
3. A linear differential operator E of order n:
n dn-k E = E aTe(x) dxn-k
Te=O
E* = (_I)R E.
(1)
(2)
In order to construct such an operator we may fix arbitrarily the
coefficients
a2.(x) E A cn-2 8-1,(8 = 0, 1, ... , [; ])and then obtain the
others coefficients
a2 r+1(x) by means of
_ ')' (2,-28+1) , 22 ,-28+2 (n-28) a2r+1(X) - :=OT _ 8 + I B2r-
2s+2 2r'-28 + 1 a28 (x), (3)
(r = 0, 1, ... , [n ; I ]),
where B2 ,-2 8 +2 are the Bernouilli's numbers (see § 3.1).
4. Given a function f(x) E OR[a, b] and a linear differential
operator E of order n [see (1.1.1) and (1.1.2)] we want construct a
solution u(x) of the
26 1. Additional results on linear differential equations
differential equation E(u) = 0 which interpolate f(x) in the sense
that we must get
u(k)(Xi) = j<k)(X() , (i = 1,2, ... ,rn; k = 0, 1, ... ,Pi) ,
(1) with
a < Xl < Xa < ... < Xm < b , m :::: 1, Pi:::: 0
,
m + PI + P2 + ... + Pm = n .
Give the condition for the existence and the unicity of u(x) [with
any f(x)], the expression of u(x) and, using the Green-Lagrange
identity (1.2.2), an integral expression of the remainder
R(x) = f(x) - u(x) . (2)
Elementary quadrature formulae and a general procedure for
constructing them
2.1. The elementary quadrature formulae on a finite interval.1) In
this chapter, as in the last, we shall consider exclusively
real-valued functions. Let [a, b] be a finite interval on the axis
x and n a positive integer; we shall consider an integral of the
type
b
J g(x) u(x) dx (2.1.1) a
where the functions g(x) (weight functions which are assumed non
zero on a set of positive measure) and u(x) (argument function)
satisfy the following hypotheses:
g(x) E L[a, b] , u(x) E A en-Ira, b] . (2.1.2)
Let us fix in [a, b] a certain number m 2: 1 of points Xv x2 , •••
, Xm (nodes) in such a way that
(2.1.3)
and let us fix also a linear differential operator E of order n
[the same n as in (2.1.2)] expressed by a formula of the type
(1.1.1) with coefficients which satisfy the hypotheses (1.1.2). We
shall call elementary quadrature formula of the integral (2.1.1)
relative to the nodes Xl' x2 , ••• , Xm and to the linear
differential operator E, any formula of the type
b n-1 m J g(x) u(x) dx = L; L; Ahi u(h)(Xj) + R[u(x)] ,
(2.1.4)
a h=Oi=l
where the constants coefficients A/Ii are independent of u(x) and
the linear functional R[u(x)] (remainder) is null when u(x) is a
solution of the homogeneous linear differential equation E[u(x)] =
O.
In other words (2.1.4) is intended to be accompanied by the
condition
E(u) = 0 => R(u) = 0 . (2.1.5)
In practice, every formula of the type (2.1.4) is used considering
the linear functional
n-1 m A(u) = L; L; A"i u(h)(xd
h=Oi=l
(2.1.6)
2) For sake of uniformity we shall sometimes write Xo instead of a
and Xm+l in stead of b.
28 2. Elementary quadrature formulae
as an approximate value of the integral (2.1.1) and the other
functional R(u) as the corresponding error of approximation. The
error is null, that is the quadrature formula is exact, when u(x)
satisfies the differential equation E(u) = 0. 1)
In the literature we find innumerable examples of formulae of the
type (2.1.4). Most of them use in A(u) only the values u(x/) of the
argument function at the nodes Xi and not (when n> 1) values of
the derivatives U'(Xi), ... , u(n-l)(Xi); but this means only that
in such formulae the coefficients Ali' ... ' An-I, i turn out to be
zero. Moreover almost all classical formulae are exact when u(x) is
a polynomial or a trigonometric polynomial; this corresponds to the
fact that such formulae are relative to the following particular
differential operators:
(exact formula for polynomials of degree n - 1)
d • (d2 ) E=-[] -2+k2 dx k=l dx
(n = 2'/1 + 1; exact formula for trigonometric poly nomials of
order '/I).
With this as introduction, we are in a position to arrive quickly
at a simple procedure to get a formula of type (2.1.4) under
condition (2.1.5).
Let us consider the non-homogeneous linear differential
equation
E*[<p(x)] = g(x) , (2.1.7)
where E* denotes the adjoint operator of E [see (1.1.5)]. Now we
fix arbitrarily m - 1 solutions
<PI(X), <P2(X), ... , <Pm-leX) E A en-l[a, b]
(2.1.8)
of (2.1.7) and consider also the two solutions <Po(x), <Pm(x)
ofthe same equation (2.1.7) which are respectively determined by
the initial conditions null at the points a, b; namely <p~k)(a)
= 0, <p<::')(b) = 0, (h = 0, 1, ... ,n - 1) and therefore,
recalling (1.3.17) and (1.3.18):
'" b
<Po (x) = - J K(~, x) g(~) d~ , <Pm(x) = J K(~, x) g(~) d~ •
(2.1.9) a '"
We now make use of the Green-Lagrange identity (1.2.2),
substituting vex) = <Pi(X), (i = 0, 1, ... , m).
Considering that E*(<pt) = g, we get
(i = 0, 1, ... , m)
and therefore, integrating on the interval [Xi> Xi+1]:
J g u dx = - 1: U(h) E:-k- 1(<Pt) + J <Pi E(u) dx , "'HI (n-l
]"'Hl "'HI • k=O ••
(i=O,l, ... ,m).
1) The solutions of E(u) = 0 are not the only functions u(x) for
which (2.1.4) is exact; we shall make precise later the whole of
such functions (see Remark 4).
2.1 The elementary quadrature formulae on a finite interval
29
Summing with respect to the index i, we arrive at
b m [n-l ]XHI m XHI ! g u dx = - i~ h~O U(h) E;_h_l(CP;) Xi + i~ xI
CPi E(u) dx . (2.1.10)
Bearing in mind that from cp~h)(a) = 0, cpC::,>(b) = °
follows
[E;-h-l(CPO)]X~xo = 0, [E;-h-l(CPm)]X~Xm+1 = 0, (h = 0, 1, ... , n
- 1) ,
we may then write
m Xi+l
This formula is precisely of type (2.1.4) with
A"i = {E;-h-l [CPi(X) - CPi-l(X)]}X~Xi' (h = 0, 1, ... , n - 1; i =
1,2, ... , m), (2.1.12)
m Xi+l
R[u(x)] = ~ J CPi(X) E[u(x)] dx , (2.1.13) i=O Xi
and from these relations it is evident that the two desired
conditions are satisfied [coefficients A"i independent of u(x) and
validity of (2.1.5)].
Thus we have the following rule for constructing an elementary
quadrature formula of type (2.1.4) relative to the nodes Xl' X2'
••• , Xm and to the linear differential operator E:
1) we consider the linear differential equation E*(cp) = g and its
solutions CPo(x) and CPm(x) defined by (2.1.9);
2) we fix as we like m - 1 other solutions CPI(X), ... , CPm-l(X)
of the same equation E*(cp) = g;
3) we compute the coefficients Ahi by means of (2.1.12); 4) we
write the expression for the remainder R(u) using (2.1.13).
This rule is not only reasonably simple; it is also the only one
possible, as is proved by the following theorem:
Theorem 2.1.1. If, having fixed m nodes Xl' x2' ••• , Xm and the mn
constants Atli , the linear functional
b n-l m R(u) = J g u dx - ~ ~ Ani U(h)(X;) (2.1.14)
a h~Oi~l
30 2. Elementary quadrature formulae
is null when u is a solution of the homogeneous linear differential
equation E(u) = 0, then there are uniquely determined m -
Isolution8 rpl(X), ... , rpm-I(X) of the differential equation
E*(rp) = g which, together with the two rpo(x) and rpm (X) given by
(2.1.9), make valid (2.1.12) and (2.1.13). Proof - Equations
(2.1.12) may also be written
[E:-h-l(tpi)]X~Xi = Ahi + [E:-h-l(rpi-l)]X~Xi ' (h = 0, 1, ... , n
- 1; i = 1,2, ... , m) ,
and therefore, to have them satisfied, itis necessary that rpl(X)
be the solution of E*(rp) = g which is characterized by the
following initial conditions at the point Xl: I)
[E:-h-l(rpl)]X~Xl = Ahl + [E:-h-I(q;O)]X~Xl ' (h = 0, 1, ... , n -
1) ,
and successively that rp2(X) be the one characterized by the
following initial conditions at the point x2 :
[E:-h-l(rp2)]X~X' = Ah2 + [E:-h-l(rpl)]X~X. ' (h = 0, 1, ... , n -
1) ,
and so on for rp3(X), ... ,rpm-l(X), Then we see that rpm (X) must
coincide with that solution of E *(tp) = g which satisfies the
following initial conditions at the point Xm:
[E:-h-l(rpm)]X~Xrn = A hm + [E:-h-l(rpm-I)]X~Xrn' (h = 0,1, ... , m
- 1) .
It remains only to see that the solution tpm(x) coincides with that
considered in the statement of the theorem, namely that defined by
the initial conditions
[E:-h-l(tpm)]x~b = 0, (h = 0, 1, ... ,n - 1) (2.1.15)
and we have also to prove (2.1.13). To this purpose we observe
that, making use of the solutions rpl(X), ... ,
... , rpm-l(X), tpm(x) just introduced, we may do again the same
computation which led to (2.1.10). However, we cannot pass from
(2.1.10) to (2.1.11) since we do not yet know whether tpm(x)
satisfies (2.1.15); therefore instead of (2.1.11) we must
write
b n-l m J g u dx = L: L: [E:_h_ 1 (rpi - tpi-l)]X~XI U(h)(Xi) a
h~Oi~l
n-l m XHI
- L: [E:-h-l(tpm)]x~b u(h)(b) + L: J rpi E(u) dx . (2.1.16) h~O i~O
Xi
Taking into account equation (2.1.14) and the fact that the
solutions rpo, tpl' ... . . . ,rpm have been chosen so as to
satisfy (2.1.12), (2.1.16) may also be written as
n-l m XHI
R(u) = - L: [E:-h-l(rpm)]x~b u(h)(b) + L: J rpi E(u) dx. (2.1.17)
h~O i~O XI
1) Please note that, here and in the following, to assign at a
point of [a, b] the values of E!-h-I(/fJ), (h = 0, 1, ... , n - 1)
is the same thing as to give the values of /fJ(h) and that if the
former are all null, the latter too are null and viceversa [see
formula (1.3.20)].
2.1 The elementary quadrature formulae on a finite interval
31
We now recall that by hypothesis R(u) must be null when u is a
solution of the differential equation E(u) = o. Therefore, denoting
by (ul , Us, ... , un) a fundamental system of solutions of that
equation, it follows from (2.1.17), putting u = Uj, (j = 1, 2, ...
, n), that
n-l 1: u~h)(b) [E:-h-l(CPm)]x~b = 0 , h~O
(j = 1, 2, ... , n) .
Since the Wronskian det (u~h)(b)) is different from zero, we
conclude that (2.1.15) holds and finally that (2.1.17) reduces to
(2.1.13), q.e.d.
On the basis of this theorem, the above stated rule allows us to
find as particular cases of (2.1.4) all the known quadrature
formulae1) and to get several others. We intend to come back to
this in § 2.4. Now we add some remarks on the above mentioned
rule.
Remark 1. If we wish to be effectively present in the quadrature
formula (2.1.4) all the prefixed nodes Xi (i = 1,2, ... , m) we
require that, for every value of the index i at least one of the
coefficients Ahi be non-zero. Keeping into account (2.1.12) it is
obvious that this is equivalent to requiring that the function
tpi(X) - CPi-l(X) must not be identically zero. We achieve our
objective by choosing the solution CPl(X) different from CPo(x);
CPI(X) different from CPl(X); ... ; CPm-2(X) different from
CPm-3(X) and finally tpm-l(X) different from CPm-2(X) and from
CPm(x).
Remark 2. Having fixed the nodes Xl' XI' ... , Xm and the linear
differential operator E, we may write the quadrature formula in
oo(m-l)n different ways, since (m - 1) n is the number of arbitrary
constants on which the m - 1 solutions CPI(X), .•. , CPm-l(X) of
the differential equation E*(cp) = g of order n depend.
Remark 3. If Xl = a, (2.1.12) and [E:-h-l(CPo)]x~a = 0, (h = 0, 1,
... , n - 1), imply
Xl
besides in the expression (2.1.13) for R(u) the first term J CPo
E(u) dx vanishes. a
Therefore, if Xl = a, it is useless to consider the solution
CPo(x). Analogously, if Xm = b we have
A hm = [E:-h- 1 (cpm - CPm-l)]x~b = - [E:-h-l(CPm-l)]x~b , b
while in R(u) the last term J cpm E(u) dx vanishes; therefore if Xm
= b it 0I:m
is useless to consider the solution CPm(x).
1) At least the most common formulae, which approximate the
integral (2.1.1) with a linear functional of the argument function
u(x) of type (2.1.6). There exist a few examples of quadrature
formulae of other types which do not fall within the theory here
stated.
32 2. Elementary quadrature formulae
Remark 4. From (2.1.13) it is clear that R(u) = 0 for all and only
the func tions u for which E(u) E L[a, b] is orthogonal on [a, b]
to the so-called influence function W(x) defined as
W(x) = <Pi(X) in [Xi, Xi +1] , (i = 0,1, ... , m),
(2.1.18)
that is, for which b
f fP(~) F(~) d~
E[ u(x)] = F(x) - a b W(x) ,1) .<2.1.19) f (/)2(~) d~
a
where F(x) is an arbitrary function summable on [a, b]. Hence, the
quadra ture formula (2.1.4) is exact if and only if the argument
function u(x) is the solution of the linear differential equation
(2.1.19) where F(x) E L[a, b] is arbitrary and W(x) is given by
(2.1.18). Note that the condition (2.1.5) made at the be ginning
corresponds to the particular choices F(x) = 0 or F(x) =
W(x).
2.2. The elementary quadrature formulae on the interval [a, +
00).2) We shall use here the same terminology and notations as in §
1.4. Let us consider an integral of the type
+00 J g(x) u(x) dx (2.2.1) a
where the functions g(x) (weight function) and u(x) (argument
function) satisfy the following hypotheses:
g(x) E L1oc[a, +00) , u(x) E A Cfo-;;1[a, +00) , (2.2.2)
g(x) u(x) E I[a, +00) . (2.2.3)
Let us fix some points xl> x2' ••• , Xm (nodes) such that
(2.2.4)
and a linear differential operator E of order n expressed by a
formula of the type (1.1.1) where the coefficients ak(x) satisfy
the hypotheses
ao(x) = 1; ak(x) E A Cfo-;;k-l[a, +00), (k = 1,2, ... , n -
1);
an(x) E L1oc[a, +00) . (2.2.5)
We shall call elementary quadrature formula for the convergent
integral (2.2.1) relative to nodes xl> x2, ••• , Xm and to the
linear differential opemtor E, any formula of the type
+00 n-l m J g(x) u(x) dx = 2: 2: Ahi U(h)(Xi) + R[u(x)] (2.2.6) a
h=O i=l
b
1) We have f fP2(~) d~ > 0 since the case in which every
solution tpi(X) of the a
equation E*(tp) = g may be identically zero in [Xi' Xi+1] is
excluded since the weight function g(x) was supposed different from
zero on a set of positive measure.
2) See A. GHIZZETTI [2], [3], [4].
2.2 The elementary quadrature formulae on the interval [a, + (0)
33
whose coefficients Alii are independent of u(x) and such that
E(u) = ° ~ R(u) = 0 . (2.2.7)
To construct such a formula it is sufficient to give a rule
analogous to the one of § 2.1, however with some clarifications and
remarks.
If we introduce a fundamental system (uv u2 , ••• , un) of the
solutions of E(u) = 0, it is obvious that in order that the
condition (2.2.7) make sense, the hypotheses
g(X) Ui(X) E I[a, +00) , (i = 1,2, ... , n) , (2.2.8)
is indispensable. With this condition established, and fixing one
point b with b > XII"
we may perform, with respect to the interval [a, b], a computation
ana logous to that of § 2.1.
Namely, we denote by CPo(x) that solution of the differential
equation E *[cp(x)] = g(x) which satisfies the initial conditions
null at the point X = a and by CPl(X), ... , CPm(x) m other
solutions arbitrarily fixed.
We apply the Green-Lagrange identity (1.2.2), putting vex) = CPi(X)
(i = 0, 1, ... , m). This leads to
d n-l CPi E(u) - g u = -d E U(h) E!-h-1(CPi) , (i = 0, 1, ... , m -
1, m) ,
xh=o
from which, integrating them on the intervals [Xi, Xi+d (i = 0, 1,
... , m - 1), [xm' b] respectively and summing the resulting
equations, we obtain
! g u dx = _ ~i:l [~1U(h) E!_h_1(CPi)]>:HI _ [~1U(h)
E!_h_1(cpm)]b a .=0 h=O >:1 h=O >:m
m-l "'HI b
+ I J CPi E(u) dx + J cpm E(u) dx , i=O Xi lim
from which, with a simple transformation and use of the fact that
cp~h)(a) = 0, (h = 0, 1, ... , n - 1),
b n-1 m J g u dx = I I [E!-h-1 (CPi - CPi-1)]",=",/ U(h)(Xi) a
h=Oi=1
- [h~IU(h) E!-h-1(cpm)1=b+ ~~11+lcpi E(u) dx +",[ f{Jm E(u) dx.
(2.2.9)
This relation, by means of a passage to the limit with b -'>- +
00, will evidently lead to a formula of the type (2.2.6), if it is
possible to choose f{Jm(x) in such a way that
lim [~IU(h) E!-h-1(CPm)] = 0 , b-++oo h=O x=b
cpm E(u) E I[a, + 00). (2.2.10)
It is possible to construct examples (see A. GHIZZETTI [2]) which
show that (2.2.3) and (2.2.8) are generally not sufficient to
assure that the differential
3 Ghizzetti/Ossicini: Quadrature
34 2. Elementary quadrature formulae
equation E*(qJ) = g possesses a solution satisfying (2.2.10). On
the other hand it does not seem easy, confining ourselves to the
field of integrals which are simply convergent and avoiding any
particularization of the operator E, to arrive at a formulation of
hypotheses which can assure the validity of the procedure but not
be too restrictive. Therefore we think it appropriate to confine
ourselves to giving a general theorem for the case of absolutely
convergent integrals.
In the case of simply convergent integrals, we think it better to
study (2.2.10) case by case, that is for any fixed choice of the
operator E.
The theorem for absolutely convergent integrals is immediately
derived from theorem 1.4.1 and can be stated as follows:
Theorem 2.2.1. Under hypotheses (2.2.2) and letting (uv ... , un),
(VI' ... , vn) be two associated systems of solutions of the
differential equations E(u) = 0, E*(v) = 0, we assume moreover that
the weight function g(x) satisfy the following hypotheses
g(x) Ui(X) E L[a, +00) , (i = 1,2, ... , n) , (2.2.11 )
and that the argument function u(x) satisfy also +00
E[u(x)] Vi(X) J ig(~) uM)i d~ E L[a, +00), (i = 1,2, ... , n).
(2.2.12)
Then g(x) u(x) E L[a, +00) (2.2.13)
and, if as solution qJm(x) of E*(qJ) = g we take that V(x) which
satisfies the initial conditions null at the point +00 [see
(1.4.9)], (2.2.10) is satisfied amd (2.2.6) follows from (2.2.9)
together with
Ahi = [E:- h - 1 (qJi - qJi-l)]X~xl' (h = 0, 1, ... , n - 1; i = 1,
2, ... , m) , (2.2.14)
m Xi+l
R(u) = J: J qJ! E(u) dx, (with Xo = a, xm +1 = + 00). (2.2.15) i=O
Xi
We have then the following rule to construct an elementary
quadrature formula of type (2.2.6) relative to the nodes Xl' x2 ,
••• , Xm and to the linear differential operator E [when for
instance the hypotheses (2.2.2), (2.2.11) and (2.2.12) are
satisfied]:
1) we consider the linear differential equation E*(qJ) = g and its
solutions x +00
qJo(x) = - J K(~, x) g(~) d~ , qJm(x) = J K(~, x) g(~) d~; (2.2.16)
a
2) we fix arbitrarily m - 1 other solutions qJl(X), ... , qJm-l(X)
of the same equation E*(qJ) = g;
3) we compute the coefficients Ahi by means of (2.2.14) ;
4) we write the expression of the remainder R(u) using
(2.2.15).
2.2 The elementary quadrature formulae on the interval [a, + 00)
35
As in the preceding § 2.1 this rule is essentially the only
possible one, as shown by the following theorem:
Theorem 2.2.11. Having fixed the linear differential operator E, we
assume that the functions g(x), u(x) satisfy the hypotheses
(2.2.2), (2.2.11) and (2.2.12). Having then fixed the m node8 Xl'
x2, ... , Xm and the mn constants A hi, if the linear
functional
+00 n-l m R(u) = J g u dx - L: L: Ahi u(h)(xd
a h~O i~l
(2.2.17)
is null when u is a solution of the differential equation E(u) = 0,
then m - 1 solutions 971(X), ... , 97m-l(X) of the differential
equation E*(97) = g are uniquely determined, which, together with
the two solutions 97o(x) and 97m(x) defined by (2.2.16), imply
(2.2.14) and (2.2.15).
Proof. As in the proof of theorem 2.1.I, first we see that in order
that (2.2.14) hold, the above mentioned solutions 971(X), 972(X),
... , 'Pm-I(X) must be neces sarily those determined by the
following initial conditions respectively at the points Xl' x2, •••
, Xm-l;
[E:-h-I(97I)]"'~"', = Ahl + [E:-h-I('PO)]Z~"" '
[E:-h-I(97m-I)]"'~"',n_l = A h, m-I + [E:-h-I(97m-2)]"'~"'m_l' (h =
0, 1, ... , n - 1) ,
I (2.2.18)
and moreover that the solution 97m must coincide with the one
determined by these initial conditions at the point Xm:
[E:-h-I(97m)]"'~"'m = A hm + [E:-h-I(97m-d]"'~"'m , (2.2.19)
(h = 0, 1, ... , n - 1) .
With the solutions 971>"" 97m of E*(97) = g defined by (2.2.18)
and (2.2.19) we may write (2.2,9) and pass to the limit as b --r +
00. In virtue of theorem l.4.I, hypotheses (2.2,11), (2,2,12)
assure that g(x) u(x) E L[a, +(0); the passage to the limit in
(2.2.9) yields, considering also that we have already proved the
validity of (2.2.14):
+00 n-I m m-I "'1+1 J g u dx = L: L: Ahi u(h)(x;) + L: J 97i E(u)
dx a h~O i~1 i~O "'I
+ lim {- [~IU(h) E:-h - 1(97 m )] + J 97m E(u) ax } , b-++oo h~O
",~b Xm
that is for (2.2.17)
36 2. Elementary quadrature formulae
Since R(u) must be zero when u coincides with the solutions u l ,
... , u" of E(u) = 0, we find
n-1 lim 1: U)hl(x) E!_h_1[tpm(X)] = 0, (j = 1,2, ... , n).
(2.2.21)
"' ..... +00 h=O
tpm(x) =ig V((X) [Pi - j g(~) uM) d~] , (2.2.22)
the value to the limit given in (2.2.21) is
",~:oo ~:U)hl(X)i~ E!_h_1[V((X)] [Pi - j g(~) uM) d~]
-"'::00 i~ ~i - ! g(~) uM) d~] hn~1U~hl(X) E!_h_I[Vi(X)] .
That is, on the basis of theorem 1.3.III,
"'::00 i~1 [Pi - j g(~) uM) d~] 15(1 = Pi - !~(~) UM) d~ ,
therefore we must have, for (2.2.21)
+00 Pi = J g(~) Uj(~) d~ , (j = 1,2, ... , n) .
a
Substituting this expression for PI in (2.2.22) we deduce that for
the solution tpm(x) we have " +00
tpm(x) = 1: Vi(X) J g(~) uM) d~ . i=1 '"
Now we see from (1.4.8) and (1.4.9) that tpm(x) is exactly that
solution which satisfies the initial conditions null at the point
+00. With the notations of theorem 1.4.1 we have namely tpm(x) =
V(x); using then the two other .conclusions of that theorem, we
conclude that
lim ["j;1U(h) E!-h-1(tpm)] = 0 , b ..... +oo h=O ",=b
b +00 lim J tpm E(u) dx = J tpm E(u) dx ,
b-++oo X,n Xm
therefore (2.2.20) reduces to (2.2.15) q.e.d. Note that formulae
(2.2.6), (2.2.14) and (2.2.15) in the case of the interval
[a, +(0) are quite similar to (2.1.4), (2.1.12) and (2.1.13)
relative to the case of the finite interval [a, b]. Remarks 1, 2,
and 3 of the preceding § 2.1 hold here too. Remark 3 is to be
limited to the fact that if Xl = a it is useless to consider the
solution tpo(x). As to Remark 4, it still stands that (2.2.6) is
exact for all and only the functionsu(x) for whichE(u) is in [a,
+(0) orthogonal to the influence function 4>(x) determined by
(2.1.18), but gen erally (2.1.19) (with b = + (0) has no
meaning.
2.3 Elementary quadrature formulae on the interval (- 00, + 00)
37
2.3. Elementary quadrature formulae on the interval (- 00, + 00).1)
Let us consider an integral of the type
+00 J g(X) U(X) dx -00
where the functions g(x) and u(x) satisfy the hypotheses
g(x) E L1oc(-00, +00), u(x) E A C~-;;l(_oo, +00),
g(x) u(x) E 1(-00, +00).
Having fixed the nodes Xl' X2, ••• , X", such as to have the
ordering
(2.3.1 )
(2.3.2)
(2.3.3)
(2.3.4)
and a linear differential operator E of order n, of type (1.1.1)
and whose coefficients a!:(x) satisfy the hypotheses
ao(x) = 1; a!:(x) E A C~;;-k-l(_oo, +00), (k = 1,2, ... , n -
1):
an(x) E L1oc( - 00, + 00) (2.3.5)
we may find a corresponding elementary quadrature formula +00 n-l m
J g(x) u(x) dx = 1: 1: Ahi U(h)(Xi) + R[u(x)] (2.3.6)
-00 h~O i~l
under the usual condition
E(u) = 0 =? R(u) = 0 . (2.3.7)
Keeping the notation of § 1.4, we may reason as in § 2.2 requiring
[together with (2.3.2)] the hypotheses
g(x) Ui(X) E L(-oo, +00), 1
-00 I (2.3.8)
x
E[ u(x)] Vi(X) J [g($) Ui($) [ d$ E L( - 00, 0] ,
+00
I E[u(x)] Vi(X) J [g($) uM)[ d$ E L[O, +00) , x
(i = 1,2, ... , n) , J which imply (by theorem 1.4.II) g(x) u(x) E
L( - 00, + 00). More precisely, having fixed a < Xl and b> Xm
and choosing arbitrarily m + 1 solutions gJo(x), gJI(X), ... ,
gJm(x) of E*(gJ) = g, we may write the following formula, similar
to (2.2.9):
b [ n-l ] n-l m J g u dx = 1: U(h) E;-h-l(gJO) + 1:.1: [E;-h_l
(gJi-gJi_l)]X~XiU(h)(X;)- a h~O x~a lt~O ,~l
_[~l u(h) E:-lt-l(gJm)l~b + ITO E(u) dx + :~ll;i E(u) dx +x1 gJm
E(u) dx
(2.3.9) 1) See A. GHIZZETTI [2], [3], [4].
38 2. Elementary quadrature formulae
and take the limits as a ~ - 00, b ~ + 00. Applying theorem 1.4.II
we see that with such passages to the limit, (2.3.9) yields a
formula of the type (2.3.6) if we take as lPo(X) , IPm(x) the
solutions of E*(IP) = g which satisfy the initial conditions null
respectively at the points -00, +00.
We thus arrive to the following rule, similar to the one of § 2.2,
and valid when the hypotheses (2.3.2) and (2.3.8) are
satisfied:
1) we consider the linear differential equation E*(IP) = g and its
solution lPo(x) defined by the initial conditions null at the point
-00 and thus given by
'" lPo(x) = - J K(;, x) g(;) d; (2.3.10) -00
and the solution IPm(x) which satisfies the initial conditions null
at the point +00, expressed by
+00
IPm(x) = J K(;, x) g(;) d;; (2.3.11)
2) we fix as we like m - 1 solutions 1P1(X), ... , IPm-l(X) of the
same equation E*(IP) = g;
3) we compute the coefficients Ahi by means of
(h = 0, 1, ... , n - 1; i = 1,2, ... , m) ;
(2.3.12)
4) we write the expression of the remainder R(u) in the following
way: m Z'+l
R(u) = ~ J lPi E(u) dx, (with Xo = - 00, xm+1 = + 00) . (2.3.13)
i=O "',
This rule is essentially the only possible since there exists a
theorem similar to theorem 2.2.II whose statement and proof we
leave to the reader.
Finally Remarks 1, 2, 4 of § 2.1 hold, if we remember that (2.1.19)
(with a = - 00, b = + 00) has no longer any meaning.
2.4. Some remarks on the rule for obtaining quadrature formulae. If
we examine the three rules given in § 2.1,2.2, and 2.3 for
obtaining elementary quadrature formulae on a finite interval [a,
b], on an interval [a, +00) and on the interval (-00, +00), we see
that they are formally ideni/ical, so that we may always refer to
(2.1.4), (2.1.9), (2.1.12), (2.1.13) with the understanding that a
can also be finite and b = + 00, or a = - 00, b = + 00. The only
difference among the three cases is in the formulation of the
necessary hypotheses regarding the weight function g(x) and the
argument function u(x). In the case of infinite intervals these
hypotheses call in also two associated systems [~(x), ... ,
u,,(x)], [vI(x), ... , v,,(x)] of solutions of the differential
equations E(u) = 0, E*(v) = 0. We know that the rules given are the
most general possible, wherefrom we may derive systematically all
the known quadrature formulae, choosing appropriately case by case
the nodes Xl' ••• , Xm and the linear differential operator
E.
2.4 Some remarks on the rule for obtaining quadrature formulae
39
Such deductions substantially do not present anything new and
generally are of no difficulty. 1)
On the contrary we may say that in the set of all quadrature
formulae supplied by the above given systematic procedure, the
classical ones are of no particular importance. We may also suggest
that it is better to make the most of the procedure by getting,
case by case, a quadrature formula
b
particularly suited to the calculation of a given integral J F(x)
dx with a
F(x) E L[a, b] taking advantage, not only from the arbitrary choice
of the nodes Xv ... , Xm and of the linear differential operator E,
of order n, but also from the numberless ways we may write F(x) =
g(x) u(x) with g(x) E L[a, b] and u(x) E A Cn - 1[a, b].
This point of view is not yet widespread and therefore, for the
conveni ence of the reader, we shall gather in Chapter 4 several
examples of de ductions of classical formulae, together with
others that we consider new. For this purpose it will be necessary
to state in Chapter 3 some notes on certain auxiliary elements
(Bernoulli numbers and polynomials, orthogonal polynomials, etc.).
In applying the general procedure it is useful to keep in mind some
observations which can be of great utility, above all when we try
to obtain quadrature formulae with certain pre-established quali
fications (for instance that certain coefficients Aht be zero). We
shall present these observations in the remainder of this § 2.4 and
in the two successive ones.
The given rule for obtaining the formula
b n-l m J g(x) u(x) dx = 2: 2: Ahi u(h)(Xt) + R(u) , [E(u) = 0 ~
R(u) = 0] ,
a h=Oi=l (2.4.1)
with a, b finite or infinite, considers first the solutions Ifo(x),
Ifl(X), ... , Ifm(x) of the differential equation E*(If) = g, then
the computation of the co efficients A by means of
Ahi = [E:_ h _ 1 (lfi - lfi-l)J.=,."
and finally the expression of the remainder written
m XI+l
(2.4.2)
(2.4.3)
Sometimes it is more convenient to compute first the coefficients A
hi ,
considering n linearly independent solutions u1(x), ... , un(x) of
the linear differential.equation E(u) = 0 and writing the system of
n linear equations
1) Some difficulties may arise when we try to deduce from the
expression of the remainder R(u) formulae of majoration and
evaluation of the remainder itself. In such operations it is often
necessary to study the sign of the influence function 4>(x) then
making use of very delicate technique.
40 2. Elementary quadrature formulae
in mn unknowns A hi ;
b n-I m J g(x) Uj(x) dx = L.; L.; Ahi U)h)(X) , a h~O i~1
(j = 1,2, ... , n) . (2.4.4)
From this system we may derive for instance the n coefficients A
01' An, ... , An-I, I in terms of the remaining (m - 1) n which can
be fixed arbi. trarily (see § 2.1, Remark 2) or with any particular
criteria. Furthermore we may apply the following theorem:
Theorem 2.4.1. Assuming that the coefficients Ahi of (2.4.1) are
known, the functions To(x), Tl(X), ... , Tm(x) necessary to write
(2.4.3), are given by the formula
x
Ti(X) = - IK(~, x) g(~) d~ + X: jfIAhj [:;', K(~, X)l~xj'
(2.4.5)
a (i = 0, 1, ... , m)
where K(x, ~) is the resolvent kernel relative to the operator E
[see (1.3.4)].
Proof. For i = 0 equation (2.4.5) is true since it coincides with
the first equation of (2.1.9); for i = m it cannot immediately seen
that (2.4.5) coincides with the second one of (2.1.9), that is
that
a
(2.4.6)
But it can easily proved because if in (2.4.1) we denote by ~ the
variable of integration and put u(~) = K(~, x) [solution of E(u) =
0, whence R(u) = 0], we obtain
b I g(~) K(~, x) d~ = :~: JI Ahi [:;h K(~, X)l~xi a
which is equivalent to (2.4.6). In order to finish the proof of
(2.4.5) it will be sufficient to prove that
(2.4.2) is satisfied. For this purpose we observe that from (2.4.5)
we may derive
and therefore for (1.3.4)
n-I n Ti(X) - Ti-l(X) = L.; Ahi L.; u~h)(Xi) Vj(X);
h~O j~l
it follows, using (1.3.9), that n~1 n
[E:~k~1 (rpi - rpi~1n"~Xi = L; Afti L; Ujh)(X;) [E:~k~1
(VI)]x~x;
which is (2.4.2) q.e.d.
il~O j~1
We next note that in (2.4.1) the condition
E(u) = O~ R(u) = 0
E(u) = E(ii) ~ R(u) = R(i/,) ,
41
(2.4.7)
(2.4.8)
where u(x) denotes another function satisfying the same hypotheses
of u(x). Finally note that (2.4.7) requires that the quadrature
formula (2.4.1) be
exact for the functions u(x) belonging to a certain n-dimensional
linear space. Instead of defining the space by means of the
operator E, we can define it by its linearly independent n points;
namely (2.4.7) can be replaced by the condition that the quadrature
formula be exact for n assigned functions ul(x), ... , u,,(x)
having Wronskian U(x) '* 0 in the interval of integration. We then
have the result
u u' ... ,
(2.4.9)
2.5. Gauss problem.!) Let us consider again the elementary
quadrature formula
b n~1 m f g(x) u(x) dx = L; L; Ahi U(h)(Xi) + R(u) , a h~O
i=1
[E(u) = O~ R(u) = 0]. (2.5.1)
In its right member we find the values, at the nodes Xl' X 2' ••• ,
x m , of the function u and of its derivatives up to order n - 1;
we know (§ 2.1, Remark 2) that in the coefficients Aili there are
altogether (m - 1) n arbitrary para meters. Therefore the question
is whether, having fixed an integer p (with 1 <p < n - 1), it
is possible to make use of the arbitrary nature of these parameters
to drop from the formula the values U(h)(Xi) of the derivatives of
order higher than n - p - 1, that is whether there can exist a
formula of the type
b n~p~1 m
f g(x) u(x) dx = L; L; Ahi U(hl(X;) + R(u) , a h=O i~l
') See A. GHIZZETTI [I].
[E(u) = O~ R(u) = 0].
42 2. Elementary quadrature formulae
If (UI , UZ' ••• , un) are a fundamental system of solutions of
E(u) = 0 it is clear that a necessary and sufficient condition that
(2.5.2) hold is that the m (n - p) coefficients Ahi satisfy the
following system of n linear equations
n-p-l m b
L: L: Ani UY')(Xi) = J g(x) Uj(X) dx , h~O i~l a
(j = 1, 2, ... , n) . (2.5.3)
To discuss this system of equations, let us consider the transposed
homo geneous system which, denoting by CI , C2, ••• , Cn the
relevant unknowns, may be written
n :E U~h)(xtl cj = 0 , j~l
(h=O,I, ... ,n-p-l; i=I,2, ... ,m). (2.5.4)
If the rank of the matrix (ujh)(Xi)) with m (n - p) rows and n
columns, is equal to n, 1) the system (2.5.4) has no non-trivial
solutions and therefore the system (2.5.3) is certainly consistent
and has oom(n-p)-n solutions.
Let us now assume that the matrix has rank n - q (with q ~ 1).2) In
this case the system (2.5.4) has q linearly independent non-trivial
solutions (cl1 , C2 ,' ••• , cn,)' (r = 1, 2, ... , q) and by a
well-known criterion the system (2.5.3) is consistent if and only
if the q conditions
n b
L: cjr J g(x) Uj(x) dx = 0 , (1" = 1,2, ... , q); (2.5.5) j~l
a
are satisfied. In this case (2.5.3) has oom (n-p)-(n-q) solutions.
Now we shall try
to present these results in a more illuminating form. To this
purpose we shall consider the following problem: determine a
solution of the differential equation E(u) = 0 which together with
its derivatives up to the order n - p - 1 is zero at each of the
points Xl' X2, ••. , X",.
This is an homogeneous boundary problem expressed by the
formula
E(u) = 0; u(h)(xtl = 0, (h = 0, 1, ... , n - p - 1; (2.5.6)
i = 1,2, ... , m) . n
Since the general solution of E(u) = 0 is given by u(x) = L: cj
Uj(x), with j~l
Cj arbitrary constants, the boundary conditions are expressed by
the fol lowing system of m (n - p) equations in n unknowns Cv ...
, Cn:
n :E Cj U)h)(Xi) = 0, (h = 0, 1, ... , n - p - 1; i = 1, 2, ... ,
m) , j~l
which coincides with the system (2.5.4).
1) This is possible only if n :::;; m (n - p). 2) \Ve must have n -
q :::;; m (n - p). Moreover the matrix considered contains
the m matrices (n - p) X n formed by the first n - p rows of the
W'ronskian of u1' ••• , Un
computed at the points XI (i = 1,2, ... , m); these matrices have
rank n - p and therefore we certainly have n - q ;;:; n - p. In
summary, we may write
n - p :::;; n - q :::;; m (n - p) .
2.6 Tchebychef problem 43
Therefore, saying that the system (2.5.4) has no non-trivial
solutions is equivalent to saying that the problem (2.5.6) admits
only the solution u(x) = 0; similarly, saying that the system
(2.5.4) has q linearly independent solutions (cIr , C2" ••• , cn,)
is equivalent to saying that the problem (2.5.6) has q linearly
independent solutions
n U,(x) = L: c,• u,(x) ,
j=1 (r = 1,2, ... , q) . (2.5.7)
In the latter case, the necessary and sufficient conditions (2.5.5)
that the quadrature formula (2.5.2) exist, can be written as
b
J g(x) U,(x) dx = 0 , (r = 1, 2, ... , q) . (2.5.8) a
Therefore we may state the following theorem:
Theorem 2.5.1. Given the nodes Xl' X2, ••• , X"" the linear
differential operator E of order n and the positive integer p <
n - 1, consider the homogeneous differential problem (2.5.6). If
this problem has no non-trivial solutions [whence n < m (n - p)]
it is possible to write a quadrature formula of the type (2.5.2) in
oom(n-p)-n different ways. If on the other hand the problem (2.5.6)
has q linearly independent solutions U,(x) [r = 1, 2, ... , q with
n - m (n - p) < q < p] then the formula (2.5.2) may apply
only if the q conditions (2.5.8) are satisfied; if so there are
oom(n-p)-n+q possible formulae of form (2.5.2).
In the conditions stated in this theorem, the equation (2.5.8) may
appear as conditions for the weight g(x) (see § 2.7, Problem 4);
but this point of view has little importance since generally the
weight g(x) is given.
However we may fix only the number m of the nodes and consider
their abscissae Xl' x2, ••• , xm, all or in part, as undetermined
parameters. In this case, (2.5.8) appear as conditions for the
nodes; they show us how we have to choose these points to make it
possible to get the quadrature formula (2.5.2) required. I) This
problem, presented in these terms, will be called Gaus8 problem
since it is a wide generalization of a particular problem studied
by Gauss which originated the quadrature formulae having his name
(see § 4.6).
2.6. Tchebychef problem.2) Now let us consider another problem
regarding quadrature formulae which, in certain particular cases,
have been studied by Tchebychef. In analogy with Gauss problem [see
(2.5.2)] we try to
1) It may be that there many groups of nodes (Xl"'" Xm) satisfying
(2.5.8). In that case it is obvious that the latter statement of
theorem 2.5.1 (existence of oom(n-p)-n+q quadrature formulae) is to
be applied to each of such groups (see § 2.7, Problems 2,3).
a) See A. GHIZZETTI [1].
44 2. Elementary quadrature formulae
get, for a fixed p (0 < P < n - 1), a quadrature formula of
the type b n-p-1 m J g(x) u(x) dx = 2: Ah 2: U(h)(X;} + R(u) ,
[E(u) = 0 ~ R(u) = 0] ,
a h~O i~l
(2.6.1)
that is, one in which the coefficients of the values U(h)(Xi), of
the derivatives
of the same order of the function u(x), are always equal to each
other. Reasoning as in the preceding § 2.5 we find for the n - p
coefficients Ah the following system of n linear equations
n-p-1 m b L Ah 2: U~h)(Xi) = J g(x) Uj(x) dx , (j = 1,2, ... , n) ,
h~O i~l a
(2.6.2)
whose homogeneous transpose can be written n m
2: ci 2: U~h)(X;} = 0, (h = 0, 1, ... , n - p - 1) . (2.6.3) j~l
i~l
If the rank of the matrix (~u}h)(Xd) (with n - p rows and n
columns)
comes out to be equal to n (this is possible only if p = 0), the
system (2.6.3) has no non-trivial solutions, therefore the system
(2.6.2) is consistent and admits of one and only one solution.
Instead let us assume that the matrix has rank n - q (with q >
1, 0 < n - q ~ n - p). In this case the aystem (2.6.3) has q
linearly independent solutions (cl " c2 " .•• , cnr) (r = 1, 2, ...
, q) and the system (2.6.2) is consistent if and only if the q
conditions
n b
2: ciT J g(x) Uj(X) dx = 0, (r = 1,2, ... , q) , (2.6.4) j~l
a
are satisfied, resulting oo(n-p)-(n-q) solutions of (2.6.1). These
results can be presented in a different way, considering the
following homogeneous boundary problem:
E(u) = 0; m
(h = 0, 1, ... , n - p - 1). (2.6.5)
n Since the general solution of E(u) = 0 is u(x) = 2: Cj Uj(x)
assuming the
j~l
boundary conditions satisfied, we get for the arbitrary constants
ci precisely the system (2.6.3). Therefore when we say that (2.6.3)
has no non-trivial solutions we mean that (2.6.5) admits only the
solution u(x) = 0; to say that (2.6.3) has q linearly independent
solutions (cIr , c2" ••• , cnr) means that the problem (2.6.5) has
q linearly independent solutions
n UT(x) = E Cjr Uj(x) ,
j~l
(r = 1,2, ... , q) , (2.6.6)
2.7 Problems 45
We have then proved the following theorem:
Theorem 2.6.1. Given the nodes Xl> X 2, ••• , X m, the linear
differential operator E of order n and the non-negative integer p
< n - 1, if the problem (2.6.5) has no non-trivial solutions
(this is possible only if p = 0) we may get a quadrature formula of
the type (2.6.1) in only one way.
If on the other hand problem (2.6.5) has q linearly independent
solutions Ur(x), (r = 1,2, ... , q with q 2: 1, P < q < n) 1)
then it is possible to con struct a formula (2.6.1) (in ooq-p
different ways) if and only if the q conditions (2.6.7) are
satisfied.
We may repeat here the remarks stated in the preceding § 2.5:
(2.6.7) may be considered as conditions on the weight g (but this
has a little im portance) or as conditions on the choice of the
nodes.
2.7. Problems.
1. Examine the possibility of devise a quadrature formula of the
type b
J u(x) dx = AOl u(x1 ) + A02 u(x2 ) + Aoa u(xa) + R(u) (1) a
[with a < Xl < x2 < xa < b] which must be exact when
u(x) is a polynomial of degree :s 3.
2. The Gauss problem of § 2.5 can be generalized by substituting
(2.5.2) by the formula
b m n-P/-l J g(x) u(x) dx = L: L: AM U(h)(Xi) + R(u) , (1) a i~l
h~O
E(u) = 0 =:> R(u) = 0,
where the maximum order n - Pi - 1 of the derivatives of the
argument function u(x), which must be considered in the node Xi,
depends on the node Xi' Prove the following theorem (which is an
extension of theor. 2.5.1): given the nodes Xl> x2' ••. , X m,
the linear diffential operator E of order nand the integers PI>
P2' .•. , Pm (with 0 <Pi < n - 1), consider the boundary
problem:
E(u) = 0; U(h)(Xi) = 0, (i = 1,2, ... , m; h = 0, 1, ... , n - Pi -
1). (2) m
If this problem has only the trivial solution u(x) - 0 [whence n
< m - L: pd
m mn- Epi-n
it is possible to write a quadrature formula of the type (1) in 00
i~l
1) It may happen that q = n, namely that all the solutions of E(u)
= ° satisfy m
the boundary conditions 2,' u(h)(Xi) = 0, (h = 0, 1, ... , n - p -
1). Example: n = 2, i~l
d2
m = 2, P = 0, E = dX2 + 1, Xl = 0, x2 = n.
46 2. Elementary quadrature formulae
different ways. If on the other hand the problem (2) has q linearly
independent m
solutions Ur(x) [r = 1, 2, ... , q with n - m n + 2: Pi < q <
min Pi], i=1 - -.
then the formula (1) is possible if and only if these q conditions
are satisfied: b
J g(x) Ur(x) dx = 0 , a
(r = 1,2, ... , q);
m mn-I;PI-n+q
(3)
in such a case this formula is possible in 00 i=1 different
ways.
3. Examine the possibility of devise a quadrature formula of the
type 1
J U(x) dx = AOI U(X1) + An U'(Xl) + A21 U"(Xl) + AOIl U(xlI ) +
R(u) , (1) -1
which must be exact when u(x) is a polynomial of degree <
5.
4. Determine the weight functions g(x) in correspondence of which
it is possible to devise a quadrature formula of the type
1 .
J g(x) u(x) dx = AOI u(- 1) + AOIl u(O) + A03 u(l) + R(u) , (1)
-1
which most be exact when u(x) is a polynomial of degree <
5.
CHAPTER 3
Special functions
a.l. llernouilli polynomials and numbers. The Bemouilli polynomials
Bn(x) are defined by the following expansion (valid for \z\ < 2
n):
eZ _ 1 = 1: Bn(x) n! ,1) z eX Z 00 ztl I B 1(x) = X _ ~n~o B 2(x) =
X2 _ X + ~ , ... ,]
(3.1.1 )
therefore we have z 00 zn 1 00 zn
-ez---l - 1: B - = 1 - --- z + 1: B - n=O n n! 2 n=2 n n!
(3.1.3)
. z 1 z cosh z/2 and smce .-1 - 1 + 2 z = 2 . h /2 - 1 is an even
function, we deduce that e - sm z
Ba = B5 = B7 = ... = 0 , (3.l.4)
while the other Bernouilli numbers B 2, B 4 , B 6 , ••• are all non
zero. Now note that from the identity
we may deduce, expanding every term in power series of z:
00 n+l DC nOOn
1: xn :,-- = 1: Bn(x + 1):' - 1: Bn(x):' n=O n. n=O n. n=O n.
or also 00 nOOn
1: n xn- 1 :, = 1: [Bn (x + 1) - Bn(x)] ~,- ; n=l n. n=l n.
therefore we have
Bn (x + 1) - Bn(x) = n xn- 1 , (n = 1,2, ... ) (3.l.5)
and particularly, putting x = 0 and using (3.l.2)
Bn(1) = B n , (n = 2, 3, ... ) . (3.l.6)
1) That Bn(x) come out to be effectively a polynomial of degree n,
it immediately follows from the formulae given furtherwards [for
instance from (3.1.8)].
48 3. Special functions
From equation (3.1.1) there follows, differentiating with respect
to x
Z2 eX Z 00 I zn
eZ _ 1 = L Bn(x) ;I ; n~l .
(3.1.7)
on the other part, always for (3.1.1), the function occurring III
the first co zn+l co ~n
member has value "B (x)-- = L n B (x) ~ so that from the n~ n n!
n~l n -1 n!'
comparison with (3.1.7) we deduce
B~(x) = n Bn_1(x) , (n = 1,2, ... ) . (3.1.8)
From this equation with further differentiations we obtain B~(x) =
n X B~_l(X) = n (n - 1) B n_ 2(x), B~'(x) = n (n - 1) B~_2(X) = n
(n - 1) (n - 2) X Bn _ 3(x), ... and in general
(k) __ ~ Bn (x) - (n _ k)! Bn_k(x) , (k = 0, 1, 2, ... , n) .
(3.1.9)
Supposing k < n - 2, that is n - k > 2, if we put x = ° or x
= 1 in (3.1.9), using (3.1.2) and (3.1.6), we obtain
nlk) _ (k) _ n! D" (0) - Bn (1) - (n _ k)! B n- k , (k = 0, 1, ...
, n - 2), (3.1.10)
while in the case of k = n - 1, since (3.1.9) becomes B~n-l)(x) =
n! (x - {-),
we obtain 1 1 B~n-l)(O) = - "2 n! , B~n-1)(1) = "2 n! .
(3.1.11)
We get another interesting consequence of (3.1.8) integrating it
between ° and x; then we have
x
Bn(x) - Bn = n J Bn-M) d~ , (n = 1,2, ... ) . (3.1.12) o
From it, changing n by n + 1 and putting x = 1, using (3.1.6) and
supposing n ~ 1 we get:
1
J Bn(x) dx = ° , (n = 1, 2, ... ) . (3.1.13) o
Let us now deduce a property of symmetry or semisymmetry of the
poly-
nomials Bn(x) with respect to the point x = ~ . We write again
(3.1.1),
a first time changing x by 1 - x and a second time changing z by -
z, thus obtaining the two formulae
z e(1-"')z co zn Z e-XZ co, zn eZ _ 1 = nfo Bn (1 - x) n! ' - e-Z _
1 = n~ (- 1)n Bn(x) n! ;
it is easy to see that the first members are equal to each other
and therefore we have the formula
Bn (1 - x) = (- 1)n Bn(x) , (n = 0, 1,2, ... ) (3.1.14)
which expresses the wanted property.
3.1 Bernouilli polynomials and numbers 49
Finally let us consider the polynomial
P 2n(x) = B 2n(x) - B 2n , (n = I, 2, ... ) , (3.1.15)
on the interval [0, I]. Using (3.1.2), (3.1.4), (3.1.6), (3.1.10)
we see that
P 2n(0) = P 2n(l) = 0,
p~2:-1)(0) = ~2:-1)(1) = 0 ,
(k = 1,2, ... , n);
wherefrom and from the Rolle theorem it is easy to deduce that
P2n(x) does not vanish for 0 < x < 1. In fact if P2n(x)
vanishes (besides at the points x = 0, x = 1 also) at a point
internal in [0, I], its.derivative p; .. (x) should vanish (besides
at the points x = 0, x = I) at least at two internal points, p~ ..
(x) at least at three internal points, P~~(x) (besides at x = 0, x
= I) at least at two internal points and so on until the absurd
conclusion that p~2nn-2)(x) (polynomial of second degree) should
vanish at least at three points internal in [0, I].
This reasoning proves also that P;n(x) vanishes only at one point
internal 1
in [0, I], which point has necessarily abscissa 2in virtue of the
symmetry
property expressed by (3.1.14). Therefore P 2n(x) is always
poeitive (or nega-
tive) for 0 < x < 1 and gets its maximum (or minimum)
For computing Pn(~) we write (3.1.1) with x = I:
z eZ 00 zn eZ _ 1 = 2: Bn(l) n! ;
.. =0
(3.1.17)
we observe then that the first member may also be written
1 2 z e 2 2 z e2z ( ~2z )
2 e2z - 1 + e2z - 1
1 so that, applying again (3.1.1) with x = 2 or x = 1 and 2 z
instead of z, we obtain
(3.1.18)
From the comparison between (3.1.17) and (3.1.18) obviously follows
that
Bn(l) = 2n- 1 l Bn (~) + Bn(I)] , whence we have Bn (~) = - (I -
2nl_1)
XBn(l) and finally, using (3.1.6):
Bn(~)= - (1-2nl_1)Bn, (n=2,3, ... ). (3.1.19)
From equations (3.1.15) and (3.1.19) then follows that the above
mentioned maximum value (or minimum) of P2n(x) in [0, I] is given
by
P 2n (~ ) = - 2 (I - 2:" ) B2t1 , (n = I, 2, ... ) . (3.1.20)
4 Ghizzetti/Ossicini: Quadrature
50 3. Special functions
We may speak of maximum if B2n < 0, of minimum if B2n > 0. In
the first case the polynomial P2n(X) hasin [0,1] the minimum (equal
to zero) at the extreme points x = ° and x = 1; from it and from
the fact for (3.1.10) we have P~n(O) = P~n(l) = 2 n (2 n - 1) B2n -
2 follows B2n - 2 > 0. Ana logously in the other case we obtain
B2n - 2 < 0. We conclude that B2n - 2
and B 2n have always opposite signs and therefore, being B2 > 0,
that B 2n
has the sign of (- 1)"-1; there follows that P2n(x) has for ° <
x < 1 the sign of (- It·
3.2. Euler functions. The Euler function of second kind or gamma
function is defined for iX > ° by the following formula
+00 r(iX) = J e-X X,,-1 dx,
o (iX> 0) ,
r(I) = 1 ,
Then we have +00 +00
r(iX + 1) = J e-X x" dx = [- x" e-X]:oo + iX J e-X X,,-l dx o
0
that is r(iX + 1) = iX r(iX) .
For n positive integer, a repeated application of this formula
yields
(3.2.1)
(3.2.2)
(3.2.3)
(3.2.4)
r(iX + n) = (iX + n - 1) (iX + n - 2) ... (iX + 1) iX r(iX) ,
(3.2.5)
I and particularly putting iX = 1 or iX = -2- and using (3.2.2) and
(3.2.3):
r (n + 1) = n! (3.2.6)
r(n+ ~)=(n- ~)(n- :) ... ~-. ~V;=2~nn~:V;. (3.2.7)
From equation (3.2.4) follows too
r(iX + 1) = iX (iX - 1) (iX - 2) ... (iX - n + 1) r (iX - n + 1),
(iX> n -1)
whence the binomial coefficients (:) may be expressed as
follows
( ",) 1 r(ex + 1) n = n! r(ex _ n +1)' (iX> n - 1) .
(3.2.8)
The Euler function of first kind or beta function is defined by the
formula 1
B(iX, fJ) = J X,,-1 (1 - x)fJ- 1 dx , o
(iX > 0, fJ > 0) . (3.2.9)
3.2 Euler functions 51
Now we prove that this function is connected to the gamma function
by the following relation
In fact for (3.2.1) we have
r(ex) r({3) B(CI., fJ) = r (ex + (3)
+00 +00
(3.2.10)
T(CI.) T(fJ) = J e- X X",-1 dx . J e-Y yf3- 1 dy = = J J e-(x+y)
X",-1 yf3- 1 dx dy , o 0 Q
where Q denotes the quadrant x 2: 0, y > ° of the plane x y.
Operating in this double integral the changing of variables x = u
v, y = (1 - 1t) v, which transforms the quadrant Q into the strip S
defined by ° < u < 1, v > 0, we get
T(CI.) T(fJ) = J J e-V U",-1 v"'+f3-1 (1 - u)f3-1 du dv s 1
+00
= J U",-1 (1 - u)f3-1 du . J e- v 1,,,,+f3-1 dv; o 0
but for (3.2.1) and (3.2.9) the latter expression has value B(CI.,
fJ) T(CI. + fJ) and (3.2.10) follows.
Finally we prove, for the gamma function, the following Legendre
duplication formula
r(2 CI.) = V~ 22",-1 T(CI.) T(CI. + ~). (3.2.11)
From (3.2.9) follows 1 1/2
B(CI., CI.) = J X",-1 (1 - X)",-1 dx = 2 J (x - X2)",-1 dx o
0
and operating on this integral the substitution x = (1 - Y-l='y)/2,
which implies y = 4 (x - X2):
J1 ( y)"'-1 dy 1 J1 - ~ B(CI., CI.) = 2 4 4 VI _ Y = 22",-1 y",-1
(1 - y) dy,
o 0
that is for (3.2.9)
B(CI., CI.) = 22:-1 B (CI.,~-) . For (3.2.lO) this formula may be
transformed into
[r(ex)]2 1 r(ex) r ( ~ ) r(2 ex) = 22 ",-1 ( 1 )
rex+ 2
4·
52 3. Special functions
3.3. Orthogonal polynomials connected to a given weight function.
Let a weight function p(x), measurable and non negative, be defined
on an interval [a, b], finite or infinite, being p(x) > ° in a
set of positive measure and p(x) xn E L[a, b] for n = 0, 1,2, ...
;1) therefore there exist finite all the moments
b
fln = J p(x) xn dx , (n = 0, 1, 2, ... ) , (3.3.1 ) a
of p(x) and we have flo > 0. Let then a sequence {an}n~o, 1, 2,
... of numbers all different from zero be assigned.
Then the following theorem holds:
Theorem 3.3.1. It is possible to construct in one and only one way
a sequence {Pn(X)}n~o,I,2, ... with Pn(x) polynomial of n degree
having an as coefficients of x n, in such a way that
b
J p(x) Pm (x) Pn(x) dx = ° if m 0:/= n, a
or, under an equivalent form b
J p(x) IIn~I(x) P n(X) dx = ° , a
(n = 1,2, ... ) ,
where IIn~I(X) denotes an arbitrary polynomial of degree < n -
1.
(3.3.2)
(3.3.3)
Proof. For the polynomial Po(x