Copyright 2012-2015 Manabu Kano. All rights reserved. 1

F2

Human Systems Lab., Dept. of Systems Science Graduate School of Informatics, Kyoto University

#4

2

2 f(x)

Leonhard Euler 1707-1783

Copyright 2012-2015 Manabu Kano. All rights reserved. 2

F2

3

cn n |cn|

4

x

f (x)

2TT0-T

a

-10 -8 -6 -4 -2 0 2 4 6 8 10

-10 -8 -6 -4 -2 0 2 4 6 8 10

x0

-

1

-1

Copyright 2012-2015 Manabu Kano. All rights reserved. 3

F2

5

cn c0 cn

in in

Outline 6

l l l l l l

Copyright 2012-2015 Manabu Kano. All rights reserved. 4

F2

7

bn an n -n

8

Copyright 2012-2015 Manabu Kano. All rights reserved. 5

F2

9

x2

--20

f (x)

x = 0

10

x M

Copyright 2012-2015 Manabu Kano. All rights reserved. 6

F2

11

x 1 M x M

12

Copyright 2012-2015 Manabu Kano. All rights reserved. 7

F2

13

N > M

Z b

a

NX

n=1

gn(x)dxZ b

ag(x)dx

=

Z b

a

NX

n=1

gn(x) g(x)!dx

Z b

a

NX

n=1

gn(x) g(x)

dx Z b

adx = (b a)

14

Copyright 2012-2015 Manabu Kano. All rights reserved. 8

F2

15

Outline 16

l l l l l l

Copyright 2012-2015 Manabu Kano. All rights reserved. 9

F2

17

n -n

18

bn an

n -n

Copyright 2012-2015 Manabu Kano. All rights reserved. 10

F2

bn an

19

n -n

a0/2

20

bn an

bn ana0/2

n, -n

n, -n

Copyright 2012-2015 Manabu Kano. All rights reserved. 11

F2

Outline 21

l l l l l l

22

f (x) x f (x)

Dirichlet 1805-1859

f (x) f (x)

Copyright 2012-2015 Manabu Kano. All rights reserved. 12

F2

23

l f (x)

l f (x) f(x) xz0

f (x) lim!0 f(z )

lim!0

f(z + )

Outline 24

l l l l l l

Copyright 2012-2015 Manabu Kano. All rights reserved. 13

F2

25

Gibbs 1839-1903

-2

-1

0

1

2

26

l n n f (x) n

Sn(x) n f*(x)

n Sn(x) f (x)

-2

-1

0

1

2

Copyright 2012-2015 Manabu Kano. All rights reserved. 14

F2

-2

-1

0

1

2

n

27

x0

f (x)

1

Outline 28

l l l l l l

Copyright 2012-2015 Manabu Kano. All rights reserved. 15

F2

29

2 f (x) TN(x)

E cn, dn

(1) 30

Copyright 2012-2015 Manabu Kano. All rights reserved. 16

F2

(2) 31

cn, dn

32

cn, dn f (x)

N N f (x)

Copyright 2012-2015 Manabu Kano. All rights reserved. 17

F2

33

Bessel 1784-1846

= 0

34

f (x) N

Copyright 2012-2015 Manabu Kano. All rights reserved. 18

F2

35

36

f (x) 2 f (x)

Copyright 2012-2015 Manabu Kano. All rights reserved. 19

F2

37

38

2 f (x) f (x) E N 0 f (x)

limit in the mean

Copyright 2012-2015 Manabu Kano. All rights reserved. 20

F2

Outline 39

l l l l l l

40