leb224w.dvi

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2 i Lebesgue Fourier 7 5 Lebesgue Walter. F. Riesz and B. Sz.-Nagy, Functional Analysis, Dover Publ. Inc., New York (99) ( 49 ) 2. ( 8 ) 3. A.2 Fourier Laplace (957 ) 4. (98 ) 5. G. G. Walter, Wavelets and Other Orthogonal Systems With Applications, CRC Press, Inc. (994) 6. I. Daubechies, Ten Lectures on Wavelets, SIAM (992) 7. K. (993 ) 8. (995 ) 9. (982 ). A. M. Robert, Nonstandard Analysis, Dover Publ. Inc., Mineola, New York (23)

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4 Lebesgue 7 2. Lebesgue Lebesgue Fubini Banach Hilbert ( ) Banach Hilbert L Fourier iii

5 iv 5.2 Fourier Fourier L Fourier L 2 Fourier S(R) Poisson Riesz B(X, Y) Riesz Haar Franklin Battle-Lemarie Shannon Meyer

6 v Daubechies Parseval Fourier

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8 . (set) (element) a A a A a A a A a A a A, B A B A B (subset) A B A A B A B A A B \ A A B B \ A B A B \ A A B A B N A, A 2,..., A N N n= A n N n= A n B = (B A) (B \ A), A B = (A \ B) (B A) (B \ A) Z N,, 2, 3,... Z + R C a, b a < b a < x < x (a, b) a x b, a < x b, a x < b x [a, b], (a, b], [a, b) (interval) (enumerable), 2, 3,..... (,) [ ] (,) 2, 3, 2 3, 4,(2 4 ), 3 4, 5, 2 5, 3 5, 4 5,...,2,3,...

9 2. (enumerable set) (finite set) (infinite set) (empty set) n a, a 2, a 3,..., a n {a, a 2, a 3,..., a n } {a k k N, k n} a, a 2, a 3,... {a, a 2, a 3,...} {a k k N} φ. A, A 2, A 3,... n= A n [ ] A n a n,,a n,2,a n,3,... n=a n a n,m n + m a,,a,2,a 2,,a,3,a 2,2,a 3,, (, ) [ ] n [n,n + ) A n.. A,A,A,A 2,A 2, [ ] A x x M M A (upper bound) (supremum, least upper bound) A sup A sup x A x A max max x A x. a = supa ǫ > a ǫ < x a x A [ ] a ǫ A a x A x < M supa M supa = M.2. (.2) [ ] M x > M x a [a, ) a A x x M M A (lower bound) (infimum, greatest lower bound) A inf A inf x A x A mina x A x A (bounded).2 A inf A supa B A inf A inf B supb supa [ ] x A inf A x supa

10 a, a 2,... {a n } n= {a n} n N a, a 2,..., a N {a n } N n= {a n} {a n } {a n } {a n } sup{a n } sup n N a n max{a n } max n N a n inf{a n } inf n N a n min{a n } min n N a n {a n } n= a ǫ > N N n > N a n a < ǫ (.) {a n } a (convergence) a n a lim n a n = a lim n a n a = a {a n } {a n } n= n N a n+ a n ( ) n N a n+ a n ( ).2 {a n } n= sup{a n} {a n } n= inf{a n} [ ] a = sup{a n }. ǫ > N N a ǫ < a N a n > N a N a n a a a n a a N < ǫ a n a.2. { n } n= {a n } n= {n k} k= n < n 2 < n 3 < {a nk } k= {a n} n=.3 {a n } n= a {a n k } k= a [ ] ǫ > N N n > N (.) n k k k N a nk a < ǫ.4 {a n }, {c n } a {b n } n N a n b n c n {b n } a [ ] ǫ > N N n N n > N a n a < ǫ/3, c n a < ǫ/3 b n a = b n a n +a n a b n a n + a n a < c n a n + ǫ/3 = c n a + a a n + ǫ/3 < ǫ

11 4..5 {a n }, {b n } a, b {b n a n } b a n N a n b n a b [ ] ǫ > N N n > N a n a < ǫ/2, b n b < ǫ/2 b n a n (b a) = b n b (a n a) b n b + a n a < ǫ b n a n b a b n n b a = b b n (a a n )+b n a n b n b a n a > ǫ ǫ > b a.6 {a n } n= {a n k } k= [ ] {a n } n= d, b n = c = 2 (b + d ) [b,c ] [c,d ] {a n } [b 2,d 2 ] a n n n n 2 {n k } {b n } n= d b {d n } n= b d d n b n.5 {d n b n } d b d n b n = (d b )/2 n d b = k N b k a nk d k.4 a nk b (non-standard analysis) N,R (illimited number) n (illimited integer) (.) (infinitesimal number) a b a b N,R (standard number) n N a n a, lim n a n = a n a n a, a n a.7 n a n a ǫ > a n a ǫ n [ ] (i) a n n (ii).6 b a ǫ b a n b n [ ] a n a (.) n a n a.7 ǫ > a n a ǫ n N N n > N (.) {a n } n= {a n } n= lim sup{a n} lim sup n a n {a n } n= lim inf{a n} lim inf n a n u l {a n } n= () u, l {a n } n= (2) {a n } a l a u

12 {a n } n= u l k N u k = sup n k a n, l k = inf n k a n {u k } k= u {l k } k= l l k l k+ l u u k+ u k, k N (.2) a {a n } n= l a u {a n } n= a u = l = a [ ] k N {a n } n= {a n} n=k u k = sup n k a n l k = inf n k a n. 2 l k l k+ u k+ u k {l k } l {u k } u u k l k.5 u l u, l {a n } (), (2) u {a nk } k= ǫ k > {ǫ k } k= k ǫ k ǫ k = 2 k n = k. n k u nk + ǫ k < a n u nk +, n n k + n n k+.3.5 {u nk + ǫ k }, {u nk +} u.4 {a nk } k= u l {a n } n= {a n k } k= {l n}, {u n } l nk a nk u nk {l nk } k= {u n k } k=.3 l u {a n k } k= a.5 l a u {a n } n= ǫ > N N n N, m N, n > N m > N a n a m < ǫ (.3) Cauchy (Cauchy sequence).3 {a n } n= Cauchy [ ] {a n } a ǫ > N N n > N a n a < ǫ/2 n > N m > N a n a m = (a n a) (a m a) a n a + a m a < ǫ.9 {a n } n= Cauchy a R N N (.3) n > N a n a < 2ǫ.3 [ ] Cauchy {a n } n= ǫ = N N n > N a n a N+ < max{max{a n } N n=,a N+ +} {a n } n= min{min{a n} N n=,a N+ } {a n } n=

13 6. [.6 ] {a n } n=.6 {a nk } k= a K N k > K a nk a < ǫ n > N, k > K, n k > N a n a a n a nk + a nk a < 2ǫ [.8 ] {a n } n=. 8 u k, l k, u, l (.3) u N+ l N+ (.3) u N+ l N+ ǫ < 2ǫ (.2) u l < 2ǫ u = l l n a n u n.4 a n u = l {a n } n= n s n = n k= a k (series) k= a k n {s n } n= k= a k k= a k k= a k k= a k a k a k k= a k k= a k k= a k (absolute convergence) [ ] k= a k k= a k n s n t n {t n } n=.2 t = sup{t n} ǫ > N N t t N < ǫ n > m N n n s n s m = a k a k = t n t m t t m t t N < ǫ k=m+ k=m+ {s n } n= Cauchy.4 [ ] k= a k s n s n l= a k(l) k(),k(2),k(3),...,2,3,... n s n s n = n l= a k(l) n N N = max l n k(l) s n s N s {s n} s s s n N N N {,2,...,n} {k(),k(2),...,k(n )} s n s N s s s s s s = s l= a k(l) s [ ] k= a k k N a + k a k a + k = max{a k,}, a k = max{ a k,} a k = a + k a k, a k = a + k + a k a + k a k, a k a k k= a k k= a+ k, k= a k.5 a k = a + k a k (.4) k= k= k=

14 .3. 7 l= a k(l) a k(l) = a + k(l) l= l= l= a k(l) (.4).3 x,(x, y), (x, y, z) n n (x (), x (2),..., x (n) ) n R n x = (x (), x (2),..., x (n) ) y = (y (), y (2),..., y (n) ) ( n k= x (k) y (k) 2 ) /2 x y.3. n R n 2 a = (a (),a (2),...,a (n) ) b = (b (),b (2),...,b (n) ) a (k) < b (k), k =,2,...,n a (k) < x (k) < b (k), k =,2,...,n x = (x (),x (2),...,x (n) n ) k= (b(k) a (k) ) n R n R n E A B E a E δ B(a, δ) x a < δ x E a δ A {x k } k= x E lim x k x = (.5) k x (converge) lim k x k = x x A A A A (closed set) E A (complementary set) A c = E \ A A A c (open set) A a δ > B(a, δ) A [ ( reductio ad absurdum) ] δ ǫ > B(a,ǫ) A {ǫ n } n N ǫ n >, ǫ n x n A c, x n a < ǫ n {x n } x n a A c a A c a A

15 8. δ > a δ a (neighbourhood) A A (closure) Ā A A A = Ā.3.2 [,] A Ā = [,] A B = A B A (dense) x A B {x k }.3.3 [,] [,] [ ] x [,] k N x k = x x [,] x k = [ k x]/ k [t] t A A A (compact).3.4 [a,b ] [ ] {x k } k=.6 (.2) A M A x A x < M (bounded) A A n x f(x) A A c = R n \ A x f(x) A (support) (compact support) x R f(x) b R x > b (x < b) f(x) = ( ) n e x e χ e (x) e (characteristic function).3.5 (a,b) χ (x) {, a < x < b χ (a,b) (x) = (.6), otherwise n R n A ǫ O, O 2, O 3,... A k= O k (null set, set of zero measure) (almost everywhere) a.e.

16 e χ e (x) χ e (x) =, a.e..3.7 [ ] A,A 2,... ǫ > A k ǫ/2 k k= ǫ/2k = ǫ.3.8 [,] Heine-Borel n R n [ ] [a,b ] [a,b ] [a,c ] [c,b ] [a,b ].4 x R n f(x) M > x f(x) < M (.7) (bounded) x R n f(x) x R n, ǫ > δ > x x < δ x f(x) f(x ) < ǫ (.8) (continuous) x R n f(x) ǫ > δ > x x < δ x, x (.8) (uniformly continuous).5 R n [ ] R [a,b] f(x) ǫ > [a,b] x δ(ǫ,x) x [a,b] (x δ(ǫ,x),x + δ(ǫ,x)) f(x ) f(x) < ǫ [a,b] x [a,b] Heine-Borel (.3)

17 . [a,b] K K x, x 2,..., x K I, I 2,..., I K K δ(ǫ,x k ) δ = δ(ǫ) x x [a,b] x x < δ I k I k I k+ I k f(x) f(x ) f(x) f(x k ) + f(x k ) f(x ) < 2ǫ x I k x I k+ I k, I k+ x x x I k x x I k+ f(x) f(x ) f(x) f(x ) + f(x ) f(x ) < 4ǫ [ ] x [a,b] x I k K f(x k ) M f(x) f(x) f(x k ) + f(x k ) < ǫ + f(x k ) M + ǫ f(x) x R n ξ x f(ξ) f(x) [ ] {ξ k } k= k ξ k x f(ξ k ) = sup {y y x ξk x } f(y) f(x) f(x) x (.8) k f(ξ k ) f(x) f(ξ k ).7 ǫ > f(ξ k ) > ǫ k N N k > N f(ξ k ) < ǫ δ = ξ N x x x < δ x (.8) {f n (x)} n= x a n = f n (x) {a n } n= {f n (x)} n= f(x) = lim n f n (x) f(x) {f n (x)} n= ǫ > N N n > N R n I x f n (x) f(x) < ǫ (.9) I f(x) (uniform convergence) n sup x I f n (x) f(x).4. n N x R f n (x) = x n I I = (,) f n (x) f(x) = sup x I f n (x) f(x) = I I 2 I 2 = (,.5) sup x I2 f n (x) f(x) =.5 n I 2 k= f k (x) s n = n k= f k (x) {s n (x)} n= {s n (x)} n= Weierstrass n= f n (x) f n (x) n= a n a n f n (x) < a n

18 .4..6 {f n (x)} n= f(x) [ ] f n (x) f(x) ǫ > N N n > N x (.9) f n (x) ǫ > δ > x y < δ f n (x) f n (y) < ǫ f(x) f(y) f(x) f n (x) + f n (x) f n (y) + f n (y) f(y) < 3ǫ f(x) x R f(x) x f(x + h) f(x) lim h h (.) d dx f(x) f (x) f(x) x f (x) f(x) x f (x) x f(x) f(x) n n f (n) (x) = d dx f(n ) (x) f = f () (x), f (x) = f (2) (x) (mean-value theorem) x R f(x) [a, b] (a, b) ξ (a, b) f(b) f(a) b a = f (ξ) (.) x R f(x), g(x) [a, b] (a, b) ξ (a, b) [f(b) f(a)]g (ξ) = [g(b) g(a)]f (ξ) (.2) [(.2) ] h(x) = [f(b) f(a)][g(x) g(a)] [g(b) g(a)][f(x) f(a)] h(a) = h(b) = x [a,b] h(x) = x h (x) = h(x) > h(x) < x (a,b) h(x) x ξ h(x) h (ξ) = [(.) ] (.2) g(x) = x (.)

19 2. Taylor x R f(x) n a R x ξ (a, x) ξ (x, a) ξ n f(x) = f(a) + k= f (k) (a) k! (x a) k + f(n) (ξ) (x a) n (.3) n! [ ] F(x) = f(x) f(a) n f (k) (a) k= k! (x a) k, G(x) = (x a) n F(a) = F (a) = = F (n ) (a) =, G(a) = G (a) = = G (n ) (a) = (.2) F(x) F(x) F(a) = G(x) G(x) G(a) = F (ξ ) G (ξ ) = F (ξ ) F (a) G (ξ ) G (a) = = F (n ) (ξ n ) G (n ) (ξ n ) = F (n ) (ξ n ) F (n ) (a) G (n ) (ξ n ) G (n ) (a) = F (n) (ξ n ) G (n) (ξ n ) = f(n) (ξ n ) n! a < x a < ξ n < ξ n < < ξ < x a > x a > ξ n > ξ n > > ξ > x.5 R f(x) x < ξ x R, ξ R f(x) f(ξ) (.4) (non-decreasing function) (non-increasing function) (monotonic function) [a, b] f(x) a, a, a 2,..., a n, a n a = a < a < a 2 <... < a n < a n = b n f(a k ) f(a k ) k= (.5) T(a, b) f(x) (a, b) (total variation) (function of bounded variation).7 [ ] f (x) f 2 (x) [a,b] f(x) = f f 2 (x) (.5) f(a k ) f(a k ) = f (a k ) f 2 (a k ) f (a k ) + f 2 (a k ) f (a k ) f (a k ) + f 2 (a k ) f 2 (a k )

20 .5. 3 T(a,b) f (b) f (a) + f 2 (b) f 2 (a) [ ] x [a,b] T(a,x) T(x) ξ x < ξ b f(ξ) f(x) T(ξ) T(x) T(x) f(x) T(ξ) f(ξ), T(x) + f(x) T(ξ) + f(ξ) N(x) = [T(x) f(x) + f(a)], 2 P(x) = [T(x) + f(x) f(a)] 2 N(x) P(x) T(x) = + f(x) = f(a) + P(x) N(x). [a,b] f(x) x = y f(y+) = lim h + f(y + h), f(y ) = lim h + f(y h) [ ] f(x) f(y+) < h < b y {h k } k= f(y + h k} k= f(a) f(y+).8 [a, b] f(x) [ ] f(x) x = y f(x). f(x) f(y+), f(y ) f(y) = f(y+) f(y ) k y k k f(y k ) k f(y j ) f(b ) f(a+) j= f(y) [f(b ) f(a+)]/ f(y). g(x) [a,b] (a,b) x min{g(x+), g(x )} g(x) max{g(x+), g(x )} G(x) max{g(x+),g(x )}, x (a,b) G(x) = g(a+), x = a g(b ), x = b (.6) E x (a,b) g(ξ) > G(x) ξ (x,b] E E (a k,b k ) g(a k +) G(b k ) (.7)

21 4. [ ] x E ξ > x G(x ) < g(ξ) (.6) g(x +) < g(ξ), g(x ) < g(ξ) x x E x E (a k,b k ) E x (a k,b k ) G(x) < G(b k ) x a k + (.6) g(a k +) G(a k +) G(b k ) a k a a k (a,b) E g(a k ) G(b k ) [ x (a k,b k ) G(x) < G(b k ) ] G(x) ǫ > G(x) [a k + ǫ,b k ] x b k G(x ) G(b k ) x E ξ > b k g(ξ) > G(x ) G(b k ) ξ b k E b k E.2. ξ > x ξ < x (.7) G(a k ) g(b k ) Lebesgue [a, b] f(x) (a, b) [ ] f(x) x (a,b) [f(x + h) f(x+)]/h h > h Λ r,λ r [f(x + h) f(x )]/h h < h Λ l,λ l Λ r,λ r, Λ l,λ l x a.e. Λ r < Λ r λ l f( x) Λ l < Λ l λ r λ l Λ l, λ r Λ r a.e. Λ r λ l Λ l λ r Λ r < a.e. Λ r = λ l = Λ l = λ r [a.e. Λ r < ] C > Λ r > C x (a,b) E x E ξ (x,b) f(ξ) f(x+) ξ x > C g(x) = f(x) Cx g(ξ) > g(x+) = G(x) E. k (a k,b k ). f(a k +) Ca k f(b k +) Cb k b k = b b k + = b C(b k a k ) f(b k +) f(a k +) k C k (b k a k ) k [f(b k +) f(a k +)] f(b ) f(a+) (.8) C E [a.e. Λ r λ l ] c,c c < C λ l < c f(x) x (a,b) E x E ξ (a,x) ξ f(ξ) f(x ) ξ x < c

22 .5. 5 g(x) = f(x) cx g(ξ) > g(x ) = G(x) g(x) S = (a,b). 2 S = k (a k,b k ) E S. 2 G(a k ) = f(a k +) ca k g(b k ) = f(b k ) cb k f(b k ) f(a k +) c(b k a k ) (.9) Λ r > C x (a k,b k ) E 2,k g(x) = f(x) Cx Λ r <. S 2,k = l (a kl,b kl ) E 2,k S 2,k (.8) C l (b kl a kl ) f(b k ) f(a k +) (.2) Λ r > C x S E 2 = k E 2,k S 2 = k S 2,k S, S, S 2 Σ, Σ, Σ 2 (.9) (.2) CΣ 2 k [f(b k ) f(a k +)] cσ cσ S = (a,b) S 2 = k k S 2,kl S 2,kl = (a kl,b kl ) S 4,kl S 4 = k l S 4,kl n =,2,... CΣ 2n cσ 2n cσ 2n 2 Σ 2n ( c C )n (b a) Λ r > C, λ l < c f(x) x Σ 2n λ l < Λ r f(x) x λ l < c < C < Λ r c, C c, λ l < c < C < Λ r λ l < Λ r x Fubini n N f n (x) [a, b] n= f n (x) f(x) f n(x) = f (x), a.e. (.2) n= [ ] s n (x) = n k= f k(x) x > x s n (x ) s n (x) = n [f k (x ) f k (x)] f(x ) f(x) (.22) k= f(x) a.e. (.22) s n(x) f (x) (.23)

23 6. (.22) f(x) s n (x) f(x ) s n (x ) (.24) k N n k f(b) s nk (b) < 2 k, f(a) s nk (a) > 2 k (.25) g(x) = [f(x) s nk (x)] k= (.25) (.24) g(x) K N g(x) K k= [f(x) s nk (x)] K [f (x) s n k (x)] g (x) k= (.23) K k f (x) = lim k s n k (x) = f n(x), n= a.e..6 {a n } n N n N a n n N p n = n k= ( + a k ) {p n } n N p p = ( + a k ) (.26) k= n (infinte product) u k = log( + a k ) p n = e k= u k k= u k p = e k= log(+a k) (.27) k= u k k= ( + a k ).9 k= ( + a k ) k= a k [ ] x < 3 x +x < 2 a < 3 log(+a) a = a ( +x )dx = a x +x dx 2 a, log( + a) = [log( + a) a] + a a 3 a 2 log( + a) 2

24 2 Lebesgue 2. Lebesgue 2. Lebesgue R n (step function) φ(x) K {a k } K k= a k A k Kk= a k A k φ(x) dx A {φ k (x)} k= [ ] φ (x) M B A ǫ k φ k (x) {φ k (x)} ǫ E k B φ k (x) < ǫ E k k E k+ E k lim k E k B \ E B {E k } k= Heine-Borel (.3) {E k} {E k } K k= B E K E φ K (x) dx < (M + A)ǫ φk (x) dx ǫ {φ k (x)} k= {φ k (x)} k= 7

25 8 2. LEBESGUE f(x) {φ k (x)} k= φ k(x) φk (x) dx f(x) dx f(x) dx = lim k φ k (x) dx (2.) f(x) {ψ l (x)} l= [ ] ψ l (x) k ψ l (x) φ k (x) ψ l (x) f(x), a.e. sup{ψ l (x) φ k (x),} [ψ l (x) φ k (x)] + k [ψ l (x) φ k (x)] +, a.e. ψ l (x) dx φ k (x) dx [ψ l (x) φ k (x)] + dx A (2.) k ψ l (x) dx f(x) dx (2.2) { ψ l (x) dx} l ψ l (x) dx f(x) dx lim l φ ψ (2.2) { ψ k (x)dx} f(x) dx lim ψ k (x) dx k ψ l (x) dx = f(x) dx lim l 2.. f(x) f(x) =, a.e. (2.) k N φ k (x) = f(x) dx =

26 2.. LEBESGUE 9 f (x) f 2 (x) {φ k (x)} k= {ψ k (x)} k= f (x) f 2 (x) [f (x) + f 2 (x)] dx = f (x) dx + f 2 (x) dx (2.3) [(2.3) ] [φ k (x) + ψ k (x)] dx = φ k (x) dx + ψ k (x) dx (2.3) (2.) f (x) f 2 (x) f (x) f 2 (x) f(x) = f (x) f 2 (x) f(x) dx = f (x) dx f 2 (x) dx (2.4) f(x) g (x) g 2 (x) f(x) = g (x) g 2 (x) f(x) = f (x) f 2 (x) = g (x) g 2 (x) f (x) + g 2 (x) = g (x) + f 2 (x) (2.3) f (x) dx + g 2 (x) dx = g (x) dx + f 2 (x) dx f (x) dx f 2 (x) dx = g (x) dx g 2 (x) dx (2.4) (2.), (2.4) Lebesgue Lebesgue (integrable) f(x) f (x) f 2 (x) f(x) = f (x) f 2 (x) L x R n L L (R n )

27 2 2. LEBESGUE 2.2 Lebesgue f(x) g(x) λ, µ λf(x)+µg(x) [λf(x) + µg(x)] dx = λ f(x) dx + µ g(x) dx (2.5) [ ] f(x) g(x) λf(x) + µg(x) [ ] 2. f (x) f 2 (x) {φ k (x)} k= {ψ k(x)} k= ζ k (x) = sup{φ k (x),ψ k (x)}, f (x) = sup{f (x),f 2 (x)} f (x) {ζ k(x)} k= f (x) [ ] {ζ k (x)} a.e. ζ k = φ k φ k+ ζ k+ (x) ζ k (x) = ψ k (x) ψ k+ (x) ζ k+ (x) {ζ k (x)} ζ k (x) f (x) f (x) > f 2 (x) x φ k (x) = ζ k (x) f (x) φ k (x) ψ k (x) = ζ k (x) f 2 (x) < f (x) φ k f (x) = f (x) ζ k (x) f (x) f 2(x) > f (x) x ζ k (x) f 2 (x) = f (x) {ζ k (x)} f (x) η (x) = inf{φ (x),ψ (x)} φ k η φ k (x) φ (x), ψ k (x) η (x) ψ k (x) ψ (x) ζ k (x) = sup{φ k (x) η (x),ψ k (x) η (x)} + η (x) φ k (x) η (x) + ψ k (x) η (x) + η (x) f (x) + f 2 (x) η (x) {ζ k (x)} f(x) f(x) f(x) dx [ ] f(x) {φ k (x)} k= {ψ k(x)} k= f (x) f 2 (x) f(x) = f (x) f 2 (x) ζ k = sup{φ k (x),ψ k (x)} f f 2 2. ζ k (x) ψ k (x) f (x) f 2 (x) f(x) dx f(x) f + = sup{f(x), } f (x) = sup{ f(x), } f(x) f(x) dx f(x) dx (2.6) [ ] f(x) {φ k (x)} k= {ψ k(x)} k= f (x) f 2 (x) f(x) = f (x) f 2 (x) f(x) f + f + (x) = sup{f(x),} = sup{f (x) f 2 (x),} = sup{f (x) f 2 (x),f 2 (x) f 2 (x)} = sup{f (x),f 2 (x)} f 2 (x) 2. sup{f (x),f 2 (x)} f + (x) f +

28 2.2. LEBESGUE 2 f(x) f f (x) = sup{ f(x),} = sup{f 2 (x) f (x),} = sup{f (x),f 2 (x)} f (x) f (x) f 2 (x) f (x) f (x) f(x) = f + (x) + f (x) f(x) dx = f + (x) dx f (x) dx f + (x) dx + f (x) dx = f(x) dx f(x) g(x) sup{f(x), g(x)} inf{f(x), g(x)} [ ] sup{f(x),g(x)} = [f(x) g(x)] + + g(x) inf{f(x),g(x)} = [f(x) g(x)] + g(x) 2. f(x) f(x) Riemann Lebesgue [ ] f(x) f + = 2 [f(x) + f(x) ] f = 2 [ f(x) f(x)] Riemann f+ (x) Riemann f (x) f(x) = f + (x) f Riemann Lebesgue 2.2. f(x) R e R \ e f(x) = Lebesgue Riemann.5 (.5) B {φ k (x)} k= φ k(x) {φ k (x)} [ ] A {φ k (x)} E ψ k (x) ψ k (x) = φ k (x) φ (x) A = A φ (x) dx ǫ ψ k (x) > A /ǫ Σ ǫ,k ψ k (x) A Σ ǫ,k ǫ ψ k (x) ψ k+ (x) Σ ǫ,k Σ ǫ,k+ k= Σ ǫ,k ǫ E k= Σ ǫ,k ǫ ǫ E {φ k (x)}

29 22 2. LEBESGUE Levi {f k (x)} k= f k(x) {f k (x)} k= f(x) { f k dx} k= f(x) dx A [ ] f k (x) {φ kl (x)} l= φ kl (x) f k (x) f l (x), k l (2.7) φ l (x) φ l (x) = supφ kl (x) k l {φ l (x)}. 2 φ l (x) φ kl (x) k l (2.7) φ kl (x) φ l (x) f l (x), k l (2.8) φ kl (x) dx φ l (x) dx f l (x) dx A, k l (2.9) fl (x) dx A B {φ l (x)} l= f(x) (2.) φl (x) dx f(x) dx l (2.8) (2.9) f k (x) f(x), a.e., f k (x) dx f(x) dx k {f k (x)}, { f k (x) dx} lim f k(x) f(x), a.e., lim f k (x) dx f(x) dx k k (2.8) (2.9) l k f k (x) a.e. f(x) fk (x) dx f(x) dx [ ] f k (x) g k (x) h k (x) f k (x) = g k (x) h k (x) k N f k (x) = f k+ (x) f k (x) f n (x) n f n (x) = f (x) + f k (x) (2.) k= ψ k (x) g k (x) h k g k (x) = g k+ (x) + h k (x) ψ k (x), h k (x) = h k+ (x) + g k (x) ψ k (x)

30 2.2. LEBESGUE 23 f k (x) = f k+ (x) f k (x) = g k (x) h k (x) (2.) g k (x) h k (x) h k+ (x)+g k {ψ kl (x)} l= h k(x) ψ kl (x) ψ k (x) l h k (x) dx ψ kl (x) dx + ψ k (x) dx < 2 k (2.2) l L ψ k (x) ψ k (x) = ψ kl (x) h k (x) (2.2) l = L h k (x) dx < 2 k n k= h k (x) dx < (2.3) {f n (x)} f k (2.) g k h k (x) h k g k (x) f n (x) A (2.), (2.) (2.3) n k= g k (x) dx A f (x) dx + (2.4) (2.3) (2.4) k= h k(x) k= g k(x) h k (x) dx = h k (x) dx, k= f(x) k= k= f(x) = f (x) + g k (x) h k (x) k= g k (x) dx = g k (x) dx (2.) (2.) f n (x) f(x) lim f n (x) dx = f(x) dx n k= k= Levi k= h k (x) k= hk (x) dx [ ] h k (x) h + k = sup{h k(x),} h k = inf{h k(x),} { n k= h+ k (x)} n= { n k= h k (x)} n= Levi A = k= hk (x) dx

31 24 2. LEBESGUE Lebesgue ( ) {f k (x)} k= f(x) g(x) f k (x) g(x), a.e., k =, 2,... (2.5) f(x) lim k f k (x) dx = f(x) dx (2.6) f(x) g(x), f(x) dx a.e. (2.7) g(x) dx (2.8) 2.2 {f k (x)} k= f(x) u(x) k N f k (x) u(x) k N u k (x) = sup{f k (x),f k+ (x),...} (2.9) u k (x) {u k (x)} k= a.e. f(x) [ ] sup{f (x),f 2 (x)} = f (x)+[f 2 (x) f (x)] + m = 3,4, sup{f (x),f 2 (x),...,f m (x)} m N u(x) Levi u (x) = sup{f (x),f 2 (x),...} = sup n f n (x) k N (2.9) u k (x) f k (x) a.e. a.e..8 (.2) {u k (x)} k= a.e. {f k (x)} f(x) 2.3 {f k (x)} k= f(x) l(x) k N f k (x) l(x) k N l k (x) = inf{f k (x),f k+ (x),...}, (2.2) l k (x) {l k (x)} k= a.e. f(x) [ ] 2.2 f k (x) f k (x) f(x) f(x) u(x) l(x) u k (x) l k (x) l k (x) (2.9) (2.2) {l k (x)} k= a.e. f(x) [ ] 2.2, 2.3 u(x) = g(x), l(x) = g(x) l k (x), u k (x) (2.2), (2.9) l k (x) f k (x) u k (x), l k (x) dx f k (x) dx u k (x) dx (2.2)

32 2.2. LEBESGUE 25 (2.5) l k (x) dx g(x) dx, u k (x) dx g(x) dx (2.22) {l k (x)}, { u k (x)} Levi f(x), f(x) lim l k (x) dx = f(x) dx, lim u k (x) dx = f(x) dx k k (2.2) (2.6) (2.7) (2.5) f(x) f(x) (2.8) (2.7) Lebesgue Lebesgue (2.5) f(x) f(x) g(x), a.e. (2.23) [ ] Lebesgue (2.22) 2.2, 2.3 (2.2), (2.9) {l k (x)}, {u k (x)} a.e. f(x).8 (.2) l k (x) f(x) u k (x), a.e. k N, (2.23) l k (x) f(x) g(x), u k (x) f(x) g(x) (2.22) Fatou {f k (x)} f(x) { f k (x)dx} k= f(x) f(x) dx lim inf k f k (x) dx (2.24) [ ] 2.3 l(x) = (2.2) l k (x) {l k (x)} k= a.e. f(x) (2.2) l l k (x) f k+l (x) l k (x) dx f k+l (x) dx. 8 (.2) { f l (x) dx} l= l k Levi 2.2 x R f(x) f(x) Riemann Lebesgue

33 26 2. LEBESGUE [ ] f(x) f + = 2 [f(x) + f(x) ] f = [ f(x) f(x)] Riemann 2 b a f(x) dx {Y k } k= Y > Y k {f k (x)} k= f k(x) = min{f(x),y k } f k (x) Riemann Levi f(x) fk dx f(x) dx a {X k } k= X > a X k {f k (x)} { f(x), x Xk f k (x) = (2.25), x > X k f k (x) Riemann f(x) = f + f (x) f + (x) f (x) f(x) = f + (x) f (x) 2.3 (measurable function) f(x) {φ n (x)} n= ǫ > N N n > N f(x)dx φ n (x)dx f(x) φ n (x) dx < ǫ (2.26) [ ] ( 2.) f(x) {φ n } n= {φ 2n } n= f (x) f 2 (x) f(x) = f (x) f 2 (x) φ n φ n (x) = φ n (x) φ 2n (x) (2.26) I I = [f (x) φ n(x)] [f 2 (x) φ 2n(x)] dx [f (x) φ n (x)] dx + [f 2 (x) φ 2n (x)] dx N n > N ǫ 2.3. x R f(x) = x f(x) g(x) x f(x) g(x) f(x) g(x) 2.3 f(x) g(x) f(x) g(x) (2.27) f(x)

34 [ ] f(x) Lebesgue ( 2.2) 2.4 f(x) f(x) f(x) g(x) + g(x) f(x) g(x) f(x)g(x) sup{f(x), g(x)}, inf{f(x), g(x)} f(x) g(x) = /f(x), a.e. {f n (x)} f(x) [ f(x),..., inf{f(x),g(x)} ] f(x) inf{f(x),g(x)} g(x) a.e. {φ n (x)} {ψ n (x)} { φ n (x) }, {min{φ n (x),ψ n (x)}} a.e. f(x), inf{f(x),g(x)} [a.e. g(x) = /f(x) ] f(x) a.e. {φ n (x)} φ n (x) = ψ n (x) = φ n (x) ψ n (x) = /φ n (x) f(x) = E a.e. {ψ n (x)} /f(x) [ {f n (x)} a.e. f(x) ] h(x) h(x) = (, ) h(x) = +x 2 g n (x) = h(x)f n(x) h(x)f(x), g(x) = h(x) + f n (x) h(x) + f(x) g n (x) g n (x) < h(x) g n (x) n g n g(x), g(x) < h(x) Lebesgue g(x) f(x) = h(x)g(x) h(x) g(x) f(x) g(x) f(x) R R L (R) g(x) f(x)g(x) f(x)g(x) g(x) (a) f(x) =, (b) f(x) = sin x, (c) f(x) = sinx x (measurable set) e (measure) m(e) e χ e (x) m(e) = χ e (x) dx m(e) = (2.28)

35 28 2. LEBESGUE R (a,b) b a ( ) [ ] χ(x) =, a.e χ(x) dx = f(x) > c x e c [ ] f c (x) f c (x) = min{f(x),c} f c+h (x) f c (x) χ c (x) = lim h h h> 2.4 e c e c e R n χ e (x) R n x f(x) f(x)χ e (x) e f(x) dx f(x) dx = f(x)χ e (x) dx (2.29) e e (a, b) e f(x) dx b a f(x) dx R n ( ) (locally integrable) R n e e R n f(x) R g(x) (a,b) χ (x) e f(x) R n R n e L (e) (a, b) L (a, b) 2.5 f(x) f(x) f(x) = f(x) dx = [ ] f(x) =, a.e f(x) dx = [ : ] Levi ( 2.2) Levi h k (x) = f(x) dx = k= h k(x) = k= h k(x) a.e. f(x) =, a.e.

36 2.4. FUBINI 29 [ : 2 ] f(x) > c x e c χ c (x) m(e c ) f(x) =, a.e. χ (x) =, a.e m(e ) = m(e ) > ǫ > ǫ m(e ǫ ) > f(x) dx = f(x) dx f(x) dx ǫ dx = ǫ m(e ǫ ) > e e ǫ e ǫ ( ) [ : ] 2.5 : 2.5 χ(x) dx = 2.5 χ(x) =, a.e. [ : 2 ] 2.5 χ(x) dx = χ(x) χ(x) {φ k (x)} φ k > φ k (x) = φ k (x) < φ k (x) = {φ k (x)} χ(x) N N 2N C(,N) χ N (x) φk (x) χ N dx C(,N) φ k (x) χ N (x) χ(x) χ N (x) Lebesgue ( 2.2) φ k (x) dx χ(x) dx = C(,N) C(,N) {φ k (x)} φ k (x) dx < 2 k C(,N) φ k (x) χ N (x) > 2 x 2 k x 2 k+ f k (x) = sup{φ k (x),φ k+ (x),...} {f k (x)} χ(x) f k (x) χ N > 2 x e kn l k φ l (x) χ N (x) > 2 x l=k 2 l+ = 2 k+2 Heine-Borel (.3) χ(x) f k (x) f k (x) χ N (x) > 2 x e N e kn k N e N N χ(x) =, a.e. 2.4 Fubini Fubini ( ) x R n, y R m f(x, y) R n+m f(x, y) dx, f(x, y) dy (2.3)

37 3 2. LEBESGUE y, x [ f(x, y) dy] dx = [ f(x, y) dx]dy = f(x, y) dxdy (2.3) f(x, y) dxdy (successive integration) [ ] f(x,y) {φ k (x,y)} [ φ k (x,y) dy] dx = φ k (x,y) dxdy lim [ φ k (x,y) dy] dx k = lim φ k (x,y) dxdy = f(x, y) dxdy (2.32) k Levi ( 2.2) F(x) φ k (x,y) dy = F(x), a.e. (2.33) lim k lim [ k φ k (x,y) dy] dx = F(x) dx (2.34) x x a.e. φ k (x,y) f(x,y) (2.33) Levi ( 2.2) φ k (x,y) dy = F(x) = f(x,y) dy, a.e. lim k (2.34) (2.32) [ f(x,y)dy] dx = f(x,y) dxdy Fubini x R n, y R m f(x, y) R n+m [ f(x, y) dy] dx (2.35) f(x, y) R n+m Fubini f(x, y) dx dy = [ f(x, y) dy] dx [ ] f(x,y) a.e. f(x,y) {φ k (x,y)} k= f(x,y) dy x x y { φ k (x,y) } k= Lebesgue ( 2.2) φ k (x,y) dy = f(x,y) dy lim k

38 (2.35) { φ k (x,y) dy} k= Lebesque φ k (x,y) dx dy = [ f(x,y) dy] dx (2.36) lim k [ φk (x,y) dy] dx = φ k (x,y) dx dy { φ k (x,y) dx dy} k= Fatou ( 2.2) f(x,y) dx dy {φ k (x,y)} k= Lebesgue f(x,y) (2.3) 2.4. e 2 x2 /σ 2 dx = 2π σ (2.37) [(2.37) ] x R e x2 I = e x2 dx I 2 = e x2 y 2 dxdy x,y R e x2 y 2 I 2 = π/2 dθ e r2 rdr = π 4 I > π e x2 dx = 2 (, ) x = x /( 2 σ) x (2.37) 2.5 f(x) Re f(x) Im f(x) f(x) dx = Re f(x) dx + i Im f(x) dx (2.38) Lebesgue

39

40 3 3. (a, b) f(x) F(x) C F(x) = x a f(t) dt + C (3.) 3. (a, b) f(x) f(x) f(x) = d x f(t) dt, a.e. (3.2) dx a (a, b) f(x) f(x) dx [ ] f(x) {φ n (x)} n= F(x) (3.) φ n (x) Φ n (x) Φ n (x) = x a φ n(t)dt+c a x < x b Φ n (x ) Φ n (x) = x x x x φ n (t) dt x f(t) dt = F(x ) F(x) x b a φ n+ (t) dt = Φ n+ (x ) Φ n+ (x) Φ n+ (x) Φ n (x) F(x) Φ n (x) x x [a,b] F(a) Φ n (a) = F(x) Φ n (x) F(b) Φ n (b) Φ n (b) F(b) Φ n (x) F(x) Φ n (x).6 (.4) F(x) [(3.2) ] Φ n+ (x) Φ n (x) x F(x) Φ (x) = lim n Φ n(x) Φ (x) = [Φ k+ (x) Φ k (x)] k= 33

41 34 3. Fubini (.5) F (x) φ (x) = k= [φ k+ (x) φ k (x)] = lim n φ n(x) φ (x) = f(x) φ (x), a.e. [ ] [ ] F(x) (a,b) T(a,b) T(a,b).5 (.5) f F ǫ(x) (a,b) T(a,b) = sup {ǫ(x)} b a ǫ(x)f(x) dx b a f(x) dx (3.3) ǫ(x)f(x) f(x) {φ n (x)} n= f(x) ǫ n(x) φ n φ n (x) < b a ǫ n (x)f(x) dx T(a,b) n ǫ n (x)f(x) f(x) Lebesgue ( 2.2) b lim n a ǫ n (x)f(x) dx = b (3.3) (3.4) a f(x) dx T(a,b) (3.4) T(a,b) = b a f(x) dx [ ] 3.. F(x) f(x) F = [ ] f(x) ǫ > δ > h < δ f(x + h) f(x) < ǫ F(x + h) F(x) h f(x) = h h [f(x + t) f(x)] dt h h f(x + t) f(x) dt < ǫ F (x) = f(x) 3. (a, b) F(x) (α, β) (a, b) β α F (t) dt = F(β) F(α) (3.5) [(3.5) ] (3.2) (3.) F(x) = x a F (t) dt + C (3.6) (3.5) [ ]

42 (a, b) F(x) F (a, b) F(x) (α, β) (a, b) β α F (x) dx F(β) F(α) (3.7) F(x) (3.5) [(3.7) ] F(x) δ > < h < δ h f h (x) = F(x + h) F(x) h β α f h (x) dx = h β+h β F(x) dx h α+h α F(x) dx F(β + δ) F(α) Lebesgue (.5) F(x) h f h (x) a.e. F (x) Fatou ( 2.2) (3.7) [ ] F(x) β α F(β) F(α) K N K {(α k, β k )} K k= Kk= (β k α k ) Kk= F(β k ) F(α k ) F(x) ǫ > δ > {(α k, β k )} K k= K (β k α k ) < δ k= (3.8) K F(β k ) F(α k ) < ǫ (3.9) k= 3..2 F(x) C > α < β F(β) F(α) C(β α) (3.) F(x) Lipshitz F(x) Lipshitz F(x) 3..3 F(x) a.e. F (x) =

43 36 3. [ ] F(x) ǫ > δ > (3.8) (3.9) F (x) F {α k,β k } K k= (3.8) (3.9) F(x) ǫ F (x) = F(x) ǫ F(x) 3.2 x (a, b) F(x) F(x) [ ] F(x) f(x) C (3.) n N f n (x) { f(x), f(x) n f n (x) = nf(x)/ f(x), f(x) > n n {f n (x)} n= f(x) ǫ > N N b a [ f(x) f N (x) ] dx < ǫ δ = ǫ/n (3.8) (3.9) K k= βk α k f(x) dx K k= βk α k f N (x) dx + K k= K N(β k α k ) + ǫ < 2ǫ k= βk α k [ f(x) f N (x) ] dx [ ] F(x) ǫ = δ > (3.8) (3.9) (a,b) δ/2 2(b a)/δ + δ (a,b) 2(b a)/δ + F(x) 2.6 F (x) F(x) (α,β) (a,b) (3.7) F (x) G(x) (3.7) G(β) G(α) F(β) F(α) H(x) = F(x) G(x) H(x) a.e. H = F (x) G (x) = F(x) G(x) H(x) a.e. H (x) = 3..2 H(x) C F(x) = G(x) + C F (x) 3.3 (a, b) f(x) b = x a = x f(x) b = x lim f(t) dt = f(t) dt (3.) x a a

44 a = b b lim f(t) dt = f(t) dt (3.2) x x [(3.) ] {X n } n= n N X n > a n X n f n (x) = χ (a,xn)(x)f(x) f n (x) n χ (a, ) (x)f(x) f n (x) χ a, (x) f(x) F(X n ) = Xn a f(t) dt = f n (t) dt Lebesgue ( 2.2) n χ (a, )(t)f(t)dt [ ] 3.4 (a, b) F(x) G(x) (α, β) (a, b) β α β F (x)g(x) dx + F(x)G (x) dx = F(β)G(β) F(α)G(α) (3.3) α [ ] 3.2, 3. F(x) G(x) F(x) < M, G(x) < M M F(β)G(β) F(α)G(α) = F(β)G(β) F(β)G(α) + F(β)G(α) F(α)G(α) M G(β) G(α) + M F(β) F(α) (3.9) F(x)G(x) 3.2, 3. (3.5) F(x) F(x)G(x), F (x) F (x)g(x) + F(x)G (x) 3..4 x R F(x) x, x G(x), G (x) F (x)g(x) dx = F(x)G (x) dx (3.4) 3.2 x R f(x) [a, b] ξ (a, b) b f(x) dx = f(ξ) (3.5) b a a

45 38 3. [ ] (3.5) A f(x) [a,b] ξ, ξ 2 f(ξ 2 ) A f(ξ ) f(x) ξ ξ 2 f(x) A x x ξ [ ] F(x) = x f(t) dt a (.4) ξ (a,b) F(b) F(a) b a = F (ξ) = f(ξ) F (x) = f(x) 3.. x R f(x) (a, b) g(x) [a, b] ξ [a, b] b a ξ b f(x)g(x) dx = g(a) f(x) dx + g(b) f(x) dx (3.6) a ξ [ ] g(x) F(x) F(x) = x f(t) dt a F(x) [a,b] ξ ξ 2 H(x) H(x) = g(a) x f(t) dt + g(b) b a x f(t) dt = F(b)g(b) F(x)[g(b) g(a)] (3.7) (3.6) H(ξ) (3.6) I g(x) g (x) I = b a f(x)g(x) dx = F(x)g(x) b x=a b a F(x)g (x) dx g (x) F(x) F(ξ ) I F(b)g(b) F(ξ )[g(b) g(a)] = H(ξ ) (3.8) (3.8) F(ξ 2 I H(ξ 2 ) H(x) ξ ξ 2 H(x) I x x ξ [ (3.8) ] f(x) φ n (x) g(x) x ψ n (x) φ n (x) ψ n (x) (a,b)

46 x, x 2,..., x K x = a, x K = b Φ(x k ) = k l= φ(x l )(x l x l ) b a φ n (x)ψ n (x) dx = = = K φ n (x k )ψ n (x k )(x k x k ) k= K [Φ n (x k ) Φ n (x k )]ψ n (x k ) k= K k= K Φ n (x k )ψ n (x k ) Φ n (x k )ψ n (x k+ ) k= K = Φ n (x K )ψ n (x K ) Φ n (x k )[ψ n (x k+ ) ψ n (x k )] k= ψ n (x) Φ n k ) b a φ n (x)ψ n (x) dx Φ n (b)ψ n (b ) min Φ n(x k ) [ψ n (x K ) ψ n (x )] <k<k n (3.8) 3.2. (a, ) f(x) X > a (a,x) x λ X lim X a f(x)sin λx dx (3.9) [ ] λ f(x) X > X > a X ξ X ξ X X f(x)sin λx dx = f(x) ξ X X sinλx dx + f(x ) ξ sinλx dx [ cos λx cos λξ ] [ cos λξ cos λx = f(x) + f(x ] ) λ λ {f(x) + [f(x) f(x )] + f(x )} λ = 2 f(x) (3.2) λ X f(x) (3.9) λ X lim X sin λx x dx (3.2) [ ] (,) (,X) (,) (,X) < a < b λ > b a sinλx x dx < 2 λa (3.22)

47 4 3. [ ] (3.2) f(x) = /x a < b λ > b sinλx dx < 3 (3.23) x a [ ] b a sinλx x dx = λb λa sin t t dx (λa,λb) (,) (, ) (,) (, ) (3.22) Riemann-Lebesgue x R f(x) λ f(x)sin λx dx =, f(x)cos λx dx = (3.24) lim λ lim λ [ ] f(x) f(x) L (R) 2.3 {φ n (x)} ǫ > N N n > N (2.26) f(x)sin λx dx = φ n (x)sin λx dx + [f(x) φ n (x)]sin λx dx φ n (x)sin λx dx + f(x) φ n (x) dx (3.25) φ n (x) K k (c k,d k ) φ n (x) a k a k M φ n (x)sin λx dx K dk K a k sinλx dx = a k cos λc k cos λd k λ k= 2KM λ c k k= (3.26) Λ > 2KM/ǫ λ > Λ ǫ (2.26) (3.26) (3.25) λ > Λ f(x)sin λx dx 2ǫ 3.3. X lim X sin λx x dx = π 2, X lim sin λx dx = π λ x 2 λ >, X > (3.27)

48 [(3.27) ] N Z + N 2 + cos nx = 2 n= N n= N e nix = sin[(n + 2 )x] 2sin 2 x (3.28) π sin[(n + 2 )x] 2sin 2 x dx = π 2 (3.29) λ = N + 2 π sin λ x x dx + π ( 2sin 2 x ) λ π sinλ sin t π x 2sin 2 x dx = dt + x x t 2xsin 2 x sinλ x dx 2 x = [,π] Riemann-Lebesgue 2 λ λ = N + 2 π/2 (3.27) X sin λx λx sin t lim dx = lim dt = π X x X t 2 [(3.27) ] X sin λx λx sint lim dx = lim dt = π λ x λ t a > f(x) ( a, a) x = 2 a sin λx lim f(x) dx = f(), λ π x a sin λx lim f(x) dx = f() (3.3) λ π a x [ < a < ] f(x) = f(x) f() (3.3) 2 lim λ π a sin λx 2 f(x) dx f() = lim x λ π a sin λx f(x) dx = x ǫ > < δ < π δ < x < δ x f(x) f(x) ǫ ( 3.2) (,δ) ξ ξ [,δ] δ sin λx f(x) x dx = ǫ δ ξ sinλx x dx 3ǫ (3.23) (δ,a) f(x)/x Riemann-Lebesgue λ ǫ [a = ] ǫ > X R X >, f(t) dt < ǫ X (3.3) (,X) (X, ) ǫ a = X

49 a > f(x) ( a,a) x = a cos λx lim f(x) dx = lim λ a x [ < a < ] (3.3) lim λ a δ ( δ a cos λx [f(x) f( x)] dx = x + a a δ ) f(x) dx (3.3) x [f(x) f( x)] x dx Riemann-Lebesgue [a = ] ǫ > X R X > 2, [ f(t) + f( t) ]dt < ǫ X (3.3) ( X,X) (, X) (X, ) ǫ a = X 3.6 x R f(x) x = lim σ σ 2π f(x)e x2 /(2σ 2) dx = f() (3.32) [ ] M = f = sup x R f(x) f(x) x = ǫ > δ > x < δ f(x) f() < ǫ (3.33) σ > < σ < σ σ 2π δ e x2 /(2σ 2) dx = 2π δ/σ e t2 /2 dt < ǫ (3.34) 2.4. (2.37) (3.34) (3.33) σ 2π /(2σ 2) dx f() = f(x)e x2 σ 2π [f(x) /(2σ 2) dx f()]e x2 δ σ 2π δ f(x) /(2σ 2) dx + 4Mǫ ǫ + 4Mǫ f() e x2 σ (3.32)

50 4 Banach Hilbert 4. ( ) (x, y) (x, y, z) n n (x, x 2,..., x n ) n R n C n n x, x 2,..., x n x = (x, x 2,..., x n ) y = (y, y 2,..., y n ) λ x y x + y = (x + y, x 2 + y 2,..., x n + y n ) (4.) x λ λx = (λx, λx 2,..., λx n ) (4.2) x y x y = x y + x 2 y x n y n (4.3) x y = x y x x x x = x = x = x x x E (linear space) (vector space) x, y, z E a, b I. x y E x + y E 43

51 44 4. BANACH HILBERT ) (x + y) + z = x + (y + z) 2) E + x = x 3) x E x E x + ( x) = 4) x + y = y + x II. ax = xa E ) a(x + y) = ax + ay 2) (a + b)x = ax + bx 3) a(bx) = (ab)x 4) x = x I. 2) (zero element) x =, ( )x = x x + ( )y = x + ( y) = x y 4.. n (zero vector) (,,...,) E x, y (x, y) (x, y) x y (scalar product, inner product) x, y, z E a, b ) (ax + by, z) = a(x, z) + b(y, z) ( ) 2) (x, y) = (y, x) ( ) 3) (x, x) ( ) x = (x, x) = (x, y) 2 (x, x)(y, y) (4.4) Schwarz (Schwarz inequality) [ ] λ (x,y) θ (x + λe iθ y,x + λe iθ y) = (x,x) + 2λ (x,y) + λ 2 (y,y) λ (x,y) 2 (x,x)(y,y) (4.4) E x x x x (norm) E (normed space) x, y E a ) x ( ) x = x =

52 4.. ( ) 45 2) x + y x + y ( ) 3) ax = a x ( ) x x = (x, x) (4.5) (4.5) Schwarz (4.4) (x, y) x y (4.6) 4. (4.5) [ ] x + y 2 = x 2 + 2Re (x,y) + y 2 x (x,y) + y 2 ( x + y ) 2 Schwarz (4.6) 4.2 x y x + n R n C n x = (x, x 2,..., x n ) x = x = n n x k 2, x = x k, k= k= x = max k, k n (4.7) n x p = ( x k p ) /p (4.8) k= p p > x = x 2 M = max k n x k M x p Mn /p x = lim p x p = M (4.7) n x x = {x k } k= (4.7), (4.8) x = x k 2, x = x p = ( k= k= x k p ) /p k= x k, x = sup x k, k< x l 2, l, l, l p x = {x k }, y = {y k } x + y = {x k + y k } x = {x k λ λx = {λx k } (4.9)

53 46 4. BANACH HILBERT x, y x + [l ] x k x, y k y x k + y k x k + y k x + y x + y x + y [l ] x + y = x k + y k ( x k + y k ) = x + y k= k= [x l 2, y l 2 x + y l 2 ] 2 x k y k x k 2 + y k 2 [l 2 ] x l 2, y l 2 (x,y) = k= x ky k 4. [x l p, y l p x+y l p ] ( x k +y k ) p ( x k + y k ) p (2max{ x k, y k }) p 2 p ( x k p + y k p ) [l p ] a >, b > log( p ap + ( p )bp ) p log ap + ( p )log bp = log(ab p ) p ap + ( p )bp ab p a = x k / x p, b = ( x k + y k )/ x + y p k= k= x k x k + y k p /( x p x + y p p ) k= x k x k + y k p x p x + y p p x y x + y p p = x k + y k p ( x k + y k ) x k + y k p ( x p + y p ) x + y p k= k= p x + y p x p + y p e C(e) f(x) f = f = sup f(x) (4.) x e x e f(x) =

54 4.. ( ) 47 [ ] f(x) g(x) C x f(x) + g(x) f(x) + g(x) f + g f + g n C n (R), C n [a, b] R, [a, b] n n n = C C C n f(x) f = sup f (k) (4.) k n f(x) = [ ] f(x) g(x) C n k n f (k) (x) C f (k) + g (k) f (k) + g (k) f + g C (R), C [a, b] C (R) = n=c n (R), C [a, b] = n=c n [a, b] (4.2) 4..2 x R e x2 C (R) 4..3 x R f(x) f(x) = { e (b a) 2 /[(x a)(b x)], a x b, otherwise (4.3) f(x) F(x) F(x) = x f(t) dt (4.4) < a < b G(x), x a x f(t) dt a G(x) = b, f(t) dt a < x b a, x > b (4.5) f(x) G(x) C (R) F(x) C (R)

55 48 4. BANACH HILBERT L f(x) f = f = f(x) dx (4.6) L ( 2.5 ( 2.3) [ ] f(x) g(x) L f + g = f(x) + g(x) dx f(x) dx + g(x) dx = f + g.2 R n A E a E δ B(a, δ) x a < δ x E A {x k } k= x lim x k x = (4.7) k x x A 4.3 {x k } k= x x < x = lim k x k [ ] x < x k x < ǫ x x k + x x k < x k + ǫ x = lim k x k x x k x x k < ǫ Cauchy {x n } n= ǫ > N N n N, m N, n > N, m > N a n a m < ǫ Cauchy Cauchy E Cauchy E (complete) Banach Banach Hilbert l, l, l p, C, L Banach l 2 Hilbert 4.2 Banach Hilbert Banach Hilbert E Cauchy E

56 4.2. BANACH HILBERT p p > l, l, l p Banach l 2 Hilbert n (4.7) x x (4.8) x p Banach (4.7) x Hilbert [l p ] 2 ǫ [ ] Cauchy {u (n) } n= u(n) = (u (n),u(n) 2,...) N N n > N, l > N k N u (n) k u (l) k ( k= {u (n) u (n) k u (l) k p ) /p = u (n) u (l) < ǫ (4.8) k } n= Cauchy u k u = (u,u 2,...) [ 2] u (n) u, u l p N K N n > N K ( u (n) k u k p ) /p < 2ǫ (4.9) k= u (n) u < 2ǫ (4.2) u (n) u l p, u = u (n) + u u (n) l p u (n) u [(4.9) ] L N l > L k K k u k < ǫ/k /p n > N, l > N, l > L K ( u (n) k u k p ) /p ( k= K k= u (n) k u (l) k p ) /p + ( K k= u (l) u (l) k u k p ) /p < 2ǫ (4.2) u (n) u (l) < ǫ [l 2, l ] p = 2, p = [l ] (4.8) sup k< u (n) k u (l) k (4.9) sup k K u (n) k u k (4.9) ǫ/k /p ǫ (4.9) K N u (n) u < 2ǫ 4.2 C(e) f(x) f = f = sup x e f(x) Banach C n (e) f(x) f = sup k n f (k) Banach [C(e) Banach ] {f l (x)} l= Cauchy ǫ > N N l > N, m > N x f l (x) f m (x) f l f m < ǫ (4.22) x {f l (x)} l= Cauchy f(x)

57 5 4. BANACH HILBERT. 9 x f l (x) f(x) < 2ǫ (4.23) f l (x) f(x).5 (.4) f(x) f l f(x) < 2ǫ f l (x) f(x) C(e) f(x) = f l (x) + f(x) f l (x) C(e) 4. 3 f < [C n (e) Banach ] C(e) f l f m, f l, f f l (x), f m (x), f(x) f (k) l (x), f m (k) (x), f (k) (x) [ ] 4.3 L f(x) f = f = f(x) dx L Banach [ ] {f n (x)} Cauchy m,m 2,... m < m 2 <... n > m k f n f mk < 2 k (4.24) f mk+ (x) f mk (x) dx = f mk+ f mk < 2 k (4.25) f mk+ (x) f mk (x) dx (4.26) k= Levi ( 2.2) f m (x) + (f mk+ (x) f mk (x)) k= a.e. {f mk (x)} k= a.e. f(x) L {f mk (x) f(x)} k= Lebesgue f m k f ǫ > r 2 r < ǫ/2 f mr f < ǫ/2 n > m r f n f f n f mr + f mr f < ǫ f n f f, f f p = [ f(x) p dx] /p f(x) p > p f(x) p e p f p f = sup x e f(x) M = sup x e f(x) ǫ > f(x) > M ǫ δ f p Mm(e) /p, f p > (M ǫ) δ /p p M ǫ f M ǫ f = M

58 4.3. L L 2 f(x) f(x) 2 f(x) L 2 f(x) L 2 f(x) 2 dx < (4.27) 4.4 f(x) L 2, g(x) L 2 f(x)g(x) f(x)+g(x) L 2 [ ] f(x)g(x) 2 ( f(x) 2 + g(x) 2 ) f(x)g(x) ( 2.3) f(x)g(x) f(x) + g(x) L 2 f(x) g(x) L 2 (f, g) (f, g) = f(x)g(x) dx (4.28) f f = (f, f) /2 (4.29) Schwarz (4.6) (f, g) f g (4.3) f(x) L 2, g(x) L 2 λ ( ) (4.29) ) f ( ) a.e. f(x) = f = 2) λf = λ f ( ) 3) f + g f + g ( ) L 2 ( 2.5 ( 2.3) 4.4 f(x) e f(x) [ ] e χ e (x) 4.4 f(t) g(t) f(x)χ e (x) χ e (x) [ ]

59 52 4. BANACH HILBERT L 2 {f n (x)} n= f n f (4.3) f(x) f(x) (mean-square convergence) Riesz-Fischer L 2 {f n (x)} n= m, n f m f n (4.32) [ ].3 (.2) {f n (x)} f(x) f n f f n f m f n f + f f m [ ] L 4.3 ( 4.2) x R n e R n Schwarz ( 4.) (4.25) f mk+ (x) f mk (x) dx m(e) /2 f mk+ f mk m(e) /2 2 k (4.33) e (4.26) f mk+ (x) f mk (x) dx m(e) /2 k= e f mk (x) e a.e. f(x) (4.33) f mk (x) f m + k f ml f ml f m + (4.34) l=2 f mk (x) 2 Fatou ( 2.2) f(x) L {f n (x)} f(x) n f n f, f n f {f n (x)} f(x) x N. Wiener The Fourier Integral & Certain of its Applications (Cambridge U.P., 23) n N k = [log 2 n] log 2 n 2 k n < 2 k+ = 2 k + 2 k, x n 2k 2 k f n (x) = n 2, k x < 2 n 2k + k 2 k, x n 2k + 2 k f n(x)dx = < 2 2 k n f(x) = f n f = f n f x x < k Z + 2 k n < 2 k+, f n (x) = n f n (x)

60 4.3. L C n L 2 [ ] f(x) L 2 f(x) {φ k (x)} ǫ > φ k f < ǫ φ k (x) L 2 g k (x) g k φ k < ǫ (4.35) C n φ k (x) c c (c,c + δ) { φk (c φ k (x) = ), c < x < c φ k (c + δ), c < x < c + δ g k (x) n = g k (x) = φ k (c ) + φ k(c + δ) φ k (c ) (x c ), c < x < c + δ (4.36) δ n < g k (x) = φ k (c ) + φ k(c + δ) φ k (c ) Φ(c Φ(x), c < x < c + δ (4.37) + δ) Φ(x) = x c δ 2 exp{ (t c )(c + δ t) } dt ( F(x) ) g k (x) φ k φ k (x) δ g k φ k (4.35) g k (x) g k f g k φ k + φ k f < 2ǫ {g k (x) C n } f(x) 4.6 2π C n (R) L 2 ( π, π) [ ] 4.5 (a,b) C n [a,b] L 2 (a,b) a = π,b = π 2π C n (R) L 2 ( π,π) φ k ( π + ) φ k (π ) 3π < x < π φ k (x) = φ k (x + 2π) c = π (c,c + δ) g k (x) c + 2π < x < π g k (x) = g k (x 2π) 4.5 f(x) L 2 (R) ǫ > δ > < r < δ r f(x + r) f(x) 2 dx < ǫ (4.38)

61 54 4. BANACH HILBERT [ ] f(x) x > R x < R C(R) f(x) ǫ > δ > < r < δ r, f(x + r) f(x) < ǫ/(2r) (4.38) [ ] ǫ > R > R f(x) 2 dx+ R f(x) 2 dx < ǫ/8 C(R) g(x) x > R x < R f g 2 < ǫ/9 x R f r (x), g r (x) f r (x) = f(r + x), g r (x) = g(r + x) δ < r < δ g r g 2 < ǫ/9 f r f f r g r + g r g + g f < 2ǫ /2 /3 + ǫ /2 /3 = ǫ /2 (4.39) f r f 2 < ǫ C n L [ ] 4.5 L 2 L f(x), g(x) L (R) h(x) = f(y)g(x y) dy (4.4) f(x) g(x) (convolution) h(x) = (f g)(x) (4.4) 4.8 h(x) f(x) L (R) g(x) L (R) h(x) L (R) [ ] f(x) {φ n (x)} g(t) 2.3, 2.4 ( 2.3) φ n (y)g(x y) y h n (x) = φ n (y)g(x y)dy (4.42) 3. ( 3.) g(x) h n (x) x h n (x) (a,b) φ n (y) y φ n (y)g(x y) Fubini b a h n (x)dx f g (4.43) (a,b) (4.43) h n(x)dx f g {h n (x)} Levi ( 2.2) h n (x) n h(x) = lim n φ n(y)g(x y)dy h(x)dx f g y {φ n (y)g(x y)} Lebesgue ( 2.2) (4.4)

62 [ ] f(x) {φ n (x)} g(t) L (R) g(x) ψ(x) g ψ = g(x) ψ(x) dx h n (x) = φ n (y)g(x y)dy, h n(x) = φ n (y)ψ(x y)dy (4.44) x g(x) ψ(x) (4.43) (a,b) b a h n(x)dx φ n ψ f ( g + ψ g ) (4.45) C n = sup y (, ) φ n (y) (4.44) h n (x) h n(x) C n g ψ b b h n (x)dx h n(x)dx (b a)c n g ψ (4.46) a a g ψ (4.45) (4.46) (4.44) h n (x) (4.43) (a,b) (4.43) 4.9 f(x) g(x) R 3 () f(x) g(x) (2) f(x) g(x) (3) f(x) g(x) (4.4) h(x) R h(x) (a, b) h(x) (), (2), (3) [ () ] (a,b) h(x) 4.8 f(x) [c,d] (4.44) g(x y) a x b, c y d, a d x y b c (4.43) g b c a d g(x) dx g(x) [c,d] f b c a d f(x) dx (4.43) h n (x), h(x), φ n (x)g(x y) (a,b) χ (x) h(x) (4.4) h(x)χ (a,b) (x) [ (2) ] f(x), g(x) [c f, ), [c g, ) f(y)g(x y) a x b, c f y, c g x y b c f, y b c g (4.43) f b c g c f f(x) dx g b c f c g g(x) dx Γ(z) Heaviside u(x) Γ(z) Re z > z Euler Γ(z) = t z e t dt (4.47) u(x) x R {, x >, u(x) =, x f(x) x f(x)u(x) = (4.48)

63 56 4. BANACH HILBERT f(x)u(x a) R q q Riemann-Liouville (fractional integral) a D q R f(x) x > a ad q R f(x) = Γ(q) x a (x y) q f(y)dy (4.49) Γ(q) xq u(x) f(x)u(x a) 4.9 ( (2)) p Riemann-Liouville (fractional derivative) a D p R f(x) x > a ad p dm Rf(x) = dx m a D p m R f(x) m p α >, f(x) = Γ(α) (x a)α p R ad p R f(x) = Γ(α p) (x a)α p (4.5) [ ] p = q < (4.49) f(x) y t = y a x a B(p,q) q Re p >, Re q > B(p,q) = t p ( t) q dt = Γ(p)Γ(q) Γ(p + q) Γ(z + ) = zγ(z) (4.5)

64 5 Fourier 5. E {f n } (linearly independent) (linearly dependent) 5.. E l, l 2, l n N e (n) E n e () = (,,,, ), e (2) = (,,,, ), e (3) = (,,,, ), E {e (n) } n= [ ] K x e () + x 2 e (2) + + x K e (K) = (x,x 2,,x K,,, ) x = x 2 = = x K = A A A A A (separable) 5..2 n R n C n x x (4.7) [ ] E R n E x = {x k } n k= x k E [ ] 5..3 l, l 2 [ ] E l, l 2 K N E x = {x k } k= k K x k k > K x k A K...2. A K. (.) K= A K E E x = {x k } k= ǫ > K N E y = {y k } k= k K y k = x k k > K y k = y x < ǫ/2 A K z = {z k } k= k K z k x k < ǫ/(2k) z x z y + y x < ǫ 57

65 58 5. FOURIER E {f n } E (basis, bases) 5..4 E l, l {e (n) } n= E [ ] E x = {x k } k= x = x k e (k) k= {f n } n= E n= a nf n E f n= a nf n f {f n } n= a n Hilbert H {f n } n= Hilbert H {f n } n= (f n, f m ) =, n m (5.) (f n, f m ) = δ nm (5.2) (orthogonal system) (orthogonal condition) (orthonormal system) (orthonormal condition) m Z δ nm n = m n Z Kronecker (Kronecker s delta) δ nm = δ mn {f n } n= e n = f n f n (5.3) {e n } n= / f n 5..5 n N f n (x) = sin nx {f n } n= L2 ( π,π) ( π,π) f n = π π sin nx} n= { 5..6 n N f n (x) f n (x) = χ [n,n+) (x) = {, n x < n +, otherwise (5.4) {f n (x)} n= L2 (R)

66 {f n } n= Hilbert H n= c n f n H f c n c n = (f, f n ) (5.5) f (f, f n ) 2 f 2 (5.6) n= (5.6) Bessel (Bessel s inequality) (5.5) c n {c n } n= l2 lim n c n = [(5.5) ] n= c N nf n n = N n= c nf n g N f N g N f Schwarz ( 4.) (g N f,f n ) g N f f n N c n = lim N (g N,f n ) = (f,f n ) [(5.6) ] c n = (f,f n ) N N N N f c n f n 2 = (f c n f n,f c n f n ) = f 2 c n 2 n= n= N c n 2 f 2 n= (5.6) n= n= 5.2 N {f n } N n= Nn= a n f n f Nn= a n f n {a n } a n = c n = (f, f n ) [ ] c n = (f,f n ) N N f a n f n 2 = f 2 [a n (f n,f) + ā n (f,f n )] + n= n= N N = f c n 2 + c n a n 2 n= n= N a n 2 a n = c n n= f, f 2,..., f N f, f 2,..., f N V N {f n } n= Nn= c n f n f V N N ( c n f n, f n= N c n f n ) = (5.7) n= E {f n } n= E f n= a n f n (complete)

67 6 5. FOURIER 5.3 {f n } n= Hilbert H f c n = (f, f n ) n= c n f n f (f, f n ) 2 = f 2 n= (5.8) f H, g H (f, f n )(f n, g) = (f, g) (5.9) n= (5.8) Parseval [ ] (5.9) {f n } n= f, g n= cnf n, n= d nf n 5. c n = (f,f n ), d n = (g,f n ) Schwarz N N N (f c n f n,g d n f n ) = (f,g) c n d n n= n= n= N N f c n f n g d n f n, n= n= N (5.9) (5.8) (5.9) g f {f n } n= Hilbert H f c n = (f,f n ) n= c nf n f {f n } n= c n Fourier L 2 {f n (x)} n= f(x) L2 (Fourier ) n= (f,f n)f n (x) f(x) 5.3 {f n } n= Hilbert H f n [ ] (5.8) f (f,f n ) n N Hilbert H [ ] 5.3 f (5.8) l l 2 [ ]

68 5.2. FOURIER 6 Schmidt (Schmidt s orthogonalization) {f n } n= Hilbert H {g n} n=, {e n } n= g = f, e = g g, g 2 = f 2 (f 2, e )e, e 2 = g 2 g 2,... g n... n = f n (f n, e k )e k, e n = g n g n, k= (5.) 5. Schmidt {f n } g n, e n k n f k f n k n e k f n = n k= (f n,e k )e k f n m > n e m 5.4 Hilbert H [ ] H {h k } k= H f n = h n Schmidt g n e n g n = {h k } k= h n n g n {e n } h k n k e n {e n } 5.2 Fourier 5.5 x R {f n (x)} n= f (x) = /2, f (x) = sin x, f 2 (x) = cos x, f 3 (x) = sin2x, f 4 (x) = cos 2x,... (5.) L 2 ( π, π) f(x) L 2 ( π, S(x) = a 2 + (a n cos nx + b n sinnx) (5.2) n= ( π, π) f(x) a n = π π π f(x)cos nx dx, b n = π π π f(x)sin nx dx (5.3) [ ] (5.) 5.7 (5.) π f = 2, f = f 2 =... = π (5.3) f(x) {f n (x)} n= (5.2), (5.3) S(x) 5.3 S(x) f(x)

69 62 5. FOURIER (5.2) f(x) Fourier (5.3) a n, b n f(x) Fourier 5.6 f(x) L ( π, π) Fourier S(x) f(x) x 2 {f(x+) + f(x )} [ ] f(x) 2π x R f(x) x.7 (.5) f(x+), f(x ) Fourier S(x) S N (x) = a N 2 + (a n cos nx + b n sin nx) (5.4) n= (5.3) 3.3 (3.28) S N (x) = π = π π π π π f(t){ N 2 + cos n(t x)} dt = π n= f(x + t) sin[(n + 2 )t] 2sin 2 t λ = N + 2, dt = π f(t) = f(x + t) + f(x t) f(x+) f(x ) (3.29) S N (x) 2 {f(x+) + f(x )} = π = π π π π π π f(t) sinλt 2sin 2 t dt f(t) f(t) sin[(n + 2 )(t x)] 2sin dt 2 (t x) [f(x + t) + f(x t)] sin[(n + 2 )t] 2sin 2 t t sin λt 2sin 2 t dt (5.5) t t = f(t) f(t) t/sin( 2 t) 3.5 ( 3.3) λ dt lim [S N(x) {f(x+) + f(x )}] = N 2 (5.4), {cos nx} n N, {sin nx} n N (trigonometric polynomial) (5.4) a N, b N N 5.2. f (x), f 2 (x) x (,2π) f (x) = x π, f 2 (x) = x2 2 πx + π2 3 2π Fourier x π = n= 2 sin nx, n < x < 2π x 2 2 π2 πx + 3 = 2 cos nx, n2 n= x 2π

70 5.2. FOURIER 63 f 2 (x) x =, x = π n 2 = π2 6, ( ) n n 2 = π2 (5.6) 2 n= n= 5.7 f(x) 2π C (R) Fourier S(x) f(x) f(x) [ ] f (x) Fourier (5.2) S (x) = (nb n cos nx na n sinnx) n= Bessel ( 4.) n 2 ( b n 2 + a n 2 ) π f (x) 2 dx (5.7) π n= π Schwarz ( 4.) a n = (n a n ) { n (n a n ) 2 } /2 n 2 n= n= n= n= (5.7) (5.6) n= a n n= b n a n cos nx + b n sinnx a n + b n Weierstrass (.3) Fourier S(x) [ ] ǫ > N N N > N π < x < π f(x) S N (x) < ǫ f S N 2π ǫ [(5.) ] 4.6 ( 4.3) 2π C (R) L 2 ( π,π) L 2 ( π,π) 2π C (R) (5.) L 2 ( π,π) f(x) L 2 ( π,π) ǫ > f g < ǫ 2π C (R) g(x) f(x) g(x) Fourier S N (x) T N (x) 5.7 N N N > N g T N < ǫ 5.2 ( 5.) f S N f T N f g + g T N < 2ǫ 5.8 f n (x) = e inx, n =, ±, ±2,... (5.8) L 2 ( π, π) f(x) L 2 ( π, π) Fourier S(x) = lim N S N(x), S N (x) = Fourier c n = 2π π π N n= N c n e inx (5.9) f(x)e inx dx (5.2)

71 64 5. FOURIER [ ] (5.3) (5.2) c = 2 a, c n e inx + c n e inx = a n cos nx + b n sinnx, n N (5.4) S N (x) (5.9) S N (x) (5.4) 5.5, 5.6, 5.7 (5.9) 5.3 (a, b) f n (x) = x n (5.2) f(x) g(x) w(x) > w(x) L (a, b) (f, g) = b a f(x)g(x)w(x) dx (5.22) (a, b) w(x) n N sup x n w(x) < x (a,b) (5.23) x {f n (x)} n= L2 (a, b) Schmidt ( 5.) g n (x), e n (x) n p n (x) = e n+ (x) n p n (x) {p n (x)} n= {p n (x)} n= (recurrence formula) xp n (x) = a n p n+ (x) + b n p n (x) + a n p n (x), n =, 2,... (5.24) a n, b n {p n (x)} n= a n = (xp n, p n+ ), b n = (xp n, p n ) (5.25) [ ] (5.) f n, e n x n, p n (x) 5. ( 5.) p n (x) n x n x n = n k= (xn,p k )p k (x) m > n p m (x) p n (x) n [ ] xp n (x) n + n+ xp n (x) = (xp n,p k )p k (x) k= (xp n,p k ) = (p n,xp k ) n > k + (5.24)

72 Legendre P n (x) (, ) w(x) = P (x) =, P (x) = x, P 2 (x) = 2 (3x2 ),... (5.26) P n () =, P n 2 = /(n + 2 ) P n(x) d n P n (x) = 2 n n! dx n(x2 ) n (5.27) Rodrigues xp n (x) = n + 2n + P n+(x) + n 2n + P n (x), n =, 2,... (5.28) (5.27) P n (x) P n 2 (5.28) P n () = P () = P () = 2. Chebyshev ( ) T n (x) (, ) w(x) = / x 2 T (x) =, T (x) = x, T 2 (x) = 2x 2,... (5.29) θ (, π) T n (cos θ) = cosnθ T 2 = π, n T n 2 = π 2 xt n (x) = 2 T n+(x) + 2 T n (x), n =, 2,... (5.3) T n (cos θ) = cos nθ T n (x) T n 2 N N θ f(θ) N f(θ) cos θ N cos nθ = T n (cos θ) 3. Chebyshev ( ) U n (x) (, ) w(x) = x 2 U (x) =, U (x) = 2x, U 2 (x) = 4x 2,... (5.3) θ (, π) U n (cos θ) = sin(n+)θ sin θ n U n 2 = π 8 xu n (x) = 2 U n+(x) + 2 U n (x), n =, 2,... (5.32)

73 66 5. FOURIER U n (cos θ) = sin(n+)θ sin θ U n (x) U n 2 N N θ f(θ) N f(θ) sin θ cos θ N sin(n + ) = sinθ U n (cos θ) 4. Hermite H n (x) (, ) w(x) = e x2 H (x) =, H (x) = 2x, H 2 (x) = 4x 2 2,... (5.33) H n (x) x n 2 n H n 2 = π /2 2 n n! H n (x) H n (x) = ( ) n x2 dn e (5.34) dx ne x2 xh n (x) = 2 H n+(x) + H n (x), n =, 2,... (5.35) (5.34) H n (x) H n 2 Hermite 6.4 Weierstrass [a, b] f(x) ǫ > n P n (x) = a k x k k= (5.36) sup f(x) P n (x) < ǫ (5.37) x [a,b] [ ] g n (x) g n (x) = 2J n ( x 2 ) n (5.38) J n = ( x 2 ) n dx (5.39)

74 δ (,) J n (δ) J n (δ) = δ ( x 2 ) n dx (5.4) J n (δ) < δ ( δ2 ) n dx < ( δ 2 ) n+, J n > ( x)n dx = /(n + ) n J n (δ)/j n ǫ > N N n > N δ g n (x) dx = J n (δ)/(2j n ) < ǫ (5.4) [ ] < a < b < f(x) [,] [a,b] P n (x) P n (x) = 2J n f(t)[ (x t) 2 ] n dt (5.42) x P n (x) x 2n t x + t P n (x) = 2J n x x f(x + t)( t 2 ) n dt = x x f(x + t)g n (t) dt (5.43) x [a,b] < b x < a < < b < x a < f(x) (.5 (.4)) ǫ > δ t < δ x f(x + t) f(x) < ǫ (5.44) δ δ < a,δ < b f(x) x [a,b] f(x) < M M > n > N P n (x) f(x) = = x x x x 4Mǫ + f(x + t)g n (t) dt f(x) f(x + t)g n (t) dt δ δ f(x)g n (t) dt f(x + t) f(x) g n (t) dt 4Mǫ + ǫ t < δ (5.44) [ δ,δ] f(x) M (5.4) 5.9 (a, b) L 2 (a, b) [ ] [a,b] f(x) L 2 (a,b) {p n (x)} n= n= c np n (x) f(x) n Weierstrass ǫ > k= a kx k f(x) n a k x k < ǫ, k= x (a,b) (5.) x n {p k (x)} n k= n k= a kx k = n k= b kp k (x) f n ( b b k p k < ǫ k= a ) /2 w(x) dx

75 68 5. FOURIER 5.2 ( 5.) c k = (f,p k )/ p k 2 f n ( b c k p k < ǫ k= a ) /2 w(x) dx [ ] 4.5 ( 4.3) C[a,b] L 2 (a,b) 5.7 ( 5.2)

76 6 Fourier 6. L Fourier f(x) L (R) t R F(t) = f(x)e ixt dx (6.) f(x) Fourier (Fourier transform) 6. Fourier F(t) t ± F(t) [ ] F(t) f(x) dx [ ] ǫ > A > A f(x) dx < ǫ, A f(x) dx < ǫ X e ix = 2 sin(x/2) X A F(t ) F(t 2 ) < 4ǫ + e i(t t2)x f(x) dx A 4ǫ + t t 2 A A A f(x) dx δ δa f(x) dx < ǫ t t 2 < δ t, t 2 F(t ) F(t 2 ) < 5ǫ F(t) [F(t) ] Riemann-Lebesgue ( 3.3) (6.) 69

77 7 6. FOURIER 6.2 f(x) L (R) Fourier F(t) f(x) x Fourier (Fourier inversion formula) [f(x+) + f(x )] = lim 2 f λ(x), f λ (x) = λ 2π λ λ F(t)e ixt dt (6.2) [ ] (6.) x x (6.2) f λ (x) t f λ (x) = 2π = π f(x )λ ) ei(x x e i(x x )λ i(x x dx = ) π f(x + t) sinλt t dt = π f(t) = f(x + t) + f(x t) f(x+) f(x ) 3.3 (3.27) f λ (x) 2 [f(x+) + f(x )] = π lim X X [f(x + t) + f(x t)] f(t) sinλt t dt f(x ) sin (x x )λ (x x ) sin λt dt t ǫ > (,X), (X, ) λ ǫ < X < X (X,X ) (3.22) X sin λt f(t) π X t dt πx X dx [ f(x + t) + f(x t) ] dt + π f(x+) + f(x ) 2 λ X X ǫ (,X) (5.5) 3. 5 ( 3.3) 6.3 x R f(x) L (R) x, x f (x)e ixt dx = it f(x)e ixt dx (6.3) [ ] 3..4 ( 3.) F(x) = f(x), G(x) = e ixt 6.4 f(x), xf(x) L (R) f(x) Fourier F(t) d dt F(t) = [ ] F(t + h) F(t) h ( ix)f(x)e ixt dx (6.4) = f(x)e ixt e ixh h dx = f(x)( ix)e ixt ixh/2 sin(xh/2) e dx xh/2 xh/2 < Lebesgue ( 2.2) h (6.4) sin(xh/2)