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). pi- pipi 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 ( pi- pipi 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 pi- pi 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). pi- pi 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)
72 Fourier
, m 2,
(3.4.30) s2Nm(f)(x) =
m1k=1
akpNk(x) + amqNm(x).
, x = 0,
|s2Nm(f)(0)| =m1k=1
akpNk(0) + amqNm(0)
am|qNm(0)|
m1k=1
ak|pNk(0)|
cam logNm Mm1k=1
ak
cam logNm Mk=1
ak.
pi am Nm, am logNm + m . k=1 ak < +, pi
(3.4.31) |s2Nm(f)(0)| +.
, lim sup |sn(f)(0)| = +. , S[f ](0) pi. 2
3.5 Askseic
1. `2(Z) pi.
2. R f : T C
f2 =(
1
2pi
2pi0
|f(x)|2dx)1/2
.
() , f R f2 = 0, f(x) = 0 x pi f .
() , f pi 0 pi , f2 = 0.
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}
({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
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