Ανάλυση Fourier Και Ολοκλήρωμα Lebesgue

Anlush Fourier

kai

Oloklrwma Lebesgue

Prqeirec Shmeiseic

Tmma Majhmatikn

Panepistmio Ajhnn

Ajna, 2012

Perieqmena

I Anlush Fourier 1

1 31.1 pi . . . . . . . . . . . . . . . . . . . . . . . . . . 41.2 L2-: . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

2 Fourier 212.1 pi . . . . . . . . . . . . . . . . . . . . . . . 212.2 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 pi . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.4 Cesa`ro Fejer . . . . . . . . . . . 392.4 Abel pi Poisson . . . . . . . . . . . . 43

2.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

3 Fourier 533.1 . . . . . . . . . . . . . . . . . . . . . . . . . 533.2 L2- Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3 pi . . . . . . . . . . . . . . . . 643.4 Fourier pi pi . . . 673.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

II Oloklrwma Lebesgue 77

4 Lebesgue 794.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 794.2 Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . . . 814.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 854.4 Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

iv

4.5 Cantor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.6 . . . . . . . . . . . . . . . . . . . . . . 994.7 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102

5 1095.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1095.2 CantorLebesgue . . . . . . . . . . . . . . . . . . . . . . . . 1165.3 pi pi . . . . . . . . 1185.4 Littlewood . . . . . . . . . . . . . . . . . . . . . . . . 1215.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6 Lebesgue 1276.1 Lebesgue pi . . . . . . . . . . 1286.2 Lebesgue . . . . . . . . . . . . 1316.3 Lebesgue: pipi . . . . . . . . . . . . . . . . . 1386.4 Riemann . . . . . . . . . . . . . . . . . . . . 1436.5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149

7 Lp 1557.1 Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1557.2 Lp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1587.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

III Metasqhmatismc Fourier 163

8 Fourier 1658.1 Fourier R . . . . . . . . . . . . . . . . . . . . . . . . 1658.2 pi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1738.3 pi Plancherel . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1808.4 pi Poisson . . . . . . . . . . . . . . . . . . . . . . . . . 1838.5 Heisenberg . . . . . . . . . . . . . . . . . . . . . 1858.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186

IV Upodexeic gia tic Askseic 193

9 Fourier 1959.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1959.2 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2069.3 Fourier . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229

v

10 Lebesgue 24910.1 Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24910.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26710.3 Lebesgue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273

Mroc I

Anlush Fourier

Keflaio 1

Eisagwg

f : [pi, pi] R Riemann . Fourier f

S[f ](x) =a02

+

k=1

(ak cos kx+ bk sin kx),

pi Fourier ak bk f pi

ak = ak(f) =1

pi

pipi

f(x) cos kx dx, k = 0, 1, 2, . . .

bk = bk(f) =1

pi

pipi

f(x) sin kx dx, k = 1, 2, . . . .

f(x) cos kx f(x) sin kx Riemann , pi - ak bk . pipi, k

|ak| 1pi

pipi|f(x)| dx |bk| 1

pi

pipi|f(x)| dx.

, {ak} {bk} . n- Fourier f

sn(f)(x) =a02

+

nk=1

(ak cos kx+ bk sin kx).

pi pi pi pi pi sn(f) f. pi pi pi-, pi pi .

4

1.1 Trigwnometrik polunuma

1.1.1 ( pi). pipipi cos kx sin kx.,

(1.1.1) T (x) = 0 +

nk=1

(k cos kx+ k sin kx),

pi n N k, k R. T n 0 pi T pi . Tn pi pi pi n. Tn pi 2pi-pi f : R R. 1.1.2. pi T (x) n pi cosx sinx n. , pi pi ( ) p(t, s) n

(1.1.2) T (x) = p(cosx, sinx).

pi pi .

1.1.3. n 1, cosnx (sin(n + 1)x)/ sinx pi cosx n.

pi. pi : n 1 pi a0,n, . . . , an1,n R

(1.1.3) cosnx = 2n1 cosn x+n1j=0

aj,n cosj x.

(1.1.3) n = 1, n = 2

cos 2x = 2 cos2 x 1.

pi (1.1.3) cos kx, pi k 2. pi

(1.1.4) cos[(k + 1)x] + cos[(k 1)x] = 2 cos kx cosx

pi

cos(k + 1)x = 2 cos kx cosx cos(k 1)x

1.1 pi 5

= 2 cosx

2k1 cosk x+ k1j=0

aj,k cosj x

2k2 cosk1 x k2j=0

aj,k1 cosj x

= 2k cosk+1 x+

kj=0

aj,k+1 cosj x

aj,k+1 R. , pi

(1.1.5) sin[(k + 1)x] sin[(k 1)x] = 2 cos kx sinx pi , n 1,

(1.1.6)sin(n+ 1)x

sinx= 2n cosn x+

n1j=0

aj,n cosj x

aj,n R ( pi ). 2 1.1.4.

(1.1.7) B = {1, cosx, cos2 x, . . . , cosn x, sinx, sinx cosx, . . . , sinx cosn1 x}.pi 1.1.3

(1.1.8) Tn span(B),pi span(B) pi pi pi B. , dim(Tn) Tn pi 2n+1, pi pi

(1.1.9) Tn = span(A),pi

(1.1.10) A = {1, cosx, cos 2x, . . . , cosnx, sinx, . . . , sinnx}. card(A) = card(B) = 2n + 1 ( card(X) pi pipi X). A . pi A Tn dim(Tn) = 2n + 1.pipi, span(B) Tn dim(span(B)) 2n+ 1, pi , ,

Tn = span(B) = span(A)., pi cosx, pi n, Tn.

6

1.1.5 ( ). f, g : [pi, pi] R Riemann .

(1.1.11) f, g = 1pi

pipi

f(x)g(x) dx

(1.1.12) f2 = f, f1/2 =(

1

pi

pipi

f2(x) dx

)1/2.

pi CauchySchwarz

(1.1.13) |f, g| f2g2. pi g, f = f, g f + g, h = f, h + g, h f, g, h Riemann , R. 1.1.6 ( ). pi:

(i) m,n = 0, 1, 2, . . . m 6= n 1

pi

pipi

cosmx cosnx dx = 0.

(ii) m,n = 1, 2, . . . m 6= n 1

pi

pipi

sinmx sinnx dx = 0.

(iii) m = 0, 1, 2, . . . n = 1, 2, . . .

1

pi

pipi

cosmx sinnx dx = 0.

(iv) m,n = 1, 2, . . .

1

pi

pipi

cos2mx dx =1

pi

pipi

sin2 nx dx = 1.

pi. . pi

2 cos cos = cos( ) + cos( + ),2 sin cos = sin( + ) + sin( ),2 sin sin = cos( ) cos( + ),

2 cos2 = 1 + cos 2, 2 sin2 = 1 cos 2. 2

1.1 pi 7

1.1.7. A = {1, cosx, cos 2x, . . . , cosnx, sinx, . . . , sinnx} .

pi.

(1.1.14) T (x) = 0 +

nk=1

(k cos kx+ k sin kx) 0,

(1.1.15) 0 = 1 = = n = 1 = = n = 0.

pipi pi 1.1.6. pi, m = 1, . . . , n

0 = T, sinmx = 01, sinmx+nk=1

(kcos kx, sinmx+ ksin kx, sinmx)

= msinmx, sinmx = m,

cos kx, sinmx = 0 0 k n sin kx, sinmx = 0 1 k n,k 6= m. m = 0 m = 0, 1, . . . , n. 2 1.1.8. f : [pi, pi] R

(1.1.16) f = sup{|f(x)| : x [pi, pi]}.

pi pi Weierstrass ( pi ).

1.1.9. f : [a, b] R . > 0 pipi p

(1.1.17) f p = max{|f(x) p(x)| : x [a, b]} < .

, pi {pm} pi f pm 0.pi 1.1.9 T

pi pi 2pi-pi :

1.1.10. f : R R 2pi-pi . > 0pi pi T

(1.1.18) f T = max{|f(x) T (x)| : x R} < .

, pi {Tm} pi f Tm 0.

8

pi. pi , pipi pi- f : , f(x) = f(x) x R. g : [1, 1] R

(1.1.19) g(y) = f(arccos y).

g , arccos y [0, pi] y [1, 1], , . pi 1.1.9, pi pi p g p < . ,(1.1.20) |f(arccos y) p(y)| < y [1, 1]. T (x) = p(cosx). T pi cosx, T T . , x [0, pi] pi y [1, 1] y = cosx, ,(1.1.21) |f(x) T (x)| = |f(x) p(cosx)| = |f(arccos y) p(y)| < . f T , pi

(1.1.22) f T = max{|f(x) T (x)| : pi x pi} < , pi .

pipi, 2pi-pi f : RR

(1.1.23) f1(x) = f(x) + f(x) f2(x) = [f(x) f(x)] sinx. f1 f2 , 2pi-pi. , pi pi T1 T2

(1.1.24) f1 T1 < 2

f2 T2 < 2.

(1.1.25) T3(x) =1

2(T1(x) sin

2 x+ T2(x) sinx),

T3 T , x [pi, pi],|2f(x) sin2 x 2T3(x)| = |f1(x) sin2 x+ f2(x) sinx T1(x) sin2 x T2(x) sinx|

|(f1(x) T1(x)) sin2 x|+ |(f2(x) T2(x)) sinx| |f1(x) T1(x)|+ |f2(x) T2(x)| <

2+

2= .

, f3(x) = f(x) sin2 x

(1.1.26) f3 T3 < 2.

1.1 pi 9

g(x) := f(x pi2

). g 2pi-pi.

pi, pi pi T4 , f4(x) = g(x) sin

2 x f4 T4 < 2 . T5(x) =T4(x + pi/2), T5 pi ( ) x R, y = x+ pi/2 (1.1.27)

|f(x) cos2 x T5(x)| = |f(x) cos2 x T4(x+ pi/2)| = |f(y pi/2) sin2 y T4(y)| < 2.

pi,

(1.1.28) f5 T5 < 2,

pi f5(x) = f(x) cos2 x. f = f3 + f5, f(x) = f(x) sin

2 x + f(x) cos2 x. T =T3 + T5. , T T

(1.1.29) f T = (f3+f5) (T3+T5) f3T3+f5T5 < 2

+

2= .

pi . 2

1.1.11. f : R R 2pi-pi (1.1.30) ak(f) = bk(f) = 0

k. , f 0.pi. pi pi pi

(1.1.31)

pipi

f(x)T (x) dx = 0

pi T . pi 1.1.10 pi {Tm} pi f Tm 0. , m

(1.1.32)

pipi

f2(x) dx =

pipi

f2(x) dx pipi

f(x)Tm(x) dx =

pipi

f(x)(f(x)Tm(x)) dx.

,

(1.1.33)

pipi

f2(x) dx pipiff Tmdx = 2piff Tm 0.

pi

(1.1.34)

pipi

f2(x) dx = 0,

10

, f , pi f 0. 2 pi pi Fourier

pi

T (x) = 0 +

nk=1

(k cos kx+ k sin kx).

pi 1.1.6 : k = 1, . . . , n

(1.1.35) ak(T ) = T, cos kx = kcos kx, cos kx = k,, k > n

(1.1.36) ak(T ) = T, cos kx = 0., k = 1, . . . , n

(1.1.37) bk(T ) = T, sin kx = ksin kx, sin kx = k,, k > n

(1.1.38) bk(T ) = T, sin kx = 0.,

(1.1.39) a0(T ) = 20.

pi , m n,

sm(T )(x) =a0(T )

2+

mk=1

(ak(T ) cos kx+ bk(T ) sin kx)

= 0 +

nk=1

(k cos kx+ k sin kx)

= T (x).

, Fourier T T :

1.1.12. T pi pin. m n (1.1.40) sm(T ) T.pi,

(1.1.41) S[T ] T.

1.2 L2-: 11

1.1.12 pi Fourier S[f ] f pi pipi pi f pi: n pi pi f sn(f) f .pi pi, 1.1.10 pi -pi . pi pi S[f ] f pi pi pi pi.

1.2 L2-sgklish: mia eisagwg

f : [pi, pi] R Riemann . sn(f)f pi 2. ,

limn f sn(f)

22 = lim

n1

pi

pipi

(f(x) sn(f)(x))2dx = 0.

pi pi , pi pi n- sn(f) Fourier f pi pi f pi Tn 2-pi. 1.2.1. f : [pi, pi] R, Riemann . n 0,

(1.2.1) f sn(f)2 = min{f T2 : T Tn}.

pi. T Tn. ,

(1.2.2) T (x) =02

+

nk=1

(k cos kx+ k sin kx)

pi k, k R.

(1.2.3) f T22 = f T, f T = f22 2f, T + T22.

pi f, T T22.

f, T = 1pi

pipi

f(x)T (x) dx

=02pi

pipi

f(x) dx+

nk=1

(k

1

pi

pipi

f(x) cos kx dx+ k1

pi

pipi

f(x) sin kx dx

)

=0a0(f)

2+

nk=1

kak(f) +

nk=1

kbk(f).

12

f T , pi ak(T ) = k bk(T ) = k pi

(1.2.4) T22 = T, T =202

+

nk=1

2k +

nk=1

2k.

T sn(f), pi f, T = 0a0(f)2 +nk=1 kak(f) +n

k=1 kbk(f) (1.2.4),

(1.2.5) f, sn(f) = sn(f)22 =a202

+

nk=1

a2k +

nk=1

b2k.

pipi,

f T22 = f22 +20 20a0(f)

2+

nk=1

(2k 2kak(f)) +nk=1

(2k 2kbk(f))

= f22 +(0 a0)2

2+

nk=1

(k ak)2 +nk=1

(k bk)2

(a202

+

nk=1

(a2k + b2k)

)

f22 (a202

+

nk=1

(a2k + b2k)

)= f22 sn(f)22.

k = ak k = 0, 1, . . . , n k = bk k = 1, . . . , n. , T sn(f). , (1.2.6) f sn(f)22 = f22 sn(f)22 f T22 T Tn. 2. pi pi pi

sn(f)22 =a202

+

nk=1

(a2k + b2k),

(sn(f)2) . pi,f22 sn(f)22 = f sn(f)22 0.

pi Bessel:

1.2 L2-: 13

1.2.2 ( Bessel). f : [pi, pi] R . n 0

(1.2.7) sn(f)22 =a202

+

nk=1

(a2k + b2k) f22,

pi ak = ak(f) bk = bk(f) Fourier f . pi,

(1.2.8)a202

+

k=1

(a2k + b2k) f22.

Bessel {ak(f)} {bk(f)} . 1.2.3 ( RiemannLebesgue). f : [pi, pi] R . ,

(1.2.9) limk

ak(f) = limk

bk(f) = 0.

pi. pi Bessel pi

k=1

a2k k=1

b2k

. {ak} {bk} 0. 2pi , (1.2.8) pi

( Parseval). pi pi .

1.2.4. f : R R , 2pi-pi . ,

(1.2.10) f22 =a202

+

k=1

(a2k + b2k).

pi. pi f sn(f)22 = f22 sn(f)22,

(1.2.11) limn f sn(f)2 = 0,

pi

(1.2.12) f22 = limn sn(f)

22 =

a202

+

k=1

(a2k + b2k).

14

1.1.10. > 0 pi T

(1.2.13) f T < 2.

,

f T2 =(

1

pi

pipi|f(x) T (x)|2dx

)1/2(

1

pi

pipif T2dx

)1/2=

2f T < . n0 T . pi 1.2.1 pi

(1.2.14) f sn0(f)2 f T2 < . , n n0

(1.2.15) sn(f)22 =a202

+

nk=1

(a2k + b2k)

a202

+

n0k=1

(a2k + b2k) = sn0(f)22,

(1.2.16) f sn(f)22 = f22 sn(f)22 f22 sn0(f)22 = f sn0(f)22,, n n0 (1.2.17) f sn(f)2 f sn0(f)2 < . pi f sn(f)2 0. 2

pi Riemann , -pi pi .

1.2.5. f : [pi, pi] R Riemann > 0. , pi g : [pi, pi] R g f g(pi) = g(pi) f g2 < .pi. > 0. pi P = {pi = x0 < x1 < 0, g(x) = f(x) pi x pi pi x0, . . . , xN . -pi xj j = 1, . . . , N 1, g pi pi g(xj ) = f(xj ). x0 = pi, pi g g(pi) = 0 g(pi + ) = f(pi + ). , xN = pi,pi g g(pi) = 0 g(pi ) = f(pi ).

g(pi) = g(pi), pi pi g pi R. pi pi pi pi f.pipi, g pi f N + 1 2 pi x0, . . . , xN . pi,

(1.2.21)

pipi|f(x) g(x)| dx 2BN 2.

pi , pi

(1.2.22)

pipi|f(x) g(x)| dx < .

(1.2.23)

pipi|f(x) g(x)| dx < 2.

(1.2.24)

pipi|f(x) g(x)|2 dx 2f

pipi|f(x) g(x)| dx,

pi pi pi

(1.2.25) f g2 < pi > 0 12pi (2)(2f) < 2 - . 2

1.2.6 ( Parseval). f : R R 2pi-pi , [pi, pi]. ,

(1.2.26) f22 :=1

pi

pipi|f(x)|2dx = a

20

2+

k=1

(a2k + b2k).

16

pi. > 0. pi 1.2.5 pi 2pi-pi g f g2 < /3. ,(1.2.27) f sn(f)2 f g2 + g sn(g)2 + sn(g) sn(f)2.

(1.2.28) sn(g) sn(f)2 = sn(g f)2 g f2 < 3.

pi,

(1.2.29) f sn(f)2 < 23

+ g sn(g)2.

pi 1.2.4 g sn(g)2 0, pi n0 N , n n0, g sn(g)2 < /3. , n n0 f sn(f)2 < . f sn(f)2 0, sn(f)22 f22 pi . 2

Parseval pi 1.1.11:

1.2.7. f : R R 2pi-pi ak(f) = bk(f) = 0 k. , f 0.pi. pi pi pi Parseval pi

(1.2.30)

pipi|f(x)|2dx = 0.

f , pi f 0. 2 pi pi

S[f ] f .

1.2.8. f : [pi, pi] R f(pi) = f(pi) = 0.pi

(1.2.31)

k=1

(|ak(f)|+ |bk(f)|) < +.

, Fourier f f . ,

(1.2.32) sn(f) f.

pi. pi pi k=1(|ak(f)| + |bk(f)|) < + pi

Weierstrass pi

sn(f)(x) = a0(f) +

nk=1

(ak(f) cos kx+ bk(f) sin kx)

1.3 17

g : [pi, pi] R

(1.2.33) g(x) = a0(f) +

k=1

(ak(f) cos kx+ bk(f) sin kx), x [pi, pi].

: k 0 n k

(1.2.34)1

pi

pipi

sn(f)(x) cos kxdx = ak(f),

, k 1 n k

(1.2.35)1

pi

pipi

sn(f)(x) sin kxdx = bk(f).

pi sn(f) g pi , k 0,

(1.2.36) ak(g) =1

pi

pipi

g(x) cos kx dx = limn

1

pi

pipi

sn(f)(x) cos kx dx = ak(f).

, , k 1,(1.2.37) bk(g) = bk(f).

f g Fourier,

1.2.7 g f . pi, sn(f) f . 2

1.3 Askseic

1. T (x) = 0 +nk=1(k cos kx + k sin kx) pi.

:

() T pi , k = 0 k = 0, 1, . . . , n.

() T , k = 0 k = 1, . . . , n.

2. : k N pi pi p(t) 2k sin2k x = p(cosx) x R.3. pi pi 1.1.6: 1, cosx, . . . , cosnx, sinx, . . . , sinnx .

4. f(x) = pi x 0 < x < 2pi, f(0) = f(2pi) = 0, pi f 2pi-pi R. Fourier f

S[f ](x) = 2

k=1

sin kx

k.

18

5. f : [pi, pi] R Riemann > 0. pi g : [pi, pi] R |g(x)| f x [pi, pi] pi

pi|f(x) g(x)| dx < .

6. f : [pi, pi] R Riemann > 0.() pi g : [pi, pi] R f g2 < .() pi 2pi-pi h : R R f h2 < .() pi pi T f T2 < .7. f : R R 2pi-pi , [pi, pi].

limt0

pipi|f(x+ t) f(x)|2dx = 0.

8. f(x) = (pi x)2 [0, 2pi] pi 2pi-pi R.

S[f ](x) =pi2

3+ 4

k=1

cos kx

k2.

pi pipi,

k=1

1

k2=pi2

6.

9. f : R R pi 2pi-pi pipi

f(x) dx = 0.

pi Parseval f f pipi|f(x)|2dx

pipi|f (x)|2dx,

f(x) = a cosx+ b sinx pi a, b R.

1.3 19

10. () k N

Ak(x) =

kj=1

sin jx.

: k > m

|Ak(x)Am(x)| 1| sin(x/2)| 0 < x < pi.

() 1 2 n 0, k

j=m+1

j sin jx

m+1| sin(x/2)| n k > m 1 0 < x < pi.11. n N M > 0. 1 2 n 0 kk M k = 1, . . . , n,

nk=1

k sin kx

(pi + 1)M x R. [pi: pi pi 0 < x < pi. , ,

nk=1

k sin kx =

mk=1

k sin kx+

nk=m+1

k sin kx,

pi m = min{N, bpi/xc}.]12. 0 < 1 f : R R 2pi-pi . pi pi M > 0

|f(x) f(y)| M |x y|

x, y R. : pi C > 0 , k Z \ {0},

|ak(f)| Ck

|bk(f)| Ck.

Keflaio 2

Seirc Fourier

2.1 Migadik morf kai paradegmata

2.1.1 ( ). T

(2.1.1) T = {z C : |z| = 1} = {ei : R}.

F : T C , f : R C

(2.1.2) f() = F (ei).

f 2pi-pi. , f : R C 2pi-pi, F : T C F (ei) = f() (pi, ei1 = ei2 pi 1, 2 R 2 = 1 + 2kpi pi k, f(1) = f(2) pi 2pi-pi f). pi 1 1 F : T C 2pi-pi f : R C.

, F f - pi ( ) 2pi, F f , F pi f pi, F pi f pi .

pi f : [a, b] C pipi , f f = u + iv, pi u(x) = Re(f(x)) v(x) = Im(f(x)), x [a, b]. f Riemann u, v Riemann ,

(2.1.3)

ba

f(x) dx =

ba

u(x) dx+ i

ba

v(x) dx.

22 Fourier

pi : f : [a, b] C ,

(2.1.4)

ba

f(x) dx

ba

|f(x)| dx.

pi , ba

f(x) dx = Rei0 , pi R =

ba

f(x) dx

0 R, pi

ba

f(x) dx

= ei0 ba

f(x) dx =

ba

ei0f(x) dx

=

ba

Re(ei0f(x)) dx ba

|ei0f(x)| dx

=

ba

|f(x)| dx.

2.1.2 ( Fourier). f : [pi, pi] C . k Z k- Fourier f

(2.1.5) f(k) =1

2pi

pipi

f(x)eikxdx.

pi (2.1.4)

(2.1.6) |f(k)| = 12pi

pipi

f(x)eikxdx 12pi

pipi|f(x)| dx f,

pi |eikx| = 1. pi, {f(k)}kZ .

Fourier f

(2.1.7) S[f ](x) =

k=

f(k)eikx.

n- Fourier f pi

(2.1.8) sn(f)(x) =

nk=n

f(k)eikx.

2.1 pi 23

, pi

(2.1.9) T (x) =

nk=n

ckeikx,

pi n 0 ck C, |k| n. F : T C , f() = F (ei)

Fourier f pi f [pi, pi],pi (2.1.5).

2.1.3 ( pi). f : [pi, pi] C - . 1.1, k 0 -

(2.1.10) ak(f) =1

pi

pipi

f(x) cos kx dx

k 1

(2.1.11) bk(f) =1

pi

pipi

f(x) sin kx dx.

: k Z \ {0},

(2.1.12) f(k) =1

2pi

pipi

f(x) cos kx dx i 12pi

pipi

f(x) sin kx dx =ak(f) ibk(f)

2,

(2.1.13) f(k) = 12pi

pipi

f(x) cos kx dx+ i1

2pi

pipi

f(x) sin kx dx =ak(f) + ibk(f)

2.

pi,

(2.1.14) f(0) =1

2pi

pipi

f(x) dx =a0(f)

2.

.

2.1.4. f : [pi, pi] C . k Z\{0}

(2.1.15) ak(f) = f(k) + f(k) bk(f) = i(f(k) f(k)).

24 Fourier

pi, a0(f) = 2f(0)

(2.1.16) sn(f)(x) =

nk=n

f(k)eikx =a0(f)

2+

nk=1

(ak(f) cos kx+ bk(f) sin kx).

, n- Fourier f 1.1.

pi. a0(f) = 2f(0), ak(f) = f(k) + f(k) bk(f) = i(f(k) f(k))pipi pi (2.1.12), (2.1.13) (2.1.14). (2.1.16)

sn(f)(x) =

nk=n

f(k)eikx

=a0(f)

2+

nk=1

f(k)eikx +

1k=n

f(k)eikx

=a0(f)

2+

nk=1

f(k)eikx +

nk=1

f(k)eikx

=a0(f)

2+

nk=1

f(k)(cos kx+ i sin kx) +

nk=1

f(k)(cos kx i sin kx)

=a0(f)

2+

nk=1

(f(k) + f(k)) cos kx+nk=1

i(f(k) f(k)) sin kx

=a0(f)

2+

nk=1

(ak(f) cos kx+ bk(f) sin kx),

pi (2.1.15). 2

pi pi pi pi : F : T C , , f : R C 2pi-pi , 2pi, sn(f)(x) =nk=n f(k)e

ikx f .

() f() = [pi, pi) pi 2pi-pi R. f pi [pi, pi]. pi Fourier f . f pi,

(2.1.17) f(0) =1

2pi

pipi

d = 0.

2.1 pi 25

k 6= 0

f(k) =1

2pi

pipi

eikd =1

2pi

pipi

[eik

ik

]d

=1

2pi

eikik

|pipi +1

2pi

pipi

eik

ikd

=1

2pi

pieikpi pieikpiik

= 12k

eikpi + eikpi

i

=(1)k+1

ik.

pi pipi

eikik d = 0. pi

(2.1.18)

S[f ]() =k 6=0

(1)k+1ik

eik =

k=1

(1)k+1eik (1)k+1eikik

= 2

k=1

(1)k+1 sin kk

.

pi , , pi pi ak(f) = 0 k Z, f pi.

(2.1.19) S[f ]() =k 6=0

bk(f) sin k.

pi , pi pipi, pi pi bk(f) pi (2.1.18).

() pi Dirichlet. n 0. n- pi Dirichlet pi

(2.1.20) Dn(x) =

nk=n

eikx, x [pi, pi].

Dn(0) = 2n+ 1. : 0 < |x| pi,

(2.1.21) Dn(x) =sin((n+ 12

)x)

sin(x/2).

pi , = eix

(2.1.22) Dn(x) =

nk=0

k +

1k=n

k =

nk=0

k +

nk=1

(1/)k.

(2.1.23)

nk=0

k =1 n+1

1 ,

26 Fourier

(2.1.24)

nk=1

(1

)k=

1

1 n1 1

=n 1

1 .

pi,

(2.1.25) Dn(x) =n n+1

1 =einx ei(n+1)x

1 eix =eix/2

eix/2ei(n+ 12

)x ei

(n+ 12

)x

eix/2 eix/2 ,

pi pi pipi (2.1.21). pi Dirichlet pi pi. pi Fourier f pi :

sn(f)(x) =

nk=n

f(k)eikx =

nk=n

(1

2pi

pipi

f(y)eikydy)eikx

=1

2pi

pipi

f(y)

(n

k=neik(xy)

)dy =

1

2pi

pipi

f(y)Dn(x y) dy.

2.3 pi -.

2.2 Monadikthta seirn Fourier

pi 1.1 1.2 2pi-pi f : RR ak(f) bk(f) , f 0. pi .

2.2.1. f : T C f(k) = 0 k Z. f 0 T f(0) = 0.pi. pi pi f pi pi . pi pi f [pi, pi] 0 = 0. [ pi : f 0, g(x) = f(x+0) 0 pi Fourier g.]

pi f(0) > 0 pi ( pipipi f(0) < 0). {pm} pi pi pi 0 pi pi

limk

pipi

pm()f() d = +.

2.2 Fourier 27

pi pi, pi f(k) = 0 k Z pipi 0 ( ).

, f , : piM > 0 |f()| M [pi, pi]. f 0, 0 < < pi/2 f() > f(0)/2 (, ).

cos cos < 1 || pi. pi, pi > 0

(2.2.1) |+ cos | < 1 /2

|| pi. pi 0 < < 2(1cos )3 . , + cos 0 | + cos | = + cos + cos < 1 /2 pi pi , + cos < 0 |+ cos | = cos 1 < 1 /2.

(2.2.2) p() = + cos , [pi, pi].

, p(0) = 1 + , pi pi 0 < <

(2.2.3) p() 1 + /2, (, ).

, m = 1, 2, . . .,

(2.2.4) pm() = [p()]m = (+ cos )m.

pm pi ( ). f(k) =0 k Z, pi

(2.2.5)

pipi

pm()f() d = 0, k = 1, 2, . . . .

(2.2.6) pipi

pm()f() d =

||pi

pm()f() d+

||

28 Fourier

(ii)

(2.2.8)

|| 0 0 < < pi/2.

(iii)

(2.2.9)

||

2.2 Fourier 29

2.2.2 ( Fourier). f : T C f(k) = 0 k Z, f 0. 2

pi pi ( ) sn(f) f Fourier f pi.

2.2.3. f : T C . pi

(2.2.15)

k=

|f(k)| < +.

, Fourier f f . ,

(2.2.16) sn(f) f.

pi. pi pi

k=|f(k)| < + pi -

(2.2.17) sn(f)(x) =

nk=n

f(k)eikx

: pi, m > n

(2.2.18) sm(f) sn(f) = maxxT|sm(f)(x) sn(f)(x)|

n

30 Fourier

f g Fourier, pi

2.2.2 pi g f . pi, sn(f) f . 2

pi pipi pi 2.2.3

k=|f(k)| : , pi ,

S[f ] f . f (pi.. pi) Fourier :

2.2.4. f : T C pi f C2(T). , pi C = C(f) > 0

(2.2.22) |f(k)| C(f)k2

, k Z \ {0}.

pi sn(f) f .

pi. k 6= 0 :

2pif(k) =

pipi

f()eikd

=

[f()

eikik

]pipi

+1

ik

pipi

f ()eikd

=1

ik

pipi

f ()eikd

=1

ik

[f ()

eikik

]pipi

+1

(ik)2

pipi

f ()eikd

= 1k2

pipi

f ()eikd,

pi pi , f f 2pi-pi,

(2.2.23)

[f()

eikik

]pipi

=

[f ()

eikik

]pipi

= 0.

pi,

(2.2.24) |f(k)| 1k2

12pi pipi

f () d C(f)k2 ,

pi

(2.2.25) C(f) =1

2pi

pipi|f ()| d.

2.2 Fourier 31

pi pi 2.2.3 pi

k=1

1k2 < +. 2

2.2.5. pi pi 2.2.4 :

() f : T C pi,

(2.2.26) f(k) =1

ik

1

2pi

pipi

f ()eikd =1

ikf (k)

k 6= 0. pi pi f

(2.2.27) 2pif (0) = pipi

f () = f(pi) f(pi) = 0.

pi,

(2.2.28) f (k) = ikf(k), k Z.

() f : T C pi,

(2.2.29) f(k) = 1k2

1

2pi

pipi

f ()eikd = 1k2f (k)

k 6= 0. pi pi f

(2.2.30) 2pif (0) = pipi

f () = f (pi) f (pi) = 0.

pi,

(2.2.31) f (k) = k2f(k), k Z. 2.2.6. f : R C 2pi-pi , [pi, pi]. f C2(T)

k=

|f(k)| < +. pi ,

k=|f(k)| pi

f . pi f pi.

k=|f(k)| f pi Holder

> 1/2: , pi M > 0

(2.2.32) |f(x) f(y)| M |x y|

x, y R.

32 Fourier

2.3 Sunelxeic kai kalo purnec

f g 2pi-pi R, f g f g [pi, pi]

(2.3.1) (f g)(x) = 12pi

pipi

f(y)g(x y) dy.

x [pi, pi], .

pi . pi, g 1 f g ,

(2.3.2) (f 1)(x) = 12pi

pipi

f(y) dy.

, f [pi, pi]. pi pi , (f g)(x) , pi , f(x)g(x) f g.

pi pi Fourier f pi :

sn(f)(x) =

nk=n

f(k)eikx =

nk=n

(1

2pi

pipi

f(y)eikydy)eikx

=1

2pi

pipi

f(y)

(n

k=neik(xy)

)dy = (f Dn)(x),

pi Dn n- pi Dirichlet, pi pi

(2.3.3) Dn(x) =

nk=n

eikx.

pi sn(f) f Dn.

pi pi pi .

2.3.1. f, g h : R C 2pi-pi .:

(i) f (g + h) = (f g) + (f h).(ii) (cf) g = c(f g) = f (cg) c C.(iii) f g = g f .

2.3 pi 33

(iv) (f g) h = f (g h).(v) f g .(vi) f g(k) = f(k)g(k) k Z.

pi pi pi :, pi. pipi pi f g pi pi f g. f g f g pi Riemann. , pi pipi Fourier. , Fourier fg Fourier f g. f g f g, .pi. (i) (ii) pipi pi -.

pipi pi pi f g . pipi, pi . pi (iii), pi x R , pi u = x y,

(f g)(x) = 12pi

pipi

f(y)g(x y) dy = 12pi

x+pixpi

f(x u)g(u) du.

f, g 2pi-pi, F (u) = f(x u)g(u) pi 2pi-pi. pi,

(2.3.4)

x+pixpi

f(x u)g(u) du = pipi

f(x u)g(u) du.

,

(2.3.5) (f g)(x) = 12pi

pipi

g(u)f(x u) du = (g f)(x).

(iv) pi .

[(f g) h](x) = 12pi

pipi

(f g)(y)h(x y) dy

=1

4pi2

pipi

pipi

f(t)g(y t)h(x y) dt dy

=1

4pi2

pipi

pipi

f(t)g(y t)h(x y) dy dt

=1

4pi2

pipi

f(t)

pipi

g(y t)h((x t) (y t)) dy dt.

34 Fourier

G(u) = g(u)h(x t u) 2pi-pi. u = y t 1

2pi

pipi

g(y t)h((x t) (y t)) dy = 12pi

pitpit

g(u)h(x t u) du

=1

2pi

pipi

g(u)h(x t u) du = (g h)(x t).

pi pi pi

[(f g) h](x) = 12pi

pipi

f(t)(g h)(x t) dt = [f (g h)](x).

pi (vi)

f g(k) = 12pi

pipi

(f g)(x)eikxdx

=1

2pi

pipi

1

2pi

( pipi

f(y)g(x y) dy)eikxdx

=1

2pi

pipi

f(y)eiky(

1

2pi

pipi

g(x y)eik(xy)dx)dy

=1

2pi

pipi

f(y)eiky(

1

2pi

pipi

g(x)eikxdx)dy

= f(k)g(k).

, f g , f g . ,

(2.3.6) (f g)(x1) (f g)(x2) = 12pi

pipi

f(y)[g(x1 y) g(x2 y)] dy.

g , ., g pi, pi R. pi > 0, pi > 0 : |s t| < |g(s) g(t)| < . pi |x1 x2| < , |(x1 y) (x2 y)| < y, pi

|(f g)(x1) (f g)(x2)| 12pi

pipi f(y)[g(x1 y) g(x2 y)] dy

12pi

pipi|f(y)| |g(x1 y) g(x2 y)| dy

2pi

pipi|f(y)| dy

2pi

2pif.

2.3 pi 35

pi f g () . pi , pi pi f g .

pipi, pi f g pi pi , pi pi pi pi pi ( f g), pi pi.

2.3.2. f : T C . pi {fm}m=1 T

(2.3.7) fm f, m = 1, 2, . . . ,

(2.3.8)

pipi|f(x) fm(x)| dx 0, m.

pi. pi f pi pi ( pipi, - pi pi f).pi 1.2.5, m N fm : T C fm f

(2.3.9)

pipi|f(x) fm(x)| dx < 1

m.

, {fm} . 2pi , pi pi . -

pi {fm} {gm} fm f gm g, pi pi f g . ,

(2.3.10) f g fm gm = (f fm) g + fm (g gm).

pi {fm},

|(f fm) g(x)| 12pi

pipi|f(x y) fm(x y)| |g(y)| dy

12pig

pipi|f(y) fm(y)| dy

0 m.

pi (f fm) g 0 pi x. ,

|(fm (g gm))(x)| 12pi

pipi|fm(y)| |g(x y) gm(x y)| dy

36 Fourier

12pifm

pipi|g(y) gm(y)| dy

f2pi

pipi|g(y) gm(y)| dy

0 m, fm (g gm) 0 , pi fm gm f g . fm gm , pi f g pi . pi (v).

pi (vi). pi pi k,

|f(k) fm(k)| = 12pi

pipi(f(x) fm(x))eikxdx

12pi

pipi|f(x) fm(x)| dx,

pi pi pipi fm(k) f(k) m. gm(k) g(k). {fm gm} f g,

(2.3.11)1

2pi

pipi|(fm gm)(x) (f g)(x)| dx (fm gm) (f g) 0.

, pi pipi, pi

(2.3.12) fm gm(k) f g(k)

m. pi fm(k)gm(k) = fm gm(k), fm gm . (vi) pipi m pi pi. (iii) (iv) pi pi pi. 2

2.3.3 ( pi). {Kn}n=1 Kn : T C pi ( pi ) pi :

(i) n N,

(2.3.13)1

2pi

pipi

Kn(x) dx = 1.

(ii) pi M > 0 , n N,

(2.3.14)

pipi|Kn(x)| dx M.

2.3 pi 37

(iii) > 0,

(2.3.15) limn

|x|pi

|Kn(x)| dx = 0.

, pi: Kn(x) 0 n x. pipi, () pipi pi () pi . () Kn () , n , .

pi pi Fourier pi pi .

2.3.4. {Kn}n=1 pi f : T C . , x T pi f ,

(2.3.16) limn(f Kn)(x) = f(x).

, f pi T,

(2.3.17) f Kn f.

pi. , 2pi-pi f : R C. pi f x > 0. pi f x,pi > 0 : |y| < |f(xy)f(x)| < piM . pi () {Kn},

(f Kn)(x) f(x) = 12pi

pipi

Kn(y)f(x y) dy f(x)

=1

2pi

pipi

Kn(y)f(x y) dy f(x) 12pi

pipi

Kn(y) dy

=1

2pi

pipi

Kn(y)[f(x y) f(x)] dy.

pi,

|(f Kn)(x) f(x)| = 12pi

pipi

Kn(y)[f(x y) f(x)] dy

12pi

|y|

38 Fourier

pi pi : |y| < |f(x y) f(x)| < piM .pi () {Kn}, pi

1

2pi

|y| 0 , n N,

(2.3.21)

pipi|Dn(x)| dx c log n,

2.4 Fourier 39

pi (). pi (2.3.21) : pi pi pi - |Dn|. pi Dirichlet pi: pi .

{Dn} pi, pi 2.3.4

(2.3.22) sn(f) = f Dn f

2pi-pi f : R C. pi pi, pi {sn(f)} f pipi, pipi ( pi ).

pi {sn(f)} f , pi pi.

2.4 Ajroisimthta seirn Fourier

2.4 Cesa`ro Fejer

(2.4.1)

k=0

ck = c0 + c1 + c2 + + cn + .

n-

(2.4.2) sn =

nk=0

ck = c0 + c1 + + cn.

s C lim sn = s. pi

(2.4.3)

k=0

(1)k = 1 1 + 1 1 + ,

pi {sn} pi 1, 0, 1, 0, . . . . pi 0 1, pi pi , , 1/2, 1/2 pi pi . pi pi {n} pi n . (2.4.1),

(2.4.4) n =s0 + s1 + + sn1

n

40 Fourier

n = 1, 2, . . .. pi n n- Cesa`ro {sk} ( n- Cesa`ro

k=0 ck).

pi limnn = C,

k=0 ck Cesa`ro -

. , Cesa`ro pi .

pi (2.4.3) pi n 1/2., pi Cesa`ro 1/2. pi

k=0 ck s =

k=0 ck = s,

n s, Cesa`ro s ( ). 2.4.1 (pi Fejer). n- pi Fejer - pi

(2.4.5) Fn(x) =D0(x) +D1(x) + +Dn1(x)

n, n 1.

pi Dn pi Dirichlet. pi Fejer

Fn(x) =1

n

n1s=0

Ds(x) =1

n

n1s=0

sk=s

eikx

=

n1k=(n1)

1n

|k|sn1

1

eikx = n1k=(n1)

n |k|n

eikx

=

n1k=(n1)

(1 |k|

n

)eikx.

2.4.2. pi pi , f 2pi-pi ,

(2.4.6) n(f)(x) :=s0(f)(x) + s1(f)(x) + + sn1(f)(x)

n

n(f)(x) =(f D0)(x) + (f D1)(x) + + (f Dn1)(x)

n

=

(f D0 +D1 + +Dn1

n

)(x)

= (f Fn)(x).,

(2.4.7) n(f) f Fn.

2.4 Fourier 41

n(f) pi pi n 1, sk(f), 0 k n 1, pi .

pi pi {Fn} pi.

2.4.3. n 1, n- pi Fejer pi

(2.4.8) Fn(x) =1

n

sin2(nx/2)

sin2(x/2), x 6= 2kpi

(2.4.9) Fn(x) = n, x = 2kpi.

{Fn}n=1 pi.pi. x 6= 2kpi. , s = 0, 1, . . . , n 1,

(2.4.10) Ds(x) =sin(s+ 12

)x

sin(x/2).

,

n1s=0

sin(s+ 12

)x

sin(x/2)=

1

2 sin2(x/2)

n1s=0

2 sin(x/2) sin(s+

1

2

)x

=1

2 sin2(x/2)

n1s=0

[cos sx cos(s+ 1)x]

=1

2 sin2(x/2)(1 cosnx)

=sin2(nx/2)

sin2(x/2)

n pi

(2.4.11) Fn(x) =1

n

n1s=0

Ds(x) =1

n

sin2(nx/2)

sin2(x/2).

x = 2kpi, Ds(x) = 2s+ 1, s = 0, 1, . . . , n 1. pi,

(2.4.12) Fn(2kpi) =1 + 3 + + (2n 1)

n=n2

n= n.

42 Fourier

(2.4.13) Fn(x) 0, x R. {Fn} - pi, pi pi () (). pi pi:

(2.4.14)1

2pi

pipi

Ds(x) dx = 1

s 0,

(2.4.15)1

2pi

pipi

Fn(x) dx =1

n

n1s=0

1

2pi

pipi

Ds(x) dx = 1

n 1. () pi , pi (2.4.8), (0, pi) < |x| pi,

(2.4.16) |Fn(x)| = Fn(x) = 1n

sin2(nx/2)

sin2(x/2) 1n sin2(x/2)

1n sin2(/2)

.

(2.4.17)

|x|pi

|Fn(x)| dx 2pin sin2(/2)

0

pi pi. 2

pi 2.3.4 2.4.3 .

2.4.4. f : T C . , FourierS[f ] f Cesaro f f : f x T, (2.4.18) n(f)(x) f(x).pipi, f x T, Fourier S[f ] f Cesaro f : ,

(2.4.19) n(f) f.

pi 2.4.4 2.2.1.

2.4.5. f : T C f(k) = 0 k Z. f 0 T f(0) = 0.

2.4 Fourier 43

pi. pi pi

(2.4.20) sn(f)(0) =

nk=n

f(k)eik0 = 0

n. ,

(2.4.21) n(f)(0) =s0(f)(0) + s1(f)(0) + + sn1(f)(0)

n= 0

n. f 0, 2.4.4

(2.4.22) n(f)(0) f(0).

pi f(0) = 0. 2

Cesa`ro n(f) - pi, pi 2.4.4 pipi f : T C ( 1.1 pi pi pi, pi pipi).

2.4.6. f : T C . pi {Tn} pi f Tn 0.pi. pi 2.4.4

n(f) f.

Tn := n(f) . 2

2.4 Abel pi Poisson

k=0 ck Abel s C

0 r < 1

(2.4.23) A(r) =

k=0

ckrk

,

(2.4.24) limr1

A(r) = s.

pi A(r) Abel k=0 ck. pi

k=0 ck s Abel s. pi pi

44 Fourier

k=0 ck Cesa`ro s Abel

s. pi

(2.4.25)

k=0

(1)k(k + 1) = 1 2 + 3 4 + 5

pi Abel Cesa`ro . pi

(2.4.26) A(r) =

k=0

(1)k(k + 1)rk = 1(1 + r)2

0 r < 1, pi

(2.4.27) limr1

A(r) =1

4.

, Cesa`ro : pipi limn(sn/n) = 0.

pi pipi pipipi .

2.4.7 (pi Poisson). 0 r < 1 Pr : [pi, pi] C pi

(2.4.28) Pr() =

k=

r|k|eik.

pi Weierstrass pi - pi [pi, pi]. Pr r-pi Poisson. pi (2.4.28) pi ( )

(2.4.29) Pr(k) = r|k|, k Z.

pi pi Pr pi pi : pi

(2.4.30) Pr() =1 r2

1 2r cos + r2 .

pi = rei. ,

Pr() =

k=0

rk(ei)k +

1k=

rk(ei)k =k=0

(rei)k +

s=1

(rei)s

2.4 Fourier 45

=

k=0

k +

s=1

s =1

1 +

1

=1 + (1 )(1 )(1 ) =

1 ||2|1 |2 .

|| = r 1 = 1 rei = (1 r cos ) ir sin ,

(2.4.31) Pr() =1 r2

(1 r cos )2 + r2 sin2 =1 r2

1 2r cos + r2 .

pi {Pr}0r1 pi. - [0, 1), pi pipi- (). : {rn} [0, 1) rn 1, {Prn}n=1 pi. () pi () Pr pi pi. pi pi .

2.4.8. 0 r < 1

(2.4.32)1

2pi

pipi

Pr() d = 1,

0 < < pi

(2.4.33) limr1

|x|pi

Pr() d = 0.

pi. 0 r < 1. Pr() =k= r

|k|eik

[pi, pi],

(2.4.34)1

2pi

pipi

Pr() d =

k=

r|k|

2pi

pipi

eikd =r0

2pi

pipi

e0d = 1,

pi pipi e

ikd = 0 k 6= 0. 0 < < pi 1/2 r < 1. (2.4.35)1 2r cos + r2 = (1 r)2 + 2r(1 cos ) (1 r)2 + 2r(1 cos ) c = 1 cos > 0 || pi ( cos cos ). pi,

(2.4.36) 0 ||pi

Pr() d ||pi

1 r2c

d 2pic

(1 r2) 0

r 1. pi (2.4.33). 2

46 Fourier

2.4.9 (Abel f). f : T C . 0 r < 1 r-Abel f

(2.4.37) Ar(f)() =

k=

r|k|f(k)eik.

{|f(k)|} , Weierstrass T. Ar(f)() r-Abel Fourier S[f ] f .

(2.4.37), pi

Ar(f)() =

k=

r|k|f(k)eik

=

k=

r|k|(

1

2pi

pipi

f()eikd)eik

=1

2pi

pipi

f()

( k=

r|k|eik())d

=1

2pi

pipi

f()Pr( ) d= (f Pr)().

{Pr} pi, pi . 2.4.10. f : T C . , FourierS[f ] f Abel f f : f x T,

(2.4.38) Ar(f)(x) f(x).

pipi, f x T, Fourier S[f ] f Abel f : ,

(2.4.39) Ar(f) f.

2.5 Askseic

1. () {eikx : k Z} C- .

2.5 47

() pi 0 < 1 < 2 < < n. ei1x, ei2x, . . . , einx

C- . pi j ;

2. f : R C 2pi-pi . : a < b R, b

a

f(x) dx =

b+2pia+2pi

f(x) dx =

b2pia2pi

f(x) dx,

pipi

f(x+ a) dx =

pipi

f(x) dx =

pi+api+a

f(x) dx.

3. f : R C 2pi-pi , Riemann [pi, pi]. :

() f , f(k) = f(k) k Z S[f ] -.

() f pi, f(k) = f(k) k Z S[f ] .

() f(x+ pi) = f(x) x R f(k) = 0 pi k.() f pi pi f(k) = f(k) k Z. , pipi,pi f , .

4. f : R R 2pi-pi , Riemann [pi, pi]. a R

a(x) = f(x a). a f . a pi; Fourier a Fourier f .

5. f : R R 2pi-pi , Riemann [pi, pi]. m N

gm(x) = f(mx).

gm f . gm pi; Fourier gm Fourier f .

6. pi 2pi-pi f : R R pi [0, pi] pi

f(x) = x(pi x).

48 Fourier

pi f , pi Fourier f

f(x) =8

pi

k=0

sin[(2k + 1)x]

(2k + 1)3.

7. 0 < < pi. f : [pi, pi] R

f(x) =

{1 |x| |x| 0 |x| pi

pi f

f(x) =

2pi+ 2

k=1

1 cos kk2pi

cos kx.

8. 2pi-pi f : R R pi [pi, pi] pi

f(x) = |x|.

pi f , pi Fourier f f(0) = pi/2

f(k) =1 + (1)k

pik2, k 6= 0.

Fourier S[f ] f . x = 0

k=0

1

(2k + 1)2=pi2

8

k=1

1

k2=pi2

6.

9. [a, b] pi pi [pi, pi]. f(x) = [a,b](x) pi [pi, pi] pi f(x) = 1 x [a, b] f(x) = 0, pi 2pi-pi R. Fourier f

S[f ](x) =b a2pi

+k 6=0

eika eikb2piik

eikx.

S[f ] pi x R. x R pi S[f ](x) .

2.5 49

10. f : R C 2pi-pi, pi Cm (m- pi f (m) ). pi C(f) > 0

|f(k)| C(f)|k|m k Z \ {0}.

11. f, fn (n N) 2pi-pi, [pi, pi], pipi

limn

pipi|f(x) fn(x)| dx = 0.

fn(k) f(k) n,

pi k. , > 0 pi n0 N n n0 k Z,

|fn(k) f(k)| < .

12. f : R C 2pi-pi -. pi , pi x R pi pi

f(x) := limtx

f(t) f(x+) := limtx+

f(t).

Fourier S[f ] f Cesa`ro x: pi -,

limnn(f)(x) = limn(f Fn)(x) =

f(x) + f(x+)2

.

pi pipi pi 2.3.4.

13. f : R C 2pi-pi -. pi , pi x R pi pi

f(x) := limtx

f(t) f(x+) := limtx+

f(t).

Fourier S[f ] f Abel x: pi ,

limr1

Ar(f)(x) = limr1

(f Pr)(x) = f(x) + f(x+)

2.

pi pipi pi 2.3.4 pi

1

2pi

0pi

Pr(x) dx =1

2pi

pi0

Pr(x) dx.

50 Fourier

14. () {qk : k N} (0, 1). F : [0, 1] R

F (x) =

k=1

1

k2[0,+)(x qk)

Riemann qk (, pipi [0, 1]).

() {qk : k N} (0, 1). g(x) = sin 1x x 6= 0 g(0) = 0. G : [0, 1] R

G(x) =k=1

1

3kg(x qk)

Riemann , qk pi [0, 1].

15. M > 0 f, g : R C pi pi [M,M ]. f g : R C

(f g)(x) =

f(y)g(x y) dy.

() f g x R (f g)(x) = 0 |x| > 2M .() f g1 f1g1, pi

u1 := |u(x)| dx.

16. n N pi Dirichlet

Dn(x) =

nk=n

eikx =sin(n+ 12

)x

sin x2.

: pi c > 0

Ln :=1

2pi

pipi|Dn(x)| dx c log n

n N.

2.5 51

[pi: pi |Dn(x)| c sin((n+12 )x)

|x| , -, pi Ln c npi

pi

| sin t||t| dt

C pi C > 0 pi n.]

17. Ln 4pi2 log n C1

pi C1 > 0 pi n.

18. f : R C 2pi-pi -.

sn(f) C log(1 + n)f,pi C > 0 pi f pi n.

19. n N > 0. pi f : T C f = 1

1

pi

pipi|f(x) signDn(x)| dx <

n,

pi signu pi u ( sign 0 = 0). pi

sn(f) Ln .

20. n N

Qn(t) = n

(1 + cos t

2

)n,

pi n pi

1

2pi

pipi

Qn(t) dt = 1.

: f : R C 2pi-pi ,

f Qn f.

pi pi Weierstrass.

52 Fourier

21. n N

Gn(x) = Fn(x) sinnx,

pi Fn n- pi Fejer. : T Tn pi pi n,

T (x) = 2n(T Gn)(x)

x R. pi

|T (x)| 2nT x R. Bernstein, pi T nT T Tn.

Keflaio 3

Sgklish seirn Fourier

3.1 Qroi me eswterik ginmeno

V ( ) pi pi R pi + : V V V ( pi) : R V V( pipi) pi pi :

1. pi: x, y, z V x+y = y+x x+(y+z) =(x+ y) + z. pi, pi 0 V , x V , 0 +x = x. , x V pi () x V x+ (x) = 0.2. pipi: x, y V , R, (x) =()x, 1x = x, (x+ y) = x+ y (+ )x = x+ x.

pi , pi,

0x = 0, 0 = 0, x = (1)x. pi ( , , ). V ( ). , pipi : C V V . pi , V pi pi C.

pi pi pi R Rd d- pi (x1, x2, . . . , xd). pi ,

(3.1.1) (x1, x2, . . . , xd) + (y1, y2, . . . , yd) = (x1 + y1, x2 + y2, . . . , xd + yd).

pi pi pipi R:(3.1.2) (x1, x2, . . . , xd) = (x1, x2, . . . , xd).

54 Fourier

, Cd d- (z1, z2, . . . , zd) pi pi C pi ,

(3.1.3) (z1, z2, . . . , zd) + (w1, w2, . . . , wd) = (z1 + w1, z2 + w2, . . . , zd + wd)

pipi C

(3.1.4) (z1, z2, . . . , zd) = (z1, z2, . . . , zd).

3.1.1 ( pi pi R). V pipi R. V , : V V R(pi (x, y) V V pi pi x, y) :

(i) y, x = x, y x, y V .(ii) ax+ by, z = ax, z+ by, z x, y, z V a, b R.(iii) x, x 0 x V . , V , pi V :

(3.1.5) x = x, x1/2, x V.

x 0. , pipi, pi x = 0 pi , , x = 0, , .

( ) Rd

(3.1.6) x, y =di=1

xiyi = x1y1 + + xdyd,

pi x = (x1, . . . , xd) y = (y1, . . . , yd). pi

(3.1.7) x = x, x =x21 + + x2d,

pi pi pi x y Rd. 3.1.2 ( pi pi C). V pipi C. V , : V V C(pi (x, y) V V pi x, y) :

(i) y, x = x, y x, y V .

3.1 55

(ii) ax + by, z = ax, z + by, z x, y, z V a, b C. pi pi , , pi . ,

x, ay + bz = ax, y+ bx, z

x, y, z V a, b C.(iii) x, x 0 x V , pi pi pipi. pi V , pi,

(3.1.8) x = x, x1/2, x V.

, pipi, pi x = 0 pi , , x = 0, , .

Cd

(3.1.9) z, w =di=1

ziwi = z1w1 + + zdwd,

pi z = (z1, . . . , zd) w = (w1, . . . , wd) Cd. pi

(3.1.10) z = z, z =|z1|2 + + |zd|2.

3.1.3 (). V , , pi pi R C. x y V

(3.1.11) x, y = 0.

, x y. , pi pi pi

:

1. . x y

(3.1.12) x+ y2 = x2 + y2.

pi. pi ,

x+ y2 = x+ y, x+ y = x, x+ x, y+ y, x+ y, y= x, x+ y, y = x2 + y2,

56 Fourier

x, y = y, x = 0, x y. 2.2. CauchySchwarz. x, y V ,(3.1.13) |x, y| x y.

pi. pipi ( pi pi). pipi y = 0. x, y = 0 x V , CauchySchwarz ( pipi x V ). t R

0 x+ ty2 = x2 + ty, x+ tx, y+ ty2 = x2 + 2tRe(x, y), ty = |t| y = 0 y, x+ x, y = 2Re(x, y). Re(x, y) > 0, pi pi t , Re(x, y) < 0, pi pit +. pi, Re(x, y) = 0. , t R

0 x+ ity2 = x2 + ity, x itx, y+ ity2 = x2 + 2tIm(x, y), ty = |t| y = 0 y, x x, y = 2iIm(x, y). Im(x, y) > 0, - pi pi t , Im(x, y) < 0, pipi t +. pi, Im(x, y) = 0. pipi, pi x, y = 0.

pi y2 = y, y > 0. t = x, y/y, y pi y (x ty). ,

x ty, y = x, y ty, y = x, y x, y = 0,pi t. pi ty (x ty). x = (x ty) + ty , pi

(3.1.14) x2 = x ty2 + ty2 ty2 = |t|2y2.pi, |t| y x pi (3.1.15) |x, y| = |t| y2 x y.

3. . pi pi pi , pi (3.1.16) x+ y x+ y x, y V .pi. x, y V .

x+ y2 = x2 + 2Re(x, y) + y2 x2 + 2|x, y|+ y2 x2 + 2x y+ y2 = (x+ y)2,

3.1 57

pi pi pipi . 2

pi pi , pi Fourier, `2(Z) R Riemann f : T C. 3.1.4 ( `2(Z)). `2(Z) pi pi C (pi) pi

(3.1.17) a = (. . . , ak, . . . , a2, a1, a0, a1, a2, . . . , ak, . . .)

pi

(3.1.18)

k=

|ak|2

58 Fourier

pi pi pi

(3.1.20) a, b =

k=akbk.

(a, b) 7 a, b (, k= akbk (pi) a, b `2(Z)) pi , . pi `2(Z) . a = 0

k=

|ak|2 = 0,

ak = 0 k Z. pi, a = 0.pi, `2(Z) pi. {a(m)} `2(Z) pi

, > 0 pi m0 = m0() N a(m) a(s) < m, s m0, {a(m)} -, pi a `2(Z) limm a a(m) = 0. pi . Rd Cd pi pi pi pi pi pi . pi Hilbert.

3.1.5 ( R). R Riemann - f : T C, , Riemann f : [0, 2pi] C. R , pi (3.1.21) (f + g)() = f() + g()

pipi

(3.1.22) (f)() = f().

R : f, g R,

(3.1.23) f, g = 12pi

2pi0

f()g() d.

. pi

(3.1.24) f2 =(

1

2pi

2pi0

|f()|2d)1/2

.

R : f [0, 2pi] pipi pi , f 6= 0, f f2 = 0.

3.2 L2- Fourier 59

pi, pi pi, R pi. f : [0, 2pi] C

f() =

{ln(1/) 0 < 2pi0 = 0

f , R. {fn}, pi

fn() =

{ln(1/) 1n < 2pi0 0 1n

fn Riemann , pi {fn} R. , pi g R limn g fn = 0. pi .

3.2 L2-sgklish seirn Fourier

pi pi pi - 1.2.

3.2.1. f : T C . ,

(3.2.1) limn

1

2pi

2pi0

|f() sn(f)()|2d = 0.

pi. R f : T C, -

(3.2.2) f, g = 12pi

2pi0

f()g() d

pi 2 pi pi

(3.2.3) f22 = f, f =1

2pi

2pi0

|f()|2d.

,

(3.2.4) limn f sn(f)2 = 0.

k Z ek() = eik pi {ek}kZ . ,

ek, em ={

1 k = m0 k 6= m

60 Fourier

f : T C , ak = f(k) k Z. Fourier f f {ek}kZ:

(3.2.5) f, ek = 12pi

2pi0

f()eikd = ak.

,

(3.2.6) sn(f) =

nk=n

akek.

pi {ek} pi ak =f, ek pi f sn(f) = f

nk=n akek ek

|k| n. pi,

(3.2.7) f sn(f) n

k=nbkek

pi bk. pi pi pipi pi.

, pi bk = ak -pi

(3.2.8) f = (f sn(f)) + sn(f) =(f

nk=n

akek

)+

nk=n

akek,

pi

(3.2.9) f22 = f sn(f)22 +

nk=n

akek

2

2

.

, {ek}kZ ,

(3.2.10)

n

k=nakek

2

2

=

nk=n

|ak|2,

pi pi pi

(3.2.11) f22 = f sn(f)22 +n

k=n|ak|2.

pi pi pipi pi (3.2.7) .

3.2 L2- Fourier 61

3.2.2 ( pi). f : T C Fourier ak. pi ck, |k| n,

(3.2.12) f sn(f)2 f

nk=n

ckek

2

.

pipi, ck = ak |k| n.pi.

(3.2.13) f n

k=nckek = (f sn(f)) +

nk=n

bkek,

pi bk = ak ck, .

(3.2.14)

f n

k=nckek

2

2

= f sn(f)22 +n

k=n|ak ck|2 f sn(f)22,

nk=n |ak ck|2 = 0, ck = ak |k| n. 2

3.2.3. : pi Tn pi pi pi n, f R, pi Tn pi f sn(f). pi f pi Tn.

pi 3.2.1. pi pi pi pi f : T C.

pi , pipi, f . > 0 n0 N pi p n0 (3.2.15) f p = max

T|f() p()| < .

,

(3.2.16) f p2 =(

1

2pi

pipi|f() p()|2d

)1/2(

1

2pi

pipif p2d

)1/2< .

pi 3.2.2,

(3.2.17) f sn0(f)2 f p2 < ., n n0 sn0(f) Tn ( ). pi, pi pi 3.2.2 ( Tn ),(3.2.18) f sn(f)2 f sn0(f)2 < .

62 Fourier

pi f sn(f)2 0 pipi pi f . pipi, pi f pi , pi

g g f

(3.2.19)

2pi0

|f() g()| d < pi2

4(f + 1) .

,

f g22 =1

2pi

2pi0

|f() g()|2d

12pi

2pi0

|f() g()| (|f()|+ |g()|) d

12pi

2pi0

|f() g()| (f + g) d

2f2pi

2pi0

|f() g()| d

0

(3.4.7) |sn(f)(x)| M

n N x T. pi 3.4.3 pi

Cesa`ro S[f ], pi pi .

3.4.4. f : T C .

(3.4.8) supnn(f) f < +.

,

(3.4.9) |n(f)(x)| f n N x T.pi. n N x T

|n(f)(x)| = |(f Fn)(x)| = 12pi

pipi

f(x t)Fn(t) dt

12pi

pipi|f(x t)Fn(t)| dt

=1

2pi

pipi|f(x t)|Fn(t) dt

f 12pi

pipi

Fn(t) dt = f.

3.4 Fourier pi pi 69

pi pi pi Fn pi pi . 2

pi 3.4.3. pi pi, pi M > 0

(3.4.10) |kf(k)| M k Z. n N x T.

sn(f)(x) n+1(f)(x) =n

k=nf(k)eikx

nk=n

(1 |k|

n+ 1

)f(k)eikx

=

nk=n

|k|n+ 1

f(k)eikx.

pi

(3.4.11) |sn(f)(x) n+1(f)(x)| n

k=n

|kf(k)|n+ 1

(2n+ 1)Mn+ 1

2M.

, pi 3.4.4 pi

|sn(f)(x)| |sn(f)(x) n+1(f)(x)|+ |n+1(f)(x)| 2M + f. 2

3.4.3 Fourier f 3.4.2.

(3.4.12)k 6=0

eikx

k

Fourier |kf(k)| 1 k Z. pi, pi M > 0 , x T n N,

(3.4.13)

1|k|n

eikx

k

= |sn(f)(x)| M. :

3.4.5. n N pi

(3.4.14) fn(x) =

1|k|n

eikx

k.

pi M > 0 |fn(x)| M n x. 2

70 Fourier

(3.4.15)

1k=

eikx

k.

3.4.3 pi pi pi g : T C

(3.4.16) S[g](x) =

1k=

eikx

k.

, pi g, pi |kg(k)| 1 pi, pi pipi 3.4.5,

(3.4.17) supnsn(g) < +.

, {sn(g)(0)} pipi . ,

(3.4.18) |sn(g)(0)| =1

k=n

1

k

= 1 + 12 + + 1n c log n pi c > 0 pi n. , |sn(g)(0)| + - pi.

pi pi. , :

3.4.6. n N pi

(3.4.19) gn(x) =

1k=n

eikx

k.

pi c > 0 |gn(0)| c log n n N. 2pi 3.4.1. pi - pi {fn} {gn} .() n N

(3.4.20) pn(x) = ei2nxfn(x) =

1|k|n

ei(k+2n)x

k.

pn ( m N pi pn(m) 6= 0)pi [n, 3n]. pi, x,

(3.4.21) |pn(x)| = |ei2nxfn(x)| = |fn(x)| M,

3.4 Fourier pi pi 71

pi 3.4.5., n N

(3.4.22) qn(x) = ei2nxgn(x) =

1k=n

ei(k+2n)x

k.

qn pi [n, 2n]. pi,

(3.4.23) |qn(0)| = |gn(0)| c log n,pi 3.4.6. , pi

(3.4.24) s2n(pn) = qn.

() {ak}k=1 pi k=1 ak

. pi, pi {Nk} , pi pi :

(i) Nk+1 > 3Nk k 1.(ii) ak logNk + k .

pi, pi pi ak = 1k2 Nk = 32k , k = 1, 2, . . ..

() , f : T C :

(3.4.25) f(x) =

k=1

akpNk(x).

pi Weierstrass, -: rk(x) = akpNk(x),

(3.4.26) rk = akpNk Mak,

(3.4.27)

k=1

rk = Mk=1

ak < +.

rk ( pi), f .

() , m N,

(3.4.28) s3Nm(f)(x) =

mk=1

akpNk(x),

, pi (3.4.24),

(3.4.29) s2N1(f)(x) = a1qN1(x)

3.5 73

3. f : [0, 2pi] C

f() =

{ln(1/) 0 < 2pi0 = 0

f R. {fn},pi

fn() =

{ln(1/) 1n < 2pi0 0 1n

fn Riemann {fn} R, pi g R limn g fn2 = 0.

4. {ak}k=

ak =

{1k k 10 k 0

{ak}k= `2(Z) pi f R f(k) = ak k Z.

5. pi f : [pi, pi] R f(x) = |x| Parseval,

k=0

1

(2k + 1)4=pi4

96

k=1

1

k4=pi4

90.

pi 2pi-pi pi g : [pi, pi] R g(x) = x(pix) [0, pi] Parseval,

k=0

1

(2k + 1)6=

pi6

960

k=1

1

k6=

pi6

945.

6. : / Z, Fourier

f(x) =pi

sinpiei(pix)

[0, 2pi],

k=

eikx

k + .

74 Fourier

Parseval, pi

k=

1

(k + )2=

pi2

sin2(pi).

7. pi {fn} fn : [0, 2pi] R

limn

1

2pi

2pi0

|fn(x)|2dx = 0,

x [0, 2pi] {fn(x)} .

8.

k=2

sin kx

ln k

x R Fourier pi Riemann .

9. f : T C pi . Fourier f pi.

10. () f : T C k 6= 0.

f(k) = 12pi

pipi

f(x+ pi/k)eikxdx,

pi

f(k) =1

4pi

pipi

[f(x) f(x+ pi/k)]eikxdx.

() pi f pi Holder |f(x + h) f(x)| C|h| pi 0 < 1, pi C > 0 x, h. pi () pi M > 0

|k||f(k)| M k Z.() , 0 < < 1,

f(x) =

k=0

2kei2kx

3.5 75

pi Holder (),

f(k) =1

k k = 2s, s N.

11. () f, g : T C pi . pi 2pi0g(t) dt = 0. 2pi

0

f(t)g(t) dt

2 2pi0

|f(t)|2dt 2pi0

|g(t)|2dt.

() f : [a, b] C pi f(a) = f(b) = 0. b

a

|f(t)|2dt (b a)2

pi2

ba

|f (t)|2dt.

12. 0

sin t

tdt =

pi

2.

13. f : R C 2pi-pi, pi pi Lipshitz

|f(x) f(y)| K|x y|

x, y R, pi K > 0 .() t > 0 gt(x) = f(x+ t) f(x t).

1

2pi

2pi0

|gt(x)|2dx =

k=4| sin kt|2|f(k)|2

pi

k=| sin kt|2|f(k)|2 K2t2.

() p N. pi t = pi/2p+1,

2p1

76 Fourier

() 2p1 0

|f(k)| Mk

k Z \ {0}.15. {k} k 0. pi f : T C : pi k Z,

|f(k)| k.

16. pi Dirichlet pi

Dn(x) =|k|n

sign(x)eikx,

pi sign(x) pi x.

()

Dn(x) =cos(x/2) cos((n+ 1/2)x)

sin(x/2) pi

pi|Dn(x)| dx c log n.

() : f : T C , (f Dn)(0) C log n.

() , 0 < < 1, k=1

sin(kx)

k

x, Fourier pi .

17. > 1/2 f : R C 2pi-pi, pi pi Holder

|f(x) f(y)| K|x y| x, y R, pi K > 0 . Fourier f pi, .

Mroc II

Oloklrwma Lebesgue

Keflaio 4

Mtro Lebesgue

4.1 Eisagwg

Riemann pi : fn, f : [a, b] R. pi fn Riemann fn f , pi pi pi

(4.1.1)

ba

fn(x)dx ba

f(x)dx

( pi pi (fn) f ). , pi pi

pi f Riemann : , pi fn f . 4.1.1. Dirichlet f = Q : [0, 1] R.,

f(x) =

1, x Q [0, 1]0, x / Q [0, 1].

f Riemann . {q1, q2, . . . , qn, . . .} Q [0, 1], fn : [0, 1] R

fn(x) =

1, x {q1, . . . , qn}0, x / {q1, . . . , qn},

fn f [0, 1] fn Riemann , pipi pi .

80 Lebesgue

Riemann - pi Riemann pi: pi - pi .pipi, pi pi (-) pi Riemann (pi, Dirichlet Q).

pi , f : [a, b] R Rie-mann pi. , pi pi -pi Riemann: f , pi P = {a = x0 < x1 < < xn = b} [a, b] pi

(4.1.2) U(f, P ) L(f, P ) =n1k=0

(Mk mk)(xk+1 xk)

pipi . f pi pi .

Lebesgue pi pi pi pi . A R f : A R. pi, pi, f f(x) 0 x A, m = inf(f), M = sup(f). (4.1.3) {m = y0 < y1 < < yn = M} pi [m,M ], f pi pi

(4.1.4)

n1k=0

yk `(Bk)

pi `(Bk)

(4.1.5) Bk = {x A | yk f(x) < yk+1}. pi pi, pipi pipi pi ( Bk pi pipi pi pi ).

Lebesgue pi -, pi 1902. Lebesgue pi :

(i) pi Riemann .

(ii) pi .

(iii) pipi pi pi pi , pi pi.

4.2 Lebesgue 81

4.2 Exwterik mtro Lebesgue

pi A R, A R (A) ( +). :

(i) ([a, b]) = b a.(ii) (A+ x) = (A) x R.(iii) (An) pi R,

(4.2.1)

( n=1

An

)=

n=1

(An)

( pi).

pi , pi. pi pipi pi ,pi ( pi) pi pi pi

(4.2.2) (A B) = (A) + (B) A,B R A B = . pi : pi pi , pi pi R pi pi pi (i), (ii) (iii). . .

1. Lebesgue A R (A) 0 +, A.

I = (a, b) . I

(4.2.3) `(I) := b a. A R (In) pipi pi A n In, (In) A. (In) A,

n `(In) pi pi

A.

(4.2.4) (A) n

`(In)

A. , .

82 Lebesgue

4.2.1 ( Lebesgue). A R. A

(4.2.5) (A) = inf{

n

`(In) : (In) A

}.

4.2.2. () pi, `() = 0 , pi pi -. (In) A pi pipi pi () , pi pi pi pipi pi . , (In)n=1 ,

n=1 `(In) ,

(4.2.6) (A) = inf

{ n=1

`(In) | A n=1

In, In }.

() inf{+} = +. ,

(4.2.7) A n=1

In =n=1

`(In) = +,

(A) = +.() pipi , A R +. , pi R , In = (n, n), n = 1, 2, . . ..

2. Lebesgue pi pi Lebesgue.

4.2.3. A B, (A) (B).

pi. B n=1 In, A n=1 In. ,(4.2.8)

{n

`(In) : (In) A

}{

n

`(In) : (In) B

},

pi pi pi (A) (B). 2

4.2.4. A pipi pi , (A) =0.

4.2 Lebesgue 83

pi. A = {x1, x2, . . .}. > 0

(4.2.9) In =(xn

2n+1, xn +

2n+1

).

, A n In (4.2.10)

n

`(In) =n

2n .

> 0 , pi (A) = 0. 2

4.2.5. (A+ x) = (A) x R.pi. A n=1 In, A+ x n=1 Jn, pi Jn = In + x. `(I + x) = `(I) = b a I = (a, b). pi,

(4.2.11) (A+ x) n

`(Jn) =n

`(In).

infimum pi (In) A, pi

(4.2.12) (A+ x) (A).

pi pi. 2

4.2.6. ([a, b]) = b a.pi. > 0 [a, b] I := (a , b+ ). ,

(4.2.13) ([a, b]) `(I) = (b a) + 2.

pi, ([a, b]) b a. pipi (In) [a, b]

pi ,

(4.2.14) b a n=1

`(In).

1: [a, b] n=1 In. [a, b] pi, pi Heine-Borel pi pipi pi (In): pi k N

(4.2.15) [a, b] I1 I2 Ik.

84 Lebesgue

2: [a, b] (c1, d1) (ck, dk).

(4.2.16) b a b,

(4.2.17) b a < dn1 cn1 k

n=1

(dn cn).

dn1 b, dn1 (a, b], pi n2 dn1 (cn2 , dn2). dn2 b, dn2 (a, b], pi n3 dn2 (cn3 , dn3). , pi ns b < dns ( pipi (cn, dn):pi n cn < b < dn).

pi n1, . . . , ns cn1 < a, b < dns

(4.2.18) cn2 < dn1 < dn2 , cn3 < dn2 < dn3 , . . . , cns < dns1 < dns .

,

kn=1

(dn cn) (dns cns) + (dns1 cns1) + + (dn2 cn2) + (dn1 cn1)

(dns cns) + (cns cns1) + + (cn3 cn2) + (cn2 cn1)= dns cn1> b a.

pi 1 2 pipi

(4.2.19) b a 0, pi

((a, b)

)= b a. 2

4.3 85

4.2.9. ((a,+)) = +.

pi. N N (a,+) (a, a+N), (4.2.21)

((a,+)) a+N a = N.

, ((a+)) = +. 2

4.2.10 ( pipi ). pipi pi (An) pi R

(4.2.22) (

n

An

)n

(An).

pi. + pi .pi pi

n (An) < +. > 0 pi

(Js) nAn pi ,

s `(Js) 0,

(4.4.8)

n=1

`(In) + (X [a,+)) + (X (, a)).

4.4 Lebesgue 91

> 0. n N (4.4.9) I n = In (a,+) , I n = In (, a),

(4.4.10) I0 =(a

2, a+

2

).

pi I n, In , ( )

(4.4.11) `(In) = `(In) + `(I

n).

pi,

(4.4.12) X [a,+) I0 n=1

I n

(4.4.13) X (, a) n=1

I n .

,

(X [a,+)) + (X (, a)) `(I0) +n=1

`(I n) +n=1

`(I n)

= +

n=1

(`(I n) + `(I

n))

= +

n=1

`(In).

pi J = [a,+) .() J = (a,+),

(a,+) =n=1

[a+ 1/n,+)

pi 4.3.10 (), pi J M.() (, a) (, a] pi .() pi [a, b], [a, b), (a, b] (a, b) . pi,

(4.4.14) [a, b] = R \ ((, a) (b,+)) [a, b] pi (, a)(b,+).2

92 Lebesgue

4.4.5 (Borel -). - pi R pipi - Borel pi R (Borel -) B. ,

(4.4.15) B ={A P (R) | A A pi }.

pi Borel -, pi M - pi 4.4.4 pi Borel pi R :

4.4.6. B M. 2 4.4.7. pi R Borel, .

pi. pi R - ( pi ). B - pi , B pi , , . 2

4.4.8. () Borel - pi pi pi pi pi R. ( G-) Borel , ( F-) Borel , .

() M pi B Borel : pi pi Borel. pi pi pi Borel 0 (, ). pi pi .

() pi pi, pi pi Borel , :

4.4.9. A R. :(i) A .

(ii) > 0 pi G R A G (G \A) < .(iii) pi G- B A B

(B \A) = 0.

pi. (i) (ii). pi A , , (A) < +. > 0. pi (A) = (A), pi (In) A

n In

(4.4.16)n

(In) =n

`(In) < (A) + .

4.4 Lebesgue 93

G =n In. G , A G

(4.4.17) (A) (G) = (

n

In

)n

(In) < (A) + .

A G , G \A (4.4.18) (G) = (A (G \A)) = (A) + (G \A)pi 4.3.7. pi,

(4.4.19) (G \A) = (G \A) = (G) (A) < ,pi (4.4.17).

(A) = +. > 0. n N An = A(n, n). An , (An) < + A =

nAn. pipi pi

pipi, n N Gn An Gn (Gn \ An) < /2n. G =

nGn. , G ,

G =nGn

nAn = A

(4.4.20) G \A =(

n

Gn

)\(

n

An

)n

(Gn \An).

pi,

(4.4.21) (G \A) (

n

(Gn \An))n

(Gn \An) 0. ,

(4.6.2) AA := {x y | x A, y A}

A pi (t, t) pi t > 0.pi. pi pi 0 < (A) 0 pi G A (G) < (1 + )(A). pi G G =

k=1 Ik pipi .

Ak = A Ik. ,

(4.6.3) (G) =

k=1

`(Ik) (A) =k=1

(Ak).

pi (G) < (1 + )(A) pi : pi k N

(4.6.4) `(Ik) (1 + )(A Ik).

100 Lebesgue

= 1/3 pi pi I

(4.6.5) (A I) 3`(I)4

.

t = `(I)2 .

(4.6.6) (A I) (A I) (t, t).

, pi s (t, t) A I (A I) + s . , pi I (I + s), pi `(I) + |s|.pi

(4.6.7) 2(A I) = (A I) + ((A I) + s) `(I) + s < 3`(I)2

,

(AI) < 3`(I)4 , pi pi. pi AA (AI)(AI) (t, t).2

4.6.2. pi E R.pi. R :

(4.6.8) x y x y Q.

R

(4.6.9) Ex = {y R | y = x+ q pi q Q}.

X = {Xa | a A} -, pi pi E = {ya | a A} R pi pi ya pi Xa. , a 6= b A ya yb / Q.

{qn : n N} Q

(4.6.10) En := E + qn, n N.

En pi :

(i) n 6= m En Em = . , pi ya, yb E ya + qn =yb + qm, 0 6= ya yb = qm qn Q, pi pi pi pi E.

(ii) R =n=1En. , x R pi a A x Xa.

x = ya + q pi q Q. , pi n = n(x) N q = qn,, x = ya + qn En.

4.6 101

pi E . , En = E + qn n N (En) = (E). pi En pi pi , pi

(4.6.11) + = (R) =n=1

(En) =

n=1

(E).

pi, (E) > 0. pi Steinhaus, E E pi (t, t) pi t > 0. pi, E E pi pi pi 0: x 6= y E x y , pi pi E. pi E . 2

4.6.3. pi pi pi A R (A) > 0 pi. pi En pi (4.6.10)

(4.6.12) A =

n=1

(A En),

pi A En

(4.6.13) 0 < (A) =

n=1

(A En).

pi, pi n N (AEn) > 0 pi Steinhaus AEnA En, En En, pi (t, t) pi t > 0. pi.

4.6.4. pi, pi pi, pi pi E [0, 1], pi Steinhaus. [0, 1] :

(4.6.14) x y x y Q.

, , x y [1, 1]. [0, 1] -

(4.6.15) Ex = {y [0, 1] | y = x+ q pi q [1, 1] Q}.

X = {Xa | a A} -, pi pi E = {ya | a A} [0, 1] pi pi ya pi Xa. , a 6= b A ya yb / Q.

102 Lebesgue

{qn : n N} Q [1, 1]

(4.6.16) En := E + qn, n N. En pi :

(i) En [1, 2].(ii) n 6= m En Em = .(iii) [0, 1] n=1En. , x [0, 1] pi a A x Xa.

x = ya+q pi q Q[1, 1]. , pi n = n(x) N q = qn, , x = ya + qn En.

pi E . , En = E+qn n N (En) = (E). pi En pi pi , pi

(4.6.17) 1 = ([0, 1]) ( n=1

En

)=

n=1

(En) =

n=1

(E) 3,

pi pi 0 ( (E) = 0) +( (E) > 0). pi, E .

4.7 Askseic

1. () A R t R. (A) = (A+ t)

( pi ).

() pipi A , A+ t .

2. () A pi R. (A) < +.() A R . (A) > 0.

3. () A,B R (B) = 0, (A B) = (A).() A,B R (A4B) = 0, (A) = (B) ( A4B (A \B) (B \A) A B).

4. () A R t R. tA tA = {tx | x A}. (tA) = |t| (A).

4.7 103

() f : B R R Lipschitz C, |f(x) f(y)| C|x y| x, y B.

(f(A)) C(A)

A B.() A R (A) = 0. A = {x2 | x A} pi (A) = 0.

pi: pi pipi pi A [M,M ] pi M > 0.

5. E R 0 < (E) < + 0 < < 1. pi I

(E I) > `(I).pi: pi , > 0, Ik E

k=1 Ik

k=1 `(Ik) <

(E) + .

6. A > 0 (A I) `(I) . (Ac) = 0.

7. A,B R

dist(A,B) = inf{|x y| : x A, y B} > 0.

(A B) = (A) + (B).

8. A R 0 < (A) < +.() f : R R f(x) = (A (, x]) .() pi F F A (F ) = (A)/2.

9. A R. :(i) A .

(ii) > 0 pi F R F A (A \ F ) < .(iii) pi F- A

(A \ ) = 0.

10. (An) pi R.

lim supAn = {x R | x An pi n}

104 Lebesgue

lim inf An = {x R | pi n0(x) N x An n n0(x)}.

lim supAn =

n=1

k=n

Ak lim inf An =n=1

k=n

Ak.

11. (An) pi R. :() lim supAn lim inf An .

() (lim inf An) lim inf (An) (n=1An) < +

lim sup(An) (lim supAn).

() n=1 (An) < +, (lim supAn) = 0.

12. Borel : Q, R\Q,[0, 1] \Q, C + 1, 2C, pi C Cantor.

13. A pi X pi X A pi pi : X A \X . X -.

14. 1/4 Cantor.

15. pi pi :

(i) A R (A) = 0, A pipi pi .

(ii) A R A , (A) > 0.

(iii) A,B R, (A) < +, B A, B (B) = (A), A .

(iv) A [a, b]. , (A) = 0 pi A pi (In)

n=1 `(In) < + x A

pi pi pi In.

(v) A R (A) = 0 pi A .

4.7 105

16. A [a, b] (A) > 0. pi x, y A x y R \Q.

17. () A (A4B) = 0, B (B) =(A).

() A,B ,

(A B) + (A B) = (A) + (B).

() A,B , A B (A) = (B) < +, (B \A) = 0.() pi A,B A B (A) = (B), (B \A) > 0.18. A = {x [0, 2pi] | sinx < 1/n}. pi (n=1En) limn (En).19. f : R R.

A = {x R | f x}

Borel.pi: pi

A =

nk=1

n=1

{x R | diam[f(x 1/n, x+ 1/n)] < 1

k

}.

20. fn : R R .

B = {x R | limn fn(x) = +}

Borel.

21. f : R R . Borel B R f1(B) Borel.

pi: A = {A R | f1(A) Borel}.

22. x [0, 1) (x1, x2, x3, . . .) pi x( x pi pi pi ). pi :

(i) A1 = {x [0, 1) | x1 6= 5}.(ii) A2 = {x [0, 1) | x1 6= 5 x2 6= 5}.

106 Lebesgue

(iii) A3 = {x [0, 1) | n = 1, 2, . . . , xn 6= 5}.

23. (0, 1). pi Cantor n- /3n pi pi pi (n 1)- . C pi Cantor. :

() C pi .() C pi.() C (C) = 1 > 0.

24. {qn}n=1 Q [0, 1]. > 0

A() =

n=1

(qn

2n, qn +

2n

).

, A = j=1A(1/j).() (A()) 2.() < 12 [0, 1] \A() .() A [0, 1] (A) = 0.() Q [0, 1] A A pi.

25. {An} Lebesgue pi [0, 1]

lim supn

(An) = 1.

: 0 < < 1 pi pi {Akn} {An}

(n=1Akn) > .

26. E Lebesgue pi R (E) < . {An} Lebesgue pi E c > 0 (An) c n N.() (lim supAn) > 0.

() pi {kn} n=1

Akn 6= .

4.7 107

27. E Lebesgue pi R (E) > 1. pix 6= y E x y Z.

28. E pi R. Lebesgue E

(E) = sup{(F ) : F E,F }.() (E) (E).() pi (E) < . E Lebesgue (E) = (E).() (E) = () pi .

29. A M x R

(A, x) = limt0+

(A (x t, x+ t))2t

,

pi. (A, x) pi A x.

() (Q, x) = 0 (R \Q, x) = 1 x R.() 0 < < 1. A R (A, 0) = .

30. pi {En}n=1 pi R

(n=1En) 0 (J \ E) > 0.

33. A pi R (A) > 0. , n N, Api pi n.

Keflaio 5

Metrsimec sunartseic

5.1 Metrsimec sunartseic

pi pi Lebesgue pi pi pi A R pi [,+] pi . pi 4.1, f : A R pi

Af pi

(5.1.1)

n1k=0

yk({x A | yk f(x) < yk+1}),

pi {y0 < y1 < < yn} pi f . pi pipi

(5.1.2) Bk = {x A : yk f(x) < yk+1} . pi pi pi f : A [,+] pi (5.1.3) {x A : a f(x) < b} . .

5.1.1 (Lebesgue ). A Lebesgue pi- R f : A R. f Lebesgue , pi, a R (5.1.4) {x A : f(x) > a} = f1((a,+)) .

110

pi (a,+) 5.1.1 pi pi pipi .

5.1.2. A pi R f : A R. :

(i) f .

(ii) a R {x A : f(x) a} = f1([a,+)) .(iii) a R {x A : f(x) < a} = f1((, a)) .(iv) a R {x A : f(x) a} = f1((, a]) .pi. (i) (ii)

(5.1.5) {x A : f(x) a} =n=1

{x A : f(x) > a 1

n

}.

(ii) (iii) (5.1.6) {x A : f(x) < a} = A \ {x A : f(x) a}.(iii) (iv)

(5.1.7) {x A : f(x) a} =n=1

{x A : f(x) < a+ 1

n

}.

(iv) (i) (5.1.8) {x A : f(x) > a} = A \ {x A : f(x) a}. , , (i)-(iv) pipi pi pi-pi . 2

5.1.3. A pi R f : A R . , f . {x A : f(x) = a}, a R.pi. pi pi 5.1.2. pi, J = [a, b]

(5.1.9) f1(J) = {x A : a f(x) b} = {x A : f(x) a} {x A : f(x) b} . , a R,

(5.1.10) {x A : f(x) = a} =n=1

{x A : a 1

n< f(x) < a+

1

n

} . 2

5.1 111

5.1.4 (Borel ). A Borel R f : A R. f Borel , a R,

(5.1.11) {x A : f(x) > a} = f1((a,+))

Borel. 5.1.2 5.1.3 Borel (pi pi ).

5.1.5. () A pi R f : A R . , f . , a R {x A : f(x) > a} A, AU pi pi U R. pi, .() A : R R A . ,

(5.1.12) {x R : A(x) > a} =

R, a < 0

A, 0 a < 1

, a 1,, pipi. , Dirichlet Q .

() A pi R. f : A R . a R {x A : f(x) > a} A , .

pi. pi f . a R. T = {x A :f(x) > a} t := inf T .(i) t = T = A. , x A pi y T y < x. y A f(y) > a, f pi y < x pi f(x) f(y) > a, x T . , pipi T = A .(ii) t R pipi: t T : t A f(t) > a. , T = A [t,+).

, x A x t f(x) f(t) > a, x T . , x T x A x t t T .

t / T : T = A (t,+). , x A x > t ( infimum) pi y T t < y < x f(x) f(y) > a, x T . , x T x A x > t t T T .

pipi T = {x A : f(x) > a} A . 2

112

5.1.6 (pi ). A pi- R f, g : A R . ,

(i) f + g .

(ii) R, f .(iii) fg .

(iv) f(x) 6= 0 x A, 1/f .(v) max{f, g}, min{f, g} |f | .

pi. (i) a R. f(x) + g(x) < a, f(x) < a g(x). , pi q

(5.1.13) f(x) < q < a g(x).pi

{x A : f(x) + g(x) < a} =qQ{x A : f(x) < q g(x) < a q}

=qQ

({x A : f(x) < q} {x A : g(x) < a q}) ,

.

(ii) a R. > 0, (5.1.14) {x A : f(x) > a} = {x A : f(x) > a/}, . < 0,

(5.1.15) {x A : f(x) > a} = {x A : f(x) < a/}, . pipi, f ( = 0, pi).

(iii) pi f2 . a < 0,

(5.1.16) {x A : f2(x) > a} = A, a 0, (5.1.17) {x A : f(x)2 > a} = {x A : f(x) > a} {x A : f(x) < a}. pipi, {x : f2(x) > a} . , fg ,

(5.1.18) fg =(f + g)2 (f g)2

4.

5.1 113

(iv) a = 0, {x A : 1/f(x) > 0} = {x A : f(x) > 0}. a > 0,

(5.1.19) {x A : 1/f(x) > a} = {x A : 0 < f(x) < 1/a}.

, a < 0

(5.1.20) {x A : 1/f(x) > a} = {x A : f(x) > 0} {x A : f(x) < 1/a}.

pipi, {x A : 1/f(x) > a} .(v) a R

(5.1.21) {x A : max{f, g}(x) > a} = {x A : f(x) > a} {x A : g(x) > a}

(5.1.22) {x A : min{f, g}(x) < a} = {x A : f(x) < a} {x A : g(x) < a}.

, max{f, g} min{f, g} . , |f | = max{f,f} . 2

R

pi pi R = [,] = R {}.pi R R < x < + x R pi R Rpi (pi) [, a), [, a], (a,+], [a,+] (pia R) [,+], [,+), (,+].

pi + [, a) (a,+] -. pi R pi pi R. pipi pi (+) (+), 0 (), ()/0, ()/(). f : A R, piA pi R, pi .

pi pipi - R.

5.1.7. A Lebesgue pi R f : A R. f Lebesgue , pi , a R

(5.1.23) {x A : f(x) > a} = f1((a,+))

.

pi pi (pi) ( pi pi pi

114

pi pi pi pipi). , f :A R ,

(5.1.24) {x A : f(x) = +} =n=1

{x A : f(x) > n}

(5.1.25) {x A : f(x) = } =n=1

{x A : f(x) < n}

.

pi

A pi R. P (x) pi A Z x A pi P (x) . pi , pipi .

5.1.8. A pi R f, g : A R - f(x) = g(x) pi A. f , g .

pi. B = {x A : f(x) = g(x)} Z = {x A : f(x) 6= g(x)}. (Z) = 0, Z , B = A \ Z .

a R. ,{x A : g(x) > a} = {x B : g(x) > a} {x Z : g(x) > a}

= {x B : f(x) > a} {x Z : g(x) > a}=

(B {x A : f(x) > a}) {x Z : g(x) > a}.

B {x A : f(x) > a} B {x A :f(x) > a} f . {x Z : g(x) > a} pi 0. , {x A : g(x) > a} .

a R , g . 2

pi .

5.1.9. A pi R (fn) - fn : A [,+]. ,

5.1 115

(i) supnfn inf

nfn .

(ii) (fn) , f : A [,+] f(x) :=limn fn(x) .

pi. (i) a R

(5.1.26) {x A : supnfn(x) > a} =

n=1

{x A : fn(x) > a}

,

(5.1.27) {x A : infnfn(x) < a} =

n=1

{x A : fn(x) < a}

. , supnfn inf

nfn .

(ii) , (an) pi

(5.1.28) lim supn

an = infmN

(supkm

ak

) lim inf

nan = sup

mN

(infkm

ak

).

bm = supkm ak lim supn

an, m =

infkm ak lim infn

an.

pipi ,

(5.1.29) gm(x) = supkm

fk(x) hm(x) = infkm

fk(x),

, pi (i), gm, hm ,

(5.1.30) f(x) = infmgm(x) = sup

mhm(x).

, pi pi (i), f . 2

: pi (ii) : (fn) pipi - fn : A [,+], lim supn fn lim infn fn pi pi

(5.1.31) lim supn

fn(x) = infmN

(supkm

fk(x)

) lim inf

nfn = sup

mN

(infkm

fk(x)

),

. 2

pi pi pi 0 .

116

5.1.10. A pi R f : A [,+]. fn : A [,+] fn(x) f(x) pi A, f .

pi. B = {x A : fn(x) f(x)}. Z = A \ B, (Z) = 0 B .

a R. , {x B : f(x) > a} pi 5.1.9 fn f B, {x Z : f(x) > a} pi Z (pi 0). ,

(5.1.32) {x A : f(x) > a} = {x B : f(x) > a} {x Z : f(x) > a}

. a R , g . 2

5.2 H sunrthsh CantorLebesgue

Cn pi pi C Cantor. n N fn : [0, 1] [0, 1] . Jn1 , . . . , Jn2n1 pi [0, 1] \Cn, fn(0) = 0,fn(1) = 1, fn(x) = k2n x J

nk , pi pi

pi Cn pi . pi, C1 = [0, 1/3] [2/3, 1]. f1 1/2

(1/3, 2/3), [0, 1/3] f(0) = 0 f(1/3) = 1/2, [2/3, 1] f(2/3) = 1/2 f(1) = 1. , [0, 1] \ C2 pi pi : (1/9, 2/9) f2 1/4, (1/3, 2/3) f2 1/2, (7/9, 8/9) f2 3/4, pi C2 pi , pi f2(0) = 0 f2(1) = 1.

5.2.1 ( Cantor-Lebesgue). {fn}n=1 - f : [0, 1] [0, 1]. f pi [0, 1]. C f (f(C)) = 1.

pi. pi {fn} :(i) fn , fn(0) = 0 fn(1) = 1.

(ii) Jnk pi pi pi n- C, fn Jnk ,

fn fn+1 fn+2

Jnk .

5.2 CantorLebesgue 117

(iii)

fn+1 fn 12n, n = 1, 2, 3, . . . .

pi {fn} C[0, 1]: m > n

(5.2.1) fm fn m1k=n

fk+1 fk m1k=n

1

2k 1

2n1 0

m,n. C[0, 1] pi pi , pi f : [0, 1] R fn f .

, fn f [0, 1]. fn fn(0) = 0 fn(1) = 1, pi f , f(0) = 0 f(1) = 1. , f pi [0, 1].

, f(C) = [0, 1]. , pi {fn} pi f J pi C, pi J pi C. f pi [0, 1], y [0, 1] f(x) pi x C. pi f(C) = [0, 1] (f(C)) = 1. 2

. ([0, 1] \ C) = 1 f (x) = 0 x / C. , x / C x pi J pi f .pi, f pi x f (x) = 0. , f pi , pi pi f pi [0, 1] pi [0, 1].

pi CantorLebesgue, pi pi -pi pi Borel. .

5.2.2. A Borel R f : A R ., Borel B R, f1(B) = {x A : f(x) B} Borel.pi.

A = {B R : f1(B) Borel}.

B pi R, f1(B) A, f . A Borel, pi f1(B) Borel( ).

A - pi . A - pi , pi Borel- B pi A. pi A pi f1(B) Borel B R Borel. 2

118

5.2.3. pi Lebesgue pi Cantor, pi Borel.

pi. g : [0, 1] [0, 2] g(x) = f(x) + x, pi f CantorLebesgue. g , pi ( g1).

g(C) (g(C)) = 1. , g(C) pi C, . pi, g pi J [0, 1] \C {f(J)}+ J , . (g([0, 1] \ C)) = (J) = 1. pi (g(C)) = 1.

g(C) , pi pi M g(C). , K = g1(M) Lebesgue pi C pi . , K Borel: , pi 5.2.2 M =(g1)1(K) Borel Borel . pi, M Lebesgue . 2

5.3 Prosggish metrsimwn sunartsewn ap aplc sunart-

seic

5.3.1 (pi ). : R R pi pipi ( pi pi , pi pipi pi pi pi)., pi

(5.3.1) =

ni=1

iAi

pi n N, pi pi 1, . . . , n pi A1, . . . , An.

pi pi Ai , pi i . pi pi pi pipi pi pi ( pi pi 0). , (5.3.1) pi

(5.3.2)

{iI

i : 6= I {1, . . . , n}} {0}

( ). pi {t1, . . . , tm}

(5.3.3) Ei = { = ti} = {x R : (x) = ti},

5.3 pi pi 119

Ei , R,

(5.3.4) =

mi=1

tiEi .

pi (5.3.4) (pi ) - pi .

pi pi pi -.

5.3.2. A f : A [0,] . pi (n) pi 0 1 2 f

(5.3.5) n(x) f(x)

x A. pi A pi f .

pi. n = 1, 2, . . . Cn = {x A : f(x) 2n}

(5.3.6) Bn,k =

{x A : k

2n f(x) < k + 1

2n

}, k = 0, 1, . . . , 22n 1.

[0, 2n] 22n 2n f . f , Cn Bn,k ., pi n :

(5.3.7) n = 2nCn +

22n1k=0

k

2nBn,k .

n pi :

(i) 0 n f pi A.(ii) 0 f n 2n A \ Cn = {x A : f(x) < 2n}.(iii) n(x) = 2n f(x) =.pi (ii) (iii) pi n(x) f(x) x A. , f(x) =

(5.3.8) n(x) = 2n = f(x).

120

f(x) < , pi n0 N f(x) < 2n0 2n n n0. ,0 f(x) n(x) < 2n n n0, n(x) f(x). n f {x A : f(x) M}, M > 0.

(n) . pi

Bn,k = {x A : k/2n f(x) < (k + 1)/2n}=

{x A : 2k

2n+1 f(x) < 2k + 1

2n+1

}{x A : 2k + 1

2n+1 f(x) < 2k + 2

2n+1

}= Bn+1,2k Bn+1,2k+1.

x Bn+1,2k, n(x) = k/2n = (2k)/2n+1 = n+1(x), x Bn+1,2k+1, n(x) = k/2n < (2k + 1)/2n+1 = n+1(x). , x Cn n(x) =2n n+1(x) ( : Cn Bn+1,2n+1 , Bn+1,2n+1+1, . . . , Bn+1,22(n+1)1 Cn+1).

pipi n(x) n+1(x), n n+1. 2 f : A [,+] . 5.3.2

f+ f , pi .

5.3.3. f : A [,+] . pi (n) pi n : A R

(5.3.9) 0 |1| |2| |f |

n(x) f(x) x A. pi A pi f .

pi. pi (n) (n) pi n(x) f+(x) n(x) f(x) x A. , n = n n, n(x) f+(x) f(x) = f(x) x A.

f+ f pi B A pi f .pi, n f+ n f B, pi pi pi n f B.

pi : C = {f < 0} n 0 C n 0 A \ C n N. pi,

(5.3.10) |n| = |n n| = max{n, n} max{f+, f} = |f |.

pi pi (n) (n) , pi pi

(5.3.11) |n| = max{n, n} max{n+1, n+1} = |n+1|.

pi (5.3.10) (5.3.11) pi (5.3.9). 2

5.4 Littlewood 121

pi pi 5.3.2,pi f Cn,Bn,k , pi pi n . n f .

5.3.2 5.1.9 pi .

5.3.4. A f : A [,+]. f pi . 2

5.4 Oi treic {arqc tou Littlewood}

Littlewood pi pi pi :

(i) pipi .

(ii) .

(iii) pi , -.

, pipi pi (, pi ). pi pipi pipi . pi .

5.4.1 ( ). A pi R (A) 0 pi I1, . . . , Ik E = I1 Ik pi (E4A) < .

pi. > 0. pi () , pi (In)

A n=1

In

(5.4.1)

n=1

(In) < (A) +

2.

(In) , pi k N

(5.4.2)

n=k+1

(In) 0.pi F pi A G A V E G (G \ F ) < /2. pi f F1 = V (A \G). V,A \G

124

A f 1 V f 0 A \ G. pi f |F1 ( , pi, ). pi, pi F F1 (F1 \ F ) < /2. ,(A \ F ) < f |F .

pi pi A pi pi

(5.4.12) =

mi=1

iEi ,

pi i R Ei pi A ( pi). f : A R . pi 5.3.2 pi

(n) (5.4.12) n f A. n N pi An A (A \ An) < 2n+3 n|An . pi, pi Egorov pi B A (A \ B) < /4 n f B.

(5.4.13) U = B ( n=1

An

).

,

(5.4.14) (A \ U) (A \B) +n=1

(A \An) < 4

+

n=1

2n+3=

2.

pi, n|U ( U An n) n|U f |U ( U B). pi f |U .

U , pi , . pi pi F U (U \ F) < 2 . ,

(5.4.15) (A \ F) = (A \ U) + (U \ F) < 2

+

2= ,

pi f |U f |F . 2

5.5 Askseic

1. A f : E [,+] . , a R, fa : A [,+]

fa(x) =

f(x), f(x) aa, f(x) > a

5.5 125

.

2. f : (a, b) R pi, f .pi: f .

3. () A R (A) = 0, f : A [,+] .

() A,B (B) = 0 f : A B [,+] pi pi f |A A . f .

() A R f : A R pi A, f .

4. () pi f f2 .

() A R f : A R. f2 {x A : f(x) > 0} , f .

5. A R fn : A [,+], n N, .

L = {x A | (fn(x))n=1 }

.

6. A pi R f : A [,+] : q Q, {x A : f(x) > q} . f .

7. f : R R . B Borel, f1(B) = {x R : f(x) B} .pi: {E R | f1(E) } - pi .

8. () g : R R h : R R Borel , h g : R R Borel .() pi CantorLebesgue g :R R Lebesgue h : R R h g : R R Lebesgue .

126

9. f : [a, b] R .() f pi F- F-.

() f pi A [a, b] (A) = 0 (f(A)) = 0.10. A pi R (A) < f : A R Lebesgue . f : R R

f (t) = ({x A : f(x) > t}).() f pi . pi ;

() fk, f : A R Lebesgue fk f , fk f .11. E pi (0, 1). f : R R f(x) = xE(x). f , R \ {0} {x : f(x) = } .

pi g : R R R {x : g(x) = } ;12. ; f (a, b ) 0 < < b a, f (a, b).

13. A pi R, f : A R g : R R . g f : A R .14. (n) pi n : R R f : RR. n f , f .

15. () fn : R R Lebesgue R. :n=1 ({x : fn(x) > }) n}) 0 ({x : |fn(x)| > n}) < 1/2n.17. f : R R . f t-pi s-pi pi t, s > 0 t/s / Q, f pi .

Keflaio 6

Oloklrwma Lebesgue

pi Lebesgue. pi pi :

(i) A , AA = (A), pi A

A.

(ii) : f, g ( ) t, s R,

(tf + sg) = t

f + s

g.

(iii) : f f 0, f 0. pi , : f, g ( ) f g, f g.

(iv) . Riemann Lebesgue , pipi.

Lebesgue . , pi :

(i) 6.1 pi pi , - pipi . pi pi (i) (ii) pipi .

(ii) 6.2 f f 0. pi

128 Lebesgue

pi pi f

pi supremum pi pi

pi, , f .

(iii) 6.3 : f = f+ f, . pi pi pi .

pi, pi Lebesgue. - Lebesgue ( ).

Lebesgue pi . , Riemann - f : [a, b] R Lebesgue. 6.4 .

6.1 Oloklrwma Lebesgue gia aplc metrsimec sunart-seic

6.1.1. : R R pi . Lebesgue

{ 6= 0} = {x R : (x) 6= 0}

pipi . pi

(6.1.1) =

ni=0

iAi ,

pi 0 = 0 A0 = { = 0}, i , Ai , (Ai) < + i 6= 0 (, (A0) =). pi

(6.1.2)

=

ni=1

i(Ai).

0 = 0, pi

(6.1.3)

=

ni=0

i(Ai) =R

({ = }).

6.1 Lebesgue pi 129

6.1.2. pi =ni=1 biEi

pi Ei . ,

(6.1.4)

=

ni=1

bi(Ei).

pi. R J = {i n : bi = }. ,

(6.1.5) { = } =iJ

Ei

(6.1.6) ({ = }) =iJ

bi(Ei).

,

(6.1.7)

=

R

({ = }) =R

iJ

bi(Ei) =

iJbi(E

Ανάλυση Fourier Και Ολοκλήρωμα Lebesgue

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