Fourier Fourier Gibbs

Transcription

1 3 4 5 Fourier 3. Fourier Gibbs Fourier. Fourier Tempered disribuion

2 , Wiersrass [] [] Funcional Analysis K. Yoshida Springer [3] Real and Complex Analysis W. Rudin McGrow-Hil [4] [5] Fourier-Laplace [6] [7] & Ver 3 7//5 Ver 6/8/5 Ver 5//9

3 Fourier. Fourier f cos n, sin n cos n, sin n f cos n, sin n f. a, b a < b L a, b { } L a, b f : b a f d <, f : a, b : < f, g > b a fg d. f, g, h L a, b < f, f >, < f, f > f < f, g > < g, f > 3 < αf + βg, h > α < f, h > +β < g, h > 3 < f, αg + βh > α < f, g > +β < f, h > f. x, y xy x + y b b fg d f g d a f, g L a, b. < f, g >. x + y x + y b a a a f + g d b b a f + g d f + g d < f, g L a, b f + g L a, b α C, f L a, b αf L a, b L a, b α, β C, f, g L a, b αf + βg L a, b.3 f, g L a, b < f, g > f, g f g.4 a π, b π π π sin n cos m d, sin n d, cos n d, d π. π π π m n π m n sin n sin m d, cos n cos m d.3 π m n π π m n 3

4 π e im m n e in d π m n.4 3 a, b P n n d n P n n n! d n n.5 m n P m P n d.6 m n n + 4 L, ψ < ψ < j N {}, k Z ψ j,k j ψ j k {ψ j,k } j,k ψ j,k ψ l,m d δ j,k,l,m.5.4, 3 hin: 3.6 P n n n m d n I d n { n } dm d m { m }d [ ] d n d n { n dm d m { m }} k < n { n } k + m [ d n d n d n { n } dm+ d m+ { m }d k kc r { n } r { + n } k r r d n { n dm d m { m }} ] d n I d n { n } dm+ d m+ { m+ }d I m d n m d n m { n } dm d m { m }d.7 4

5 n > m dm+ d m+ { m } m n.7 I m+ I n n n!d n n! n + n d [ n n + n+ n! n + n n n! n + n!n! n!.6 f L a, b. d n m+ d n m+ { n } dm+ d m+ { m }d ] n + n+ d n + n+ n d n + + n d n!n! n!n + n+ n! n + n+ P n P m d n m n + n m b f f d a.8 f.7 f, g L a, b < f, g > f g, Cauchy-Schwarz.9 f + g f + g. s C i sf + g < sf + g, sf + g > s f + Rs < f, g > + g, iiiiiiii < f, g > < f, g > e iθ s ξe iθ, ξ R ξ f + ξ < f, g > + g ξ D D 4 < f, g > f g < f, g > f g 5

6 f + g f + R< f, g > + g f + f g + g Cauchy-Schwarz f + g f + g f + g.8 f n, f L a, b b f n f d a f n f n f n f lim n f n f.9 f n f a, b f n f n f nk f k {f nk }. {ϕ n } n N L a, b b m n ϕ m ϕ n d < ϕ m, ϕ n > a d n m n. d n f L a, b n N {ϕ n } n N c n < f, ϕ n >. f {ϕ n } n N. {ϕ n } n N f, g L a, b 3 f g f + g f + g,.3 N f < f, ϕ n > ϕ n, ϕ k, k,,, N.4 n N a n ϕ n n N a n.5 n 4 N f a n ϕ n N f < f, ϕ n > ϕ n n n.6 5 < f, ϕ n > f, Bessel.7 n 6

7 : < f, g > f + g f + R < f, g > + g f + g : k N N f < f, ϕ n > ϕ n, ϕ k n N < f, ϕ k > < f, ϕ n >< ϕ n, ϕ k > n < f, ϕ k > < f, ϕ k >< ϕ k, ϕ k >, {ϕ n } n N 3: 4: N a n ϕ n n N f a n ϕ n n N a n ϕ n, n N m,n N a m ϕ m n a n a m < ϕ n, ϕ m > N a n, {ϕ n } n N n N f N < f, ϕ n > ϕ n + < f, ϕ n > a n ϕ n n n N N f < f, ϕ n > ϕ n ϕ k k,,, N < f, ϕ n > a n ϕ n n N f < f, ϕ n > ϕ n n N f < f, ϕ n > ϕ n n n n N + < f, ϕ n > a n ϕ n 5: N N f N N f < f, ϕ n > ϕ n + < f, ϕ n > ϕ n N N 4 f < f, ϕ n > ϕ n < f, ϕ n > ϕ n n n N f < f, ϕ n > ϕ n n N < f, ϕ n > ϕ n n N < f, ϕ n > 3 ; n N < f, ϕ n > f n n n N + < f, ϕ n > ϕ n n 7

8 N < f, ϕ n > f n. Parseval {ϕ n } n N n f < f, ϕ n >, f < f, ϕ n > ϕ n c n ϕ n c n..3 f n n < f, ϕ n > ϕ n n.4 {ϕ n } n N {ϕ n } n N < f, g > < f, ϕ n > < g, ϕ n > n 3 f L a, b f ϕ n n N f N 4 lim N f < f, ϕ n > ϕ n n [].5.4 { } sin n cos n,, : n N π π π { } e in : n Z L π, π, π L π, π P n : n N L, n+ n N a n π b n π π f cos n d, a π π f sin n d π f d.8 c n fe in d.9 π π 8

9 . < f, < f, < f, cos n π > π sin n π > π > π π π cos n f d π π sin n f d π π π π f cos n d π π π f cos n d πa n ; f sin n d π f sin n d πb n ; π π π f d π π f d f d π π π a π f < f, ϕ n > ϕ n n π a cos n + πan + π π n a + a n cos n + b n sin n n n πbn sin n π f a + a n cos n + b n sin n. n f a n, b n Parseval f < f, ϕ n > n π a + πa n + πb n; n π a + Parseval Bessel.6.3 f n πa n + πb n n π f d a π π + a n + b n. π f d a π π + a n + b n. n { } e in : n Z π n c n e in, π f d π π < f, n.7 f L π, π e in π > πc n c n a n ib n, c n a n + ib n, n N c n.3 9

10 f πa n πb n π π f cos n d f sin n d f sin n d + π π a n π f cos n d + f sin n d + f sin nd f cos n d f sin n d f f cos n d f cos n d, b n.4 a n, b n π f sin n d.5.8 [ π, π] f a fa + lim f fa lim f a,x>a a,x<a f f fa f fa lim lim a,>a a a,<a a [ π, π] f r f r f C r - π π [ π, π] f f f π fπ [ π, π]. [ π, π] f [ π, π] f. [ π, π] f C - f [ π, π] f f f. fx [ π, π] f π fπ a n, b n f a n, b n f a n nb n, b n na n, n N f f [ ] π b n f sin n d π π [ f ] π n cos n + f cos n d π π π n + π f cos n d n π π n a n;

11 a n f cos n d π π [ f ] π n sin n f sin n d π π π n π f sin n d n π π n b n.3 fx π fx fx S n x a n S n x + n a k sin kx + b k cos kx a + a k sin kx + b k cos kx k k a n + a k sin kx + b k cos kx a k + a k + b k, sin kx, cos kx k a n + k a k + b k,. a + k n k a + 8 8,. n k n k k a + k a k + b k n k a k + b k n k n a k + b k π k k π 6, π k. Cauchy-Schwarz a + b a + b, a, b f d, n N. Weiersrass S n x gx gx fx. Parseval π f S n d a π π kn+ + a k + b k k a + n a k + b k k a k + b k n,.8 S n g Limi.4 f g d lim f S n d n π π.3 g f.4 f g [ π, π]

12 [hx ] hx π x π < x < hx x π x < x < π hx k sin kx k hx hx, a n b n π sin nd π [ ] π π cos n π n π hx sin kx k k hx cos x, x, gx x n cos nd n ; gx x gx x C - g h x cos x lim x x x h x sin lim x x x x sin lim x h x x x sin lim x lim h x x x cos θ sin θ x gx.3 gx g S h nx hx cos xs h nx cos x n k n k π π n k π h sin k d sin kx h sin k d cos x sin kx h sin k d sin kx sink + x + sink x 3

13 n k n k n k π π π S g nx π h sin k d sin kx n k π h sin k d sink + x n h sin k d sink x π k h sin k sink sink + d sin kx π h cos sin k d sin kx π S g nx g h sin n d sinn + x + π h sin n d sinn + x + π h sin n d sinn + x + π h sinn + d sin nx h sinn + d sin nx h sinn + d sin nx Bessel lim n π h sin n d lim cos n xsh nx lim n Sg nx gx gx lim n Sh nx hx cos x δ > x cos x Fourier.5 [ π, π] fx x f fx fx + + fx x f I f I x, x,, x l fx fπ f π x π fx fx gx fx l k π fx k + fx k hx x k, x x m, m,,, n x x m gx gx m ± fx m ± fx m + fx m fx m + + fx m l l k x m π fx k + fx k hx m x k k x m π fx k + fx k hx m x k ;. 3

14 gx m + gx m gx m gx m +, m,,, n gx [ π, π].3 Fourier l gx lim n Sg nx lim Snx f n π fx k + fx k S h x k n x S h xm n S h xm n x π cos x S h nx x m h π π π h x m d + h x m d + n k n k x m h π π π cos π π S h nx x m gx lim n hd + hd + S f nx S h n.4. lim n Sf nx lim n Sf nx n k n k l k π k π h x m sin n d sin nx + π π h x m cos n x d π π π h cos n x x m d π π π h x m cos n d cos nx π h cos n d cos nx x m + h sin n d sin nx x m π π π π π fx k + fx k Snx h x k l π fx k + fx k hx x k, x x m m,,, n k π fx k + fx k hx x k, x x m m,,, n x k x m fx, x x m m,,, n lim n Sf nx fx m + + fx m, x x m m,,, n ;. I f.3. limi I.3 I.3. Gibbs.3.4 Gibbs 4

15 .6 n sin kx k n+x sin sin nx sin x, n cos kx k n+x cos sin nx sin x a k e ikx cos kx + i sin kx n k e ikx eix e inx e ix ei x e inx e i x e i x e i x ei x e i n+x i sin x cos x n+x cos + isin x n+x sin i sin x n+x sin sin nx n+x + i cos sin nx i sin x n+x cos sin nx sin x n+x sin sin nx + i sin x n n cos kx + i sin kx k k n e ikx k.7 hx S h nx n k π x π < x < hx x π x < x < π k sin kx x n π n + Sh nx n sin lim n Sh nx n d.6 S h nx n cos kx k x > S h nx cos S h nx n n k n + x k sin kπ n + n+x cos sin nx sin x n k, i.e. n + n + x k sin n + π k n + π x n π n + sin π lim n Sh nx n d sin d 5

16 .8 Cauchy sin d π sin d [3]p [ sin x x ].9 Gibbs fx [ π, π] x m m,,, l f S f nx lim n Sf nx m n fx m n fx m + fx m x m n x m +. π n + lim n x mn x m gx m n lim n Sg nx m n lim n Sf nx m n lim x mn x m.7 n lim n gx mn fx m + + fx m lim hx mn x k n.8 fx m + + fx m lim n Sf nx m n fx m n l k π sin π d π fx k + fx k hx m n x k.8 l sin d x m x k hx m x k x m x k k x m π fx k + fx k hx m x k ; lim n Sf nx m n π fx sin m + fx m fxm + fx m π π fx m + fx m sin sin d d sin.3 d.859 Wilbraham-Gibbs π sin d.789 f π d.3.3 fx [ π, π] π S n x S n x a n + a k cos kx + b k sin kx k 6

17 S n x π π fd n x d fx D n d π π D n x D n x sin n + x x sin Diriche kernel f π R S n x S n x π π π π π π f π π n π f d + f cos k d cos kx + f sin k d sin kx π k π π π n π f d + fcos k cos kx + sin k sin kx d π k π n π f d + f cos kx d, π k π n + cos kx d k.6 + n k n+x cos sin nx cos kx + sin x S n x π π x s sin x f sinn + x sin x n+x + cos sin nx sin x d fd n x d π π sinn + x sin x π fd n x d π π π π π x π x+π x+π x π π fx sd n s ds fx sd n s ds fx sd n s ds, fx, D n x π-.3 S n x σ n x n + n S k x.33 fx [ π, π] π σ n x π π k fk n x d fx K n d π π 7

18 K n x sin n+ x n + x sin.9 Fejér kernel S n x σ n x n + n + π n S k x k n k π π π π f n + f sink + x n k sin x sink + x sin x n k sin k + x sin x n k sink + x sin x n k d d cosk + x + cos kx cosn + x sin n+ x sin x n + sin x ; σ n x π π f n + sin n+ sin x x d fk n x d π π σ n x fx sk n s ds.3 π π.34 K n x K n x, x [ π, π] π π K n d 3 δ > x δ x K n x.9.3 fx S n x σ n x π K n d σ n x π π 3 K n x sin n+ x n + x sin n + δ sin n, x δ > K n x x δ > n 8

19 .35 D n x K n x.34 3 D n x K n x x D x, D 8 x, D 5 x K x, K 8 x, K 5 x x n K n x x π D n x d +.36 fx [ π, π] fπ f π σ n x fx fx [ π, π].6 fx ε >, δ δε : x x < δ fx fx < ε x, x [ π, π] M max{ fx : x [ π, π]} , σ n x fx π fx K n d fx K n d π π π π π fx fx K n d π π fx fx K n d π π fx fx K n d + fx fx K n d π π x δ δ fx fx ε fx fx fx + fx M εk n d + MK n d π x δ π x δ x δ.343 N : n > N max{k n x : x δ} < ε ε K n d + M K n d ε + Mε.34 π x δ π x δ σ n x fx , a n, b n.4. 9

20 .38 fx [ π, π] C r - fx lim n nr a n, lim n nr b n a r n, b n r f r x. k a n n r k+ n r a r n, r k b r n, r k + k, b n n r k n r b r n, r k a r n, r k +.3 f r lim n ar n, lim n br n.3 lim n nr a n, lim n nr b n.39 fx a + a n cos nx + b n sin nx r N, ε > n lim n nr++ε a n, lim n nr++ε b n a + a n cos nx + b n sin nx fx C r - r n {a n }, {b n } fx fx C r - r.9 a n, b n n N, M > a n M n r++ε, b n M n r++ε a n cos nx + b n sin nx n na n sin nx + nb n cos nx nn n <, nr+ε r N na n + nb n nn. Weiersrass na n sin nx + nb n cos nx.9 fx n r r.3 nn M <.3 n r+ε na n sin nx + nb n cos nx n nn M < + n+ε fx r +.3 fx C -.38 na n sin nx + nb n cos nx na n + nb n n n

21 na n n n n n a n + + nb n n n b n {a n }, {b n } f x.3 n n a n + n b n n n n n n n + b n a n a n + b n < +. Weiersrass.9 fx a + a n cos nx + b n sin nx n fx C r - r

22 Fourier. Fourier [ L, L] L, fx [ L, L] fx fx dx <., ε λ > λ fx dx + λ., fx f. f. Ffs ˆfs π fx dx < ε. F fs ˇf π fe is d.3 fse is ds.4. sin x x dx π.3, s > sin F s π, s ± π, s <, s > fs π, s ± ˇf sin π, s < sin sin F s π π π sin e is d sin cos s π sin + s sin + s + sin s d + π d d sin s d

23 a τ. sin a d, s > π, s ± π, s < π a > a π a < ˇf π π eis ds cos s ds [ sin s ] sin.4 F F e s π e s e s π e e is d π π e +is s d e +is d e s e is d Cauchy π e x dx e +is d π π e d π e d e s e d e s.5 f R s R f s f s lim f s f d s ε > f g d < ε, λ : g, > λ.5 R g g [ λ, λ] ε > δ > : s < δ, g s g < ε, R.6 3

24 s < δ f s f d s f s g s d + g s g d + g f d < ε + g s g d + ε λ < + λε lim a n lim b n n n.6 Riemann-Lebesgue f, ˆfs, lim ˆfs s ˆfs.5 ε > f g d < ε π, λ : g, > λ ω ε λ e iω ω < ε ε λ λ ˆfs + ω ˆfs fe is+ω d fe is d π π π π + π λ f ge is+ω d + π λ f g d + π λ λ g d ε λ λ ge is+ω e is d + π g e iω d + π g f d g fe is d ˆfs lim ˆfs e iπ s.5, lim s ˆfs ˆfs π fe i+ π s s d π f π e is d s ˆfs f f π e is d π s f f π d π s f f π s d s 4

25 .7 f [, λ] λ lim s f sin s d π f+. fx f+ f+ λ lim s f sin s d.7 λ lim s f sin s d lim s λ + f+ lim s f+ lim f+ f f+ s λ λs sin s d + lim s sin s d sin d sin d f+ π λ f+ sin s d f f+.7, f θ δ > f M,, δ M sup{ f : δ} λ f sin s δ d sin s λ f d + f δ δ f λ sin s d + f δ λ f e is e is Mδ + d i δ sin s f, [δ, λ] g.6 d d sin s lim s λ f sin s d π Mδ + ĝ s ĝs δ, s.8 f [ λ, + λ] λ lim s λ fx +.7 sin s d π fx + + fx λ λ fx + sin s d λ λ fx + fx + sin s d + sin s d + λ λ fx + fx sin s d sin s d s π fx + + π fx 5

26 .9 f, lim s fx + sin s d π fx + + fx f s, λ sin s λ sin s fx + d fx + d λ sin s fx + d >λ fx + sin s >λ d fx + >λ λ d < f d λ ε > λ f d < λ.8 s > s > s ε λ sin s fx + d π λ fx + + fx < ε fx + sin s d π fx + + fx < f d + ε < ε, s > s λ sin s lim fx + d π fx + + fx s Fubini fx, y [a, b] [c, d] d b b d fx, ydx dy fx, ydy dx c a a c [a,b] [c,d] fx, ydxdy.8 [a, b], [c, d] a, c b, d d b fx, y dx dy <,.8 c a b a d fx, y dy dx <. f, f cos sx d ds π fx + + fx π fx + + fx lim s lim s sin s fx + d s fx + cos u du d f s lim fx + cos u d du s s lim f cos u x d du f cos u x d du s du 6 c

27 fx + + fx π π f cos sx d ds fcos sx cos s + sin sx sin s d π f cos s d cos sx + π As cos sx + Bs sin sx ds As, Bs As π f cos s d, Bs π f sin s d ds f sin s d sin sx ds Inversion Formula. f, fx + + fx e ixs fe is d ds π π N s lim N N f ˆf ˇˆf f π e ixs π fe is d ds π e ixs π fcos s i sin s d ds cos xs + i sin xsas ibsds As cos xs + Bs sin xs + ias sin xs Bs cos xsds As Bs i. As cos xs + Bs sin xs ds fx + + fx.3 Fourier R f, g, h Faf + bgs affs + bfgs f f Ff s isffs.9 f n, k f k k n Ff n s is n Ffs. 7

28 3 f n f 4 f g f g d Ffs F ifs. ds d n ds n Ffs F in fs. f g fτg τdτ.3 f g h f g h, f g g f.4 Ff g Ff gs πffsfgs.5 5 f, g f, g f g fsĝse is ds.6 f, g 6 f convoluion f # f.7 f g # g # f #.8 Ff # s Ffs.9 7 f, g f, g fgd fsĝsds. f d f, g. Parseval fs ds, Plancherel. Faf + bgs af + bge is d π af + bge is d π a π affs + bfgs fe is d + b π ge is d 8

29 lim f f f α lim f f + f τdτ f f τdτ α α N M f d > α N.9 f > α, > N M N M f α Ff s π f e is d π [ fe is ] + π is π isffs. fe is d fise is d, lim f 3 f 9. d ds Ffs d fe is d ds π d fe is d π ds ife is d π F ifs n k n k f d k f d + n f d + n f d + k f d f d, k n >, k [, ] f d < 4 d n ds n Ffs F in fs Fubbini φx, y R R φx, y dxdy φx, y dy dx < φx, ydx dy φx, y φx, ydy dx 9

30 f g d fτg τdτ d fτg τ dτd fτ g τ d dτ fτ g d dτ τ fτ dτ g d < f, g f g f g h f gτh τdτ fsgτ sds fs fs h τdτ gτ sh τdτ gτh s τdτ fsg h sds f g h; f g h f g h ds ds Fubbini τ s τ f g g f; f g g f fτg τdτ f τgτdτ τ τ Ff gs πffsfgs Ff gs π π π f ge is d π π fτg τdτ e is d fτg τe is dτd fτe iτs g τe i τs d fτe iτs ge is d dτ fτe iτs ĝsdτ π fsĝs dτ Fubbini τ 3

31 5 f, g f, g f g 4 f g π Ff gse is ds π π fsĝse is ds fsĝse is ds 6 Ff # s π π π f # e is d π fe is d fe is d fs f e is d f g # f g 7.6 fτg τdτ fτ g τdτ f + τ g τdτ + τ τ g # τf # τdτ g # f # fg d fg d f g.6 fsĝsds, g g #.9 fgd fsĝsds g f f d fs ds Plancherel.4 f f f n lim n f m, n, m N {}, M > ; > > f m < M f m d < M d < > > g f, g d d f g d fτg τ dτ fτ d d d g τ dτ fτg τ dτ f g 3

32 f m d < f m lim n e f f,, n N e > λ.5 f ˆf f n f n f.33 Ff.3,3 s m dn ds ˆfs d n i n+m m F n d m f s dm n d m f.6 dn lim sm s ds ˆfs n lim d m s F n d m f s f.6 f ˇˆf f f, g gs ˆfse ixs ds gs fe is d e ixs ds π f, g δ > δs s π δ gδs ˆfse ixs ds f gse is x ds d π ĝ xfd ĝfx + d; gs ˆfse ixs ds gδse is ds π δ ĝ ĝfx + d gse i δ s δ ds δ ĝ δ fx + d ĝfx + δd. δ, gs e s.4. e δs ˆfse ixs ds e fx + δd 3

33 lim e δs, lim fx + δ fx f.5 δ δ ˆf limi ˆfse ixs ds e d π π e fxd ˆfse ixs ds fx R v v Ri i,. vi Ri Ri d Ri d R îs ds [s s + s] îs s.7 f g #,f f # Es Ff f # s fs. f π Esds fse is ds π lim fs ds k n fs k e isk s k s k f [s k s k ] e is k fs k. h T x h T x sin π T x π T x L [ L, L] fx fx d < L f L fx d L [ L, L] fx, gx, < f, g > L L 33 fxgxdx.3

34 Cauchy-Schwarz < f, g > f g T > f f fx [ ˆfs, s π T, π ] T n fnt h T x nt, x R ε > N N N : N > N fx fnt h T x nt < ε, x R.3 L π T ˆf [ π T, π T ˆfs n N ] c n ˆf N n N c n e int s f. c n N.5 c n T π T π T ˆfse int s ds T π ˆfse int s ds T π f nt.6 ˆfs N n N f π T N π π T n N T π f nt e int s ˆfs T fnt e int s e is ds π N n N T fnt e int s N π π T π π T T π π T ˆfse is ds T π ˆfse is ds T π ˆfs N n N N T fnt e int s e is ds π T n N π N T fnt e int s e is ds π T n N π T fnt e int s e is ds π f Cauchy-Schwarz π ˆfs N n N T π fnt e int s e is 34

35 π ˆfs N n N T π N π T n N n N T π fnt e int s π T N n N T π fnt T π fnt e int s e is ds T π T e i nt s ds N T sin nt π π fnt T nt N n N fnt h T nt N f fnt h T nt ˆfs T.9 sin π T π n N N n N N n N [ ] T e i nt s π T π fnt i nt π T N sin nt π T fnt nt π T n N T fnt e int s N R π.8.3 χ [ π T, π T ] s sin π χ [ π π T, π T ] T s F s π s [ π T, π T ] s [ π T, π T ]..8 f L, L T < π T > L {fnt : Z} f. Poisson summaion fourmula φ 4. n Z φπn π n Z φn 4. φ n Z φ + πn, φs + n f : n Z φ + πn f π c k c k π n Z n Z fe ik d π π π π π n+π φ + πne ik d n π φ + πne ik d π n Z.5 f k Z c k e in n Z φ + πne ik d fe ik d φk π π n Z φ + πn k Z 35 π φke in

36 n Z φπn π... π F fs : n Z φπn π k Z φk fe is d F φk φ F ψ F F ψ πψ k Z k Z ψk n ZF ψπn n Z k Z ψk n Z F ψπn F ψπn.3 ˆfs, s, f,, f T N f, ft, ft,, fkt,, fn T ˆfs f.8 f, ˆfs f, <, NT.7 ˆfs, s π T s > π T.8 f, ˆfs ˆfs T f nt n,,, N c n π oherwise.5 ˆfs N n T π fnt e int s.9 f, NT f n c n e i πn NT.3 36

37 c n NT πn i fe NT d NT.7.8 c n NT πn i π fe NT d NT NT π π NT ˆf πn NT N < n N oherwise πn i π fe NT d NT ˆf πn NT.3 f N n N + π NT ˆf πn e i πn NT.3 NT.9 ˆfs {fnt } N n.3 f { ˆf πn } N NT n N + π ˆf πn T NT, n,,,, N Xn π ˆf πn N T NT n N +,, N.9 s πk NT πnk e i N π Xk T ˆf πnk N i e N, k > N N πk NT πnk i.3 kt e N fkt N N n n DFT IDFT Xk N n fnt N πnk i fnt e N.3 πnk N i e N, k > N Xne i πnk N.33 πnk i fnt e N N k Xke i πnk N.4 {fnt } N n {Xn}N n.3 ON 967 Cooley Turkey ON log N FFT.5 MRI.6 n n :n f f,,, n 37

38 m m, m,, m n,,,, n, < m, > cm f π π m, m n n :< s, > ˆfs π n f π n n m k k k fe i<m, > d d d n i<m, > cme n s k k k fe i<s, > d d d n ˆfse i<s, > ds ds ds n.7 ψ L, + f L, +, a, b R CW T f a, b a b fψ d a f f f b CW T f a, bψ C ψ a a dadb C ψ ψs ds < s f f d CW T f a, b C ψ a dadb.8 l Z { } l Z {fn} n Z : fn < l Z {fn} n Z n Z fω n Z fne iωn ω π, π] fn fωe iωn dω π π {fn} n Z, {gn} n Z l Z f gn k Z fkgn k 38

39 f gω fωĝω., n Z fn π π fω dω.9 π, π], +, Z, +, R, + II G φ : G T, φgh φgφh.34 φ G T {z C : z } G Ĝ φ, ψ φ ψg φgψg φ ψ.34 Ĝ Ĝ G T, { : π < π}, + [ ] G π, π], + φ n Z : φ e in m Z e im T T Z [ ] G Z φn ω π, π] : φn e iωn θ π, π] e iθm Z Ẑ T 3 [ ] G R φ s R : φ e is s R e is R R R fφ fgφg dg G Ĝ G Ponryagin fg f hg. Ĝ fφφg dφ G f hφ fφĥφ fkg k dk 39

40 u u c x + u y + u z u u c x + u y + u z c >, u u, x, y, z u x + u y + u z u, x, y, z x, y, z x, y, z c T T ρ ρ c K K σ ρ σρ u, x, y, z ux, y, z u u u u c x + u y 3.4 u u c x + u y 3.5 u x + u 3.6 y u c u c u x u x u x + u fx, y 3.9 y 3.7, u, x fx, u, x gx [a, b] a, b u, a, u, b, 4

41 u u, x x + [x, x + x] x x + x x x u, x K x x [x, x + x] K u, x x x + x [x, x + x] [x, x + x] K u, x K u, x + x x x K u, x + x x u K x u, x + x, x x u, x Taylor x K u, x x x K u, x x 3. x K u, x K x x u, x u, x + x x x + x u, x + x ρσ u, x x [x, x+ x] σ + 3 u +, x u, x xρσ ρσ u, x x K u u, x x ρσ, x x x c u u, x, x x c K σρ 3. x y u u x K, x + x, y, x, y y K u, x x y x x x 3.3 u u y K, x, y + y, x, y x K u, x y x y y y 3 g 4

42 u u K, x, y + y, x, y x y y y x, y + y [x, x ρσ u + x] [y, y + y], x x y x + x, y + y x u u K, x + x, y, x, y y x x x, y x + x, y [x, x + x] [y, y + y] u K x, x + u, x x y 3.4 y + [x, x + x] [y, y + y], u +, x, y u, x, y x yρσ ρσ u, x x y ρσ u u, x x y K x, x + u, x x y y u, x c u x, x + u, x y, c K σρ u u c x, 3.7 u,, u, L 3.8 u, x fx, fx, gx u, x gx 3.9 Sep u, x GF x u u G F x, G F x c GF x, u x GF x i.e. G c G F x F x 4

43 G c G F x x k F x G c G F x F x k G, F x G kc G, F x kf x 3. Sep k F x F, F L F x k > F x Ae kx + Be kx F, F L x, L A + B, Ae kl + Be kl A, B F x k F x Ax + B F, F L F x 3 k < F x A cos kx + B sin kx F A F L B sin kl kl πn, n Z k πn L sin nπ x n n L G 3. cnπ G G L, λ n cnπ L F n x sin nπ x n N 3. L G n C n cos λ n + D n sin λ n 3. u n, x G n F n x C n cos λ n + D n sin λ n sin nπ L x u n, x λ n cnπ L {λ n : n N} 3. x λ n π cn L λ n π cn T L n T ρ L L ρ T L 43

44 Sep3 u, x, u, x αu + βu α, β R u, x u, x C n cos λ n + D n sin λ n sin nπ L x 3.4 n 3.9 C n, D n fx u, x gx u, x C n cos λ n + D n sin λ n sin nπ L x C n sin nπ L x n n λ n C n sin λ n + λ n D n cos λ n sin nπ L x λ n D n sin nπ L x 3.5 n Fourier fx, gx [ L, L] f, g C n L L λ n D n L L fx sin nπ L x dx gx sin nπ L x dx, i.e. D n cnπ C n, D n 3.4 u, x L n gx sin nπ L x dx 3.6 C n, D n u, x Wiersrass u, x C n cos λ n + D n sin λ n sin nπ L x 3.7 n. Wiersrass C n cos λ n + D n sin λ n sin nπ L x n C n cos λ n + D n sin λ n sin nπ L x n C n + D n sin x, cos x n x, x L f fl, g gl fx C - gx. C n nπ L f x cos nπ dx 3.8 L 44

45 a f n C n L nπ af n, D n L cnπ bg n f a g n g L C n + D n nπ af n + L cnπ bg n n n L n π af n + L n cπ bg n n L n a f n + L b g π cπ n n Parseval. n n π 6 < n n n a f n <, Cauchy-Schwarz b g n < n C n + D n <. Wiersrass n 3.7 u, x 3.7 u n, x 3.7 u, x , x u x, x nπ L C n cos λ n + D n sin λ n cos nπ L x n u x, x nπ Cn cos λ n + D n sin λ n sin nπ 3.9 L L x n u, x λ n C n sin λ n + D n cos λ n sin nπ L x n u, x λ nc n cos λ n + D n sin λ n sin nπ 3.3 L x n nπ L C n cos λ n + D n sin λ n cos nπ L x nπ L C n + D n n n nπ Cn cos λ n + D n sin λ n sin nπ nπ L L x Cn + D n L n n λ n C n sin λ n + D n cos λ n sin nπ L x λ n C n + D n n n λ nc n cos λ n + D n sin λ n sin nπ L x λ n C n + D n n n , ,3.3 n C n, n n D n 3.33 fx C C n L L fx sin nπ L x dx nπ L n f x cos nπ L x dx 45

46 L nπ L nπ 3 [ f x sin nπ ] L L [ f x cos nπ ] L L x L L f x sin nπ L x dx f 3 x cos nπ L x dx L nπ f u x, c u,, f L u x, L c u, L L f x sin nπ L x dx L nπ 3 C n L nπ 3 L L f 3 x cos nπ L x dx; L3 3 f 3 x cos nπ L x dx nπ 3 af n 3.34 gx C - D n D n cnπ L gx sin nπ L x dx [ gx cos nπ ] L L x + [ L cnπ 3 g x sin nπ ] L L x L cnπ D n L cnπ 3 L L L L3 g x cos nπ L x dx g x sin nπ L x dx L cnπ L cnπ 3 L L g x cos nπ L x dx g x sin nπ L x dx; g x sin nπ L x dx cnπ 3 bg n Cauchy-Schwarz f 3, g Parseval n C n L3 π 3 a f 3 n L3 n π 3 n a f 3 n < n n n D n L3 cπ 3 n n n b g n n L3 cπ 3 n n n, u, x, x b g n n < fx C 3 - gx C - C n, D n u, x 3.7 u, x fx, u, x gx u, x fx fx C -.3 u, x gx gx C -.3 u, x gx 3. fx C 3 - gx C - u, x u, x u C n cos λ n + D n sin λ n sin nπ L x n C n L c L u x, u,, u, L fx sin nπ L x dx, D n cnπ u, x fx, L gx sin nπ L x dx u, x gx 46

47 3. D lamber u, x u, x {fx + c + fx c} + c x+c x c gξ dξ u, x fx, u, x gx fx, gx [ L, L] L f C - g C - u, x {fx + c + fx c} + c x+c x c gξ dξ 3.7 u, x C n cos λ n + D n sin λ n sin nπ L x n C n cos λ n sin nπ L x + D n sin λ n sin nπ L x n λ n c nπ L n n 3.6 C n, D n C n sin nπ L x + λ n + sin nπ L x λ n + D n cos nπ L x + λ n + cos nπ L x λ n C n sin nπ L x + c + sin nπ L x c + D n cos nπ L x + c + cos nπ L x c L + cnπ n n L fη sin nπ L η dη sin nπ L x + c + sin nπ L x c L gη sin nπ L η dη cos nπ L x + c + cos nπ L x c.3 fx + c + fx c + n cl L gη sin nπ x+c L η dη sin nπ L ξ dξ x c.8.3 fx + c + fx c + x+c x c n cl L gη sin nπ L η dη sin nπ L ξ dξ fx + c + fx c + c u, x fx + c + fx c + c x+c x c x+c x c gξ dξ gξ dξ; f C - g C - u, x {fx + c + fx c} + c 47 x+c x c gξ dξ

48 hx d dx x a hξ dξ hx u c {f x + c + f x c} + c g x + c g x c; u x {f x + c + f x c} + c g x + c g x c; u u c x u, x {fx + fx} fx; u, x c f x + c f x c + c c{gx + c + gx c} gx f, g [ L, L] u, fc + f c + c c c u, L fl + c + fl c + c gξ dξ ; L+c L c gξ dξ D Alember u, x fx fx c fx + c u, x c L fx + c fx c fx c fx + c u, x D Alember y x + c, s x c,, x s, y 3.36, u x u y y x + u s s x u y + u s, u u y y + u s u s c y u s, y x, y c, s x ; s c u x u x y + u s u y y x + u s y s u + x y s y x + u s s x u y + u s y + u s ; 48

49 u c u y u u y c s y + u s u y c s y y s + u s s c u y c u s y + c u s s u c u x 4c u y s u y φy i.e. u y s us, y ψs 3.36 u, x u y φy y x+c φξ dξ + ψs φξ dξ + ψx c 3.37 fx u, x x gx u, x cφx cψ x φξ dξ + ψx φx {f x + c gx} ψ x {f x c gx} 3.37 u, x x+c x+c {f ξ + c f ξ dξ + x c gξ} dξ + x c fx + c + fx c + c u, x fx + c + fx c + c f ξ dξ + x+c x c x+c x c gξ dξ {f ξ gξ} dξ c x+c gξ dξ c x c gξ dξ f ; gξ dξ c f C - g C [, L] u c u x 3.38 u,, u, L, 3.39 u, x fx, x 3.4 Sep,,3 u, x GF x 3.38 G c G F x k k 3.4 F x 49

50 G, F x G kc G, F x kf x 3.4 Sep F F L G G λ n cnπ L F n x sin nπ L x, n N, k nπ L cnπ G L cnπ G n C n e L u n, x C n sin nπ L xe λ n 3.43 u, x u, x C n sin nπ L xe λ n n fx u, x C n sin nπ L x C n L L n fx sin nπ L x dx u, x C n sin nπ L x e λ n C n L n L fx sin nπ L x dx, λ n cnπ L , x u, x u, x, x Wiersrass u, x u C n λ n sin nπ L x e λ n ; n u x nπ nπ C n sin L L x e λ C n λ n sin nπ L x e λ n C n λ n e λ n ; n n nπ C nπ n sin L L x e λ n C n nπ L n n n e x e x n n e λ n x n n! e x e x k!, x >, k N 3.45 xk 5

51 k C n b f n, λ n cnπ L C n λ n e λ n n n C n nπ L n Parseval C n λ n L cπ n b f n n L b f cπ n n 4 < ; n n e λ n C n nπ L L λ 4 n n c b f π n n n L c b f π n n 4 < x fx C -.3 R, 3.4 R, n n u c u x u, x fx, u, x : c π u, x c π x ξ fξ exp 4c lim u, x fx + dξ 3.46 ξ exp x 4c, > hea kernel 3.46 > u c u x u Fu, ω u, xe ixω dx x π u, x 6. u F Fu;.3 F u x iω Fu ω Fu A ω Fu, ω π Fu cω Fu Fu, ω A ω e cω A ω u, x e ixω dx π 5 fx e ixω dx Ffω

52 Fu, ω Ffωe c ω.4 x F e c ω c e 4c..3 5 u, x F Fu, ω F Ffωe c ω f F e c ω f π π x c e 4c u, x c x ξ fξ exp π 4c dξ u, x c ξ exp x π 4c c x c ξ exp x π 4c c ξ exp x π 4c 3 c ξ exp x π 4c + exp c x ξ x ξ π 4c 4c c ξ x ξ exp x π 4c 4c ; c x c ξ exp x π 4c c { } x c ξ x ξ exp x π 4c c { } c c ξ x ξ exp x π 4c c c { c ξ x ξ exp x π 4c 4c } 3.48 c x ξ fξ exp π 4c dξ fξ c ξ exp x π 4c dξ; x c x ξ fξ exp π 4c dξ c fξ x ξ exp π x 4c dξ 3.49 > fξ x ξ λ c ξ exp x π 4c dξ fξ x ξ λ c ξ x ξ exp x π 4c 4c dξ M x ξ λ c 4c x ξ π x ξ 4 4c + dξ M c 4 π x ξ λ x ξ + 8c λ dξ, ; x ξ 4 x ξ fξ exp x 4c dξ x ξ λ 5

53 M M x ξ λ x ξ λ x ξ λ f fξ M { fξ x ξ x ξ exp 4c 4c 4 } c dξ 4c { x ξ x ξ 4 4c 4 + } c dξ 4 x ξ + 8c λ dξ, x ξ 4 exp x ξ 4c u, x lim u, x fx + µ σ πσ µ x, σ c exp x µ σ dx u, x fx c π c π c π π π η λ + π c π x ξ exp 4c dξ x ξ fξ exp 4c dξ fx c π x ξ fξ fx exp 4c dξ fξ fx exp x ξ 4c dξ fx + cη fx exp η dξ fx + cη fx exp η dξ η >λ fx + cη fx exp η dξ x ξ exp 4c dξ η ξ x c f fx + cη fx ϵ fx + cη fx M M π 3.5 η >λ exp η π η λ ϵ + M π ϵ exp η η >λ exp η dξ M dξ + π η >λ λ dξ lim u, x fx + c π exp x ξ 4c exp η dξ 3.48 fξ δξ y 4.7 u, x c π x ξ δξ y exp 4c 53 dξ

54 c y exp x π 4c y x c π exp x y 4c u, x c x ξ fξ exp π 4c ξ fξ dξ 3.4 u x + u y 3.5 ux, y R C : ux, y x,y C fx, y unx, y u gx, y n x,y C C u C n n n n, n n n, n R C u n n u x + n u y u C n R b u fx f, fa u R u u a ux, y F xgy F F G G k, 54 k

55 u, y F Gy, ua, y F agy F, F a F F kf F, F a F n x sin nπ a x, k nπ a G nπ G a G Gy C n e nπ a y + D n e nπ a y ux, F xg G C n + D n. Gy C n e nπ a y e nπ a y C n sinh nπ a y C n sinh λ n y λ n nπ a u n x, y C n sin λ n x sinh λ n y ux, b fx C n sinh λ n b a a ux, y fx C n sin λ n x sinh λ n y 3.5 n C n sin λ n x sinh λ n b n fx sin λ n x dx, i.e. C n ux, y C n sin λ n x sinh λ n y n C n a sinh λ n b a a sinh λ n b fx sin λ n x dx a fx sin λ n x dx u n x, y ux, y fx C 3 - u, y, ua, y, ux, b fx f fa ; f u x, u,, y f C n a sinh λ n b a a u x a, u y a, 3.5 fx sin λ n x dx 55

56 ux, y x n a f x cos λ n x dx aλ n sinh λ n b a aλ f x sin λ n x dx n sinh λ n b a aλ 3 f 3 x cos λ n x dx; n sinh λ n b 3 C n λ 3 n sinh λ n b af n 3.53 u x λ nc n sin λ n x sinh λ n y; n λ nc n sin λ n x sinh λ n y λ n C n sinh λ n y n λ n C n sinh λ n b sinh x, x > sinh λ n y sinh λ n b n a f 3 n 3.53 λ n n a f 3 n < Cauchy-Schwarz Parseval λ n n n ux, y y u y λ nc n sin λ n x sinh λ n y n fx C 3 - ux, y ux, y fx C 3 - ux, y C n sin λ n x sinh λ n y n C n a sinh λ n b a fx sin λ n x dx : 6 u u c x + u y u u c x + u y 3 u 3 56

57 4 4. RLC L L di d + Ri + q v 4. C i i dq q v d R L C R v v k i k, k, v av + bv i Φ : CR C R C i ai + bi 4. Φv i, i.e. v i 4.3 CR R C R R C - 4. Φav + bv ai + bi, 4.4 Φ v Φ i Φ v k k v Φv k k Φv Tempered disribuion 4. SR { } φx : lim x xα φ β x, α, β N {} 4.6 p α,β φ sup + x α φ β x 4.7 x R SR {p α,β : α, β N {}} SR 4. T : SR C T aφ + bψ at φ + bt ψ, φ, ψ SR, a, b C 4.8 {φ m } SR : lim m p α,β φ m, α, β N {} lim m T φ m 4.9 T Tempered disribuion SR Tes funcion space SR 57

58 4.8 T 4.9 T T φ < T, φ > 4.3 R fx C R α N {} C α, N α : f α x C α + x Nα 4. f C, N : fx C + x N 4. fx x N. 4.4 f 4. T f : SR C T f φ T f 4.5 f, g T f φ fx φx dx fxφxdx 4. C + x N p N+,φ + x Cp N+, φ Cp N+, φ C + x N φx dx N+ dx + x dx + x dx πcp N+,φ lim p α,βφ m α, β lim T φ m m m T f φ T g φ φ SR fxφxdx gxφxdx fx gxφxdx fx gx fx gx f T f SR SR 4.6 T i, T SR T i T lim i T i T 4.7 T φ φ T i φ T φ φ SR 4.3 T φ φ p, φ 58

59 T Dirac T δx T φ δxφxdx φ δx δx x δx x, x > n f n x n n x, < x n n x + n, n x < lim T f n φ lim f n xφx φ n n lim f nx δx n C 4.8 ϕx exp x, x < C, x ϕx C R, ϕx, ϕ h x h ϕ x h ϕ h x, x h ϕ h x 3 ϕ h xdx R ϕ h x φx SR R ϕ h xφxdx φ 3 R R x exp ϕxdx x ϕ h x φx φ dx ϕ h x φx φ dx x h φ x ε, δ : x δ φx φ < ε h δ x h lim h ϕ h x δx 4.7 {T fn },3 ε ϕ h xdx ε ϕ h x, h 5,, T dn T dx d n T dx n, φ n < T, φ n >, φ SR < T, φ > T φ 59

60 dn T dx n p α,β φ n p α,β+n φ lim p α,βφ k lim p α,βφ n k k k d n T lim k dx n, φ k lim k n T, φ n k dn T dx n dn T dx n 4. fx dtf dx, φ fxφ xdx [fxφ x] + dt f dx T f f xφxdx T f, φ ; f xφxdx [fxφ x] 4. Heaviside x > Ux x x < dtu dx, φ Uxφ xdx dt U dx δx φ xdx [φx] φ 4. T f φ SR T Ť T, φ T, φ 4.4 Ť, φ T, ˇφ 4.5 f T f T, φ T, f φ T f f x f x T f, φ T, f φ 4.7 6

61 .,. is α φ β s ix β φ α x ix β φ α xe ixs dx π ix β φ α xe ixs dx x β φ α x dx π π + x α φ β s + x β+ φ α x π + x dx p β+,α φ π { } sup s α φ β s : s R p α,β φ α αc r x r φ β s r α, β N {} α π αc r p β+,rφ r lim p α,βφ k lim p α,β φ k, k k + x dx π π p β+,α φ; π p β+,αφ π α αc r p β+,r φ r T, φk T, φ k k f 4. x α f φ β β x xα βc k f k xφ β k β x x α βc k f k x φ β k x k k β x α βc k C k + x N k φ β k x f k x C k + x N k k β βc k C k + x Nk+α φ β k x k α β p α,β f φ αc r βc k C k p Nk +r,β kφ r k p Nk +r,β kφ p Nk +α,β kφ, r α β βc k C k p Nk +α,β kφ k α αc r r β βc k C k p Nk +α,β kφ k β α βc k C k p Nk +α,β kφ k α αc r α r α, β N {} lim p α,βφ k lim p α,βf φ k, f T, φ k T, f φ k k k k f, g f T g T f g 3 T f lim p α,βφ k lim p α,βf φ k k k 6

62 f, g < T g f, ψ > g fxφxdx gy gy gyfx ydyφxdx fx yφxdxdy f y xφxdxdy gyf φydy < T g f, φ >; T g f T g f 4.3 T f T f π T f 4.8 F f π f xe ix dx π fxe ix dx f φ SR f φ FF f φ F πf f φ πf f φ < T f, φ > < T f, φ >< T, f φ >< T, πf f φ > < π T, f φ >< π T f, φ >; T f π T f 4.4 T, S SR, a, b C FaT + bs aft + bft 4.9 FT n is n FT 4. d n ds n FT F ixn T 4. F FT T a R δx π, δx a e ia 4.3 π < δx, φ > δx φxdx φ π 6 φxdx < T π, φ >;

63 δx T π π < δx a, φ > δx a φxdx φa e iax φxdx < T π π e ias, φ >; δx a T π e e ias ias π 4.6 F πδ, Fe iax πδx a 4.4 F π F πδ φxdx π < Fe iax, φ > φxe ix dx φxe ix dx πφ < πδ, φ >; e iax φxdx π φa < πδx a, φ >; Fe iax πδx a 4.7 T x < T x, φ > {T x x R} T xdx < T xdx, φ > < T x, φ > dx T xdx 4.8 T f δ x fxdx 4.5 φ SR δ x fxdx, φ f δ x fxdx T f δ x fxdx δ x fx, φ dx δ x, φ fxdx fxφxdx < T f, φ >; Φ : CR C R Φ : SR SR RLC L di d + Ri + q v 4.6 C 63

64 i, v, q SR h : Φδ 4.7 h h δ SR φ δ x φxdx δ x δ x Φφ Φ δ x φxdx n Φ lim δ xk φx k x k x k lim n Φ lim n k n k n δ xk φx k x k x k k n Φ δ xk φx k x k x k Φ δ x φxdx h xφxdx φ h; Φ lim Φ Φ δ x h x Φφ φ h 4.8 v e iω Φ e iω Hiωe iω 4.9 Hiω e iω ω e iω Hiω arg Hiω 4.9 Hiω h Hiω πĥω φ e iω Φφ φsh sds Φe iω e iω h sds e iω s hsds e iωs hsds e iω πĥωeiω Hiω πĥω 4.4 f, Lfs F s fe s d 4.3 f s C Lfs f Rs > s s C s 64

65 s p {s C : Rs > s } Lfs F s Lfs Lfs, Rs α f g F s f L F s f Bromwich 4. Bromwich x fx Lfz s s > s fx + + fx πi s+i s i F ze zx dz 4.3 fx x z s + ix Lfs + ix fe s+ix d fe s e ix d πfe s x Lfs + ix π π Lfs + ixe ix dx π e s Lfs + ixe ix dx fe s f Lfs + ixe s+ix dx π x z s + ix f πi s+i s i π fe s xe ix dx πfe s ; Lfze z dz 4. λ R Le λ s L n s n s n e s d L n s n! s n+ e λ e s d Le λ s s λ, ] [ n e s s n e s d + [ e e λ s λ s d λ s Rs > λ 65 n s n e s d n! s n e s d n! s n+ ; ], Rs λ > ; s λ

66 3 e iλ + e iλ Lcos λs L { L e iλ s + L e iλ s } { s iλ + } s s + iλ s + λ ; s Lcos λs s + λ λ Lsin λs s + λ 4 Lδs δe s d, Lδs 4. f, g f, g λ, µ, F s Lfs Laf + bgs alfs + blgs Lfλ s λ F λ 3 Lf λ e λs F s 4 Lf + λ e λs F s 5 Le µ f F s µ λ e s fd n 6 Lf sf s f, Lf n s n F s f r s n r 7 L f F s, L n f F n s 8 L fτdτ s F s f 9 L s F σdσ f g r fτg τdτ f g Lf gs LfsLgs Le µ f 6 Lf s e µ fe s d e µ fe s µ d F s µ f e s d [ fe s] + sfe s d 66

67 f + sf s, lim fe s ; Lf s sf s f Lf s slf s f sslfs f f s Lfs sf f 8 lim L fτdτ s Lf gs fτdτ e s f ge s d [ fτdτ e s d ] fτdτ e s + s s fτg τdτ e s d fe s d s F s fτg τe s dτd ω + σ, σ τ ω τ, σ τ ω, σ, τ τ σ τ ω fτg τe s dτd fσgωe ω+σs ω, σ, τ fσgωe ω+σs dσdω fσe σs dσ Lf gs LfsLgs dσdω gωe ωs dω LfsLgs; RLC L di d + Ri + q v C i, q, v < i q v di LL + RLi + Lq Lv d C q i q iτdτ, q 68 LsF s i + RF s + F s Lvs C s F sls + R + Li Lvs Cs 67

68 F s Lis F s F s Ls + R + Cs Lvs + i F s Ls + R + Cs s Ls Lvs + Rs + C Li Ls + R + Cs Hs Ls + R + Cs L s Ls + Rs + C Lis F s Lw vs i w v wτv τdτ 4.3 Duhamel 4.6 i i w Hs s Ls + Rs + C w wτv τdτ 4.34 s Ls + Rs + C L s Ls + Rs + C w Ls + Rs + C α, β Ls + Rs + C Ls αs β, α β Ls αs β α β α Lα β s α β ; s β w L α L Ls αs β Lα β s α αe α βe β Lα β i α, β CR > 4L L, R, C α, β w αe α βe β Lα β ii α, β CR < 4L Rα R <, β α L w αe α βe β Lα β αe α αe α Lα α Rα, Iα α L β s β LIα Iαeα 68

69 Ls + Rs + C Ls α α R L <, CR 4L Ls α L s α + α s α ; w L Ls α L + L α L s α s α e α + e α L lim w Hs 4.5 w Hs Hiω h w Φ 4.3 τ < v τ Φv i v δ h Φδ Hiω Φe iω wτv τdτ wτv τdτ wτδ τdτ w; h w 4.35 wτe iω τ dτ wτe iωτ dτ e iω Lwτiωe iω Hs siω e iω ; Hiω Hs siω 4.6 Hiω Hs s iω Hiω Hs siω 4.36 Hs w h L Hs h

70 f, > s C Lfs F s fe s d 5. f Laplace F s f s C Lfs 5. s F s f L L F f L well-defined f, > Ls Ls s [ e e s s d s ] e s lim s s ; 5. Rs > Rs < lim e s + Lfs 5. a C Le a s Ra s < lim e a s e a e s d [ e a s lim a s ] e a s e s d lim e a s a s ; Ra s <, i.e. Ra < Rs s a Le a s, Ra < Rs 5.3 s a 5.3 f, g, > a, b C Lfs Lgs s Laf + bgs alfs + blgs, 5.4 a R e a + e a Lcosh as L s a + s s + a s a ; s Lcosh as s a a < Rs 5.4 e a e a Lsinh as L s a a s + a s a ; a Lsinh as s a a < Rs 5.5 a < Rs a < Rs a < Rs 7

71 5.5 a R a R Le ia s s ia s + ia s + a, Lcos as Le a fs Ria < Rs; s s + a, < Rs, Lsin as a s, < Rs a 5.6 a R 5.7 a, b R e a fe s d Le a fs Lfs a F s a fe s a d F s a Le a s a cos bs s a + b, b Lea sin bs s a, + b Rs > a 5.7 Le a s a cosh bs s a b, b Lea sinh bs s a, b Rs > a + b 5.8 Γ- Γx x e d 5.9 x > x C Rx > x > lim n e, Γx n N x e d [ x e ] + x x e d lim x e + x Γx x Γx ; Γx x Γx 5. n N Γ Γn n Γn n n Γ n n n! Γn n! 5. n + Γ n + 5. Γ Γ e d Γ π x e x x dx 5.8 a R s τ L a a e s d τ s e x dx π; x a e τ s dτ s a+ 7 τ a e τ dτ Γa + s a+ ; 5.

72 L a Γa + s a L n n! s n a > n N L n e a s n! s a n+ 5. U U U a > > a U, U a a < < a LU as U ae s d LU as e as s Ue s a d Dirac δx φ 4. δxφxdx φ e s a d e as s ; δx δx x δx x, x > n f n x n n x, < x n n x + n, n x < lim f n xφx φ n lim f nx δx 4. n δ a δx aφxdx φa Lδ as δ ae s d e as d U a δ a 4. d 7

73 5. Lfs s 5. f, > Lfs Rs > Rs s Lfs G e sτ fτ dτ lim G Lfs ε >, N : > N G Ls < ε G Ls G Ls < ε G < ε + Ls, > N G [, N] L G M L + ε + Lfs > : G M > 5.5 Lfs e s f d e s s e s f d e s s e sτ fτ dτ d [ ] e s s G + s s lim e s s G G + s s Lfs + + s s s s Lfs s s e s s G d e s s G d e s s G d Rs > Rs e s s G d; 5.6 e s s G d e s s G d e s s G d Rs > Rs Lfs e Rs Rs M M d Rs Rs e s f d 5. α inf{rs; Lfs } s Lfs α α f 5. Rs > α s Lfs 73

74 Rs > α s y fe s d < 5.3 β inf{rs; L f s } fe s d <, Rs > Rs α β x β f α β 5.4 f e, > e e e s d e s e s 4 d e s 4 e s d e s 4 e s d e > e, > e s 4 max{ s,} e d + s, α 5.5 f, > f Me k, > M >, k R Rs > k s C Lfs f Me k, > fe s d M f e Rs d Me k e Rs d e Rs k d M ; Rs > k Rs k, f β k β l, M > : f Me k, > l [, ] f f 5. Rs > k Lfs 5.7 n N lim n e f [, n f k > L n fs β β 74

75 5.8 Lerch [, ] f, g Rs > s s C Lfs Lgs s R f g, s < s R f Lfs f G Lf gs Lfs Lgs, s > s e sτ fτ dτ G, Lfs s s G Lfs e s s G d s s + n, n N s s e s s G d, Rs > s n e n G d, n N 5.8 x e log x, e n x n, d dx x n x n G log x x dx; x n G log x dx, n N 5.9 G log x [, ] x G Lfs Wiersrass 6. ε > max{ G log x px : x [, ]} < ε 5. n px px a k x k 5.9 pxg log x dx G log xg log x dx 5. ε k k n n a k x k G log x dx a k x k G log x dx ; k G log x pxg log x dx + pxg log x dx G log x pxg log x dx + pxg log x dx G log x px G log x dx + G log x dx 75

76 ε G log xg log x dx G log x G log x, x [, ] G G e sτ fτ dτ e s f f 5.8 fs F s Lfs L F s f well defined β < f Lfs f 5.9 Bromwich f, β < σ > β f πi σ+i σ i Lfse s ds σ+it πi lim Lfse s ds T σ it σ > β e σ f d < Lfσ + iω e σ+iω f d e iω e σ f d f [,, f e σ f d e σ f, Lfσ + iω e σ f d < e iω e σ f d π Fe σ fω e σ f. σ + iω s e σ f π π Fe σ fωe iω dω π Lfσ + iωe iω dω; f Lfσ + iωe σ+iω dω π f πi σ+i σ i 5. a, b R Le a cos b Lfse s ds σ+it πi lim Lfse s ds T σ it s a s a + b e a cos b σ+i s a πi σ i s a + b es ds, σ > a

77 f, g f, g, > λ, µ, F s Lfs α f Laf + bgs alfs + blgs Lfλs s λ F λ 3 Lf λs e λs F s f λ, < λ 4 Lf + λs e λs F s 5 Le µ fs F s µ 6 Rs > max{α, α, } Rs > max{α, α,, α n, } λ e s fd Lf s sf s f n Lf n s s n F s f r s n r α, α,, α n f, f,, f n 7 Rs > max{α, } r L fs F s, L n fs F n s F s, Rs > max{α, } 8 Rs > max{α, } L fτdτ s s F s f 9 lim f β β < s > β f L s F σdσ f g s fτg τdτ f g Rs > max{β, β } β, β f, g 5. λ τ Lf gs LfsLgs 5. 3 f f λ < λ λ τ 77

78 4 f Lf + λs 6 lim fe s f f + fe s fe s + λ f + λe s d λ fe s λ d fe s d e sλ e λs F s f τ dτ λ e s fd f τ dτ e s f + Lf s, s > max{α, } Lf s Lf s sf s f f e s d [ fe s] + sfe s d lim fe s f + sf s f + sf s; n Lf n s s n F s f r s n r r Lf s slf s f sslfs f f s Lfs sf f 8 A lim A lim L fτdτ s fτdτ e s fτdτ e s d [ fτdτ e s s s lim fτdτ e s, s > max{α, } s > b > max{α, } b τ fτ dτ e s e bτ e bτ fτ dτ e s { [ e bτ τ ] e bx fx dx b e s b e bx fx dx b ] + s fe s d fτdτ e s + s F s s F s e bτ τ e bτ τ e bx fx dx τ Lfb M > : τ } e bx fx dx dτ e s e bx fx dx dτ e s 5. e bx fx dx < M, τ >

79 5. s > b s > b > max{α, } τ τ e bτ e bx fx dx dτ e s e bτ e bx fx dx dτ e s e bτ M dτ e s M b eb e s M b e s b e s 7 f 5. fe s, fe s Rs > d ds F s d ds f Rs > fe s d f e s d L fs f d ds e s d f B B α f Rs > α Rs > b > α b fe s d e s b e b f d fe s d 5.4 [ e s b ] e bτ fτ dτ + [ s be s b e s b ] e bτ fτ dτ d 5.3 G β e bτ fτ dτ 5. LGs b, LGs b lim e s b G + s blgs b LGs b lim n e c, c > s blgs b LGs b fe s d Rs > α f 5.7 G d ds Lfs Lfs s s e s s G d e sτ fτ dτ, s > α G e s s G d + s s e s s G d 79

80 LGs s s s LGs s 8 s s L s s L Gτ dτ s s s s LGs s Gτ dτ G s s G s s L s s L τgτ dτ s s τe sτ fτ dτ s s L e s fs s B 8 L fs 5 n L n f F n s 9 f β β < s s fe σ d dσ s f f e σ dσ d Fubini Lf gs s F σ dσ s f ge s d ω + σ, σ τ ω τ, σ τ ω, σ, τ s e σ dσ d fe σ d dσ f f e s d L s fτg τdτ e s d τ s f e s d < fe σ dσ d σ τ fτg τe s dτd ω Rs > max{β, β } fτg τe s dτd fσgωe ω+σs ω, σ, τ fσgωe ω+σs dσdω fσe σs dσ 8 dσdω gωe ωs dω LfsLgs;

81 Lf gs LfsLgs F s 5. λ R 7 Le λ s d ds Leλ s d ds s λ L cos λ d d s Lcos λ ds ds s + λ s λ ss 3λ s + λ 8 L cos λτ dτ s Lcos λ s s s + λ s + λ cos λτ dτ λ sin L cos λτ dτ L λ sin s + λ sin 9 lim sin <, > β sin [ ] L s Lsin σ dσ σ π + Tan s s µs + λ Lf f λ, µ R Le µ s µ, L sin λ λ f s s + λ e s µs + λ L µ λ sin λ e µx λ sin λ x dx λ I s e µx e iλ x dx λ I e iλ e µ iλx dx λ I e iλ eµ iλ µ iλ e µ λ I e iλ eµ cos λ µ λ sin λ µ iλ µ + λ 5.4 Lfs F s L F s f F s f F s qs ps, α C s α n α ps L n n! sn+ e α n n! L s α n 5.3 L s L s e 8

82 s 3 L s s + 5 s s + 5 s ± i A s + i + B s i s 3 s s + 5 A + B, A i + B + i 3 A + i, B i s 3 L s L s + 5 +i i s + i + s i + i e+i + i e i e cos sin s 3 s s s + 5 s + s 3 s L s L s + 5 s + L s + e cos e sin e cos sin 3 s s + s + A s + Bs + C s + s + As + s + + s Bs + C s s + s + A + Bs + A B + Cs + A C s s + s + A + B, A B + C, A C s s + s + A 5, B 3 5, C 5 L s s + s + 5 L 5 s + 3 s L s e L s + 5 s + + L + s e e cos + e sin L log + ω s log d ds log + ω s + ω s ω s 3 ω s + ω s 8

83 Lf log + ω s s s + ω s d f L ds log + ω s L s s + ω s cos ω f 5.5 s + a a > a Lsin a s + a L s + a a sin aτ sin a τ dτ a [ ] sin aτ a cos a + a + cos ω; cos a + cos aτ dτ a cos a + sin a a s s + a a > a s Lsin a s, Lcos a + a s + a L s s + a sin aτ cos a τ dτ sin a + sin aτ dτ a a [ ] cos aτ sin a sin a a a a , s + a n, s s + a n 5.9 β f σ > β F s {s C : Rs > β} ABC C R lim R e s F s ds C R 5.5 s i s R θ A σ+i σ i e s F sds CABC e s F s ds πi CABC k Res[e s F s, s i ] i B s k O s β σ {s i : i,, k} CABC Res[e s F s, s i ] s s i e s F s C e s F s ds e s F s ds + e s F s ds πi CABC πi C R πi CA 83

84 5.5 R e s F s ds + πi C R πi f σ+it πi lim Lfse s ds T σ it {s i : i,, n} {s C : Rs < σ} σ+ir σ ir Lfse s ds n Res[e s F s, s i ] 5.7 β f σ > β f F s lim e s F s ds R C R f L F s i n Res[e s F s, s i ] {s i : i,, n} {s C : Rs < σ} C R F s 5.8 i lim max{ F s : s R} 5.6 R lim e s F s ds R C R F s qs ps, qs ps deg p > deg q lim max{ F s : s R} R qs, ps s C R C R s σ + Re iθ, π θ 3π σ + Re iθ Re iθ σ R σ 5.6 R : R > R : max{ F s : s R σ} < ε R > R F σ + Re iθ < ε 3π e s F s ds e σ+reiθ F σ + Re iθ ire iθ dθ C R π ireσ 3π Re σ e Reiθ F σ + Re iθ e iθ dθ Re σ θ π ρ Re σ ε Re σ ε π 3π π e R cos θ dθ 3π e R sin ρ dρ Re σ ε e R sin ρ dρ π 3π π e Reiθ F σ + Re iθ e iθ dθ e R cos θ F σ + Re iθ dθ 84

85 sin ρ > π ρ, ρ π Re σ [ ε e R π ρ dρ Re σ ε π R ρ] π e π R lim e s F s ds R C R F s qs, deg p > deg q ps qs lim, s ps eσ π e R ε < eσ πε lim max{ F s : s R} R 5.9 e s s + a, a > 3 s + a s ±ia gz z z n d n Res[g, z ] lim z z n! dz n gzz z n d e s s ia 3 ds s + a 3 d e s ds s + ia 3 n uv n nc r u r v n r r e s s + ia 3 6e s s + ia 4 + e s s + ia 5 ; d e s s ia 3 ds s + a 3 e ia ia 3 6e ia ia 4 + e ia ia 5 sia i e ia a 3 6e ia a 4 ie ia a 5 ; a a d e s s + ia 3 ds s + a 3 i e ia a 3 6e ia a 4 + ie ia a 5 ; s ia [ e s s ia 3 ] [ e s Res s + a 3, ia s ia 3 ] + Res s + a 3, ia a 3 sin a 6a 4 cos a + a 5 sin a 5.7 L s + a 3 a 3 sin a 6a 4 cos a + a 5 sin a 85

86 5.5 x + ax + bx r, x α, x β Lx + alx + blx Lr Rs Xs Lxs Lx slx x sxs α, Lx s Lx sx x s Xs sα β Xs s Xs sα β + asxs α + bxs Rs Xs Rs αs + a + β s + + as + b s + as + b x L Xs L Rs αs + a + β s + L + as + b s + as + b Hs s +as+b, x W Hs r x + ax + bx r 5.3 x 4. RLC L di d + Ri + q v C i, q, v < i q v di LL + RLi + Lq Lv d C q i q iτdτ, q 68 LsF s i + RF s + F s Lvs C s F sls + R + Li Lvs Cs 86

87 F s Lis F s F s Ls + R + Cs Lvs + i Ls + R + Cs Li s F s Ls Lvs + Rs + C L s Ls + Rs + C w Lis F s Lw vs i w v wτv τdτ 5.3 Duhamel 4.6 i i w Hs wτv τdτ 5.7 s Ls + Rs + C RLC L s Ls + Rs + C w Ls + Rs + C α, β Ls + Rs + C Ls αs β, α β Ls αs β α β α Lα β s α β ; s β w L α L Ls αs β Lα β s α αe α βe β Lα β i α, β CR > 4L L, R, C α, β w αe α βe β Lα β ii α, β CR < 4L Rα R <, β α L w αe α βe β Lα β αe α αe α Lα α Rα, Iα α L β s β LIα Iαeα 87

88 Ls + Rs + C Ls α α R L <, CR 4L Ls α L s α + α s α ; w L Ls α L + L α L s α s α e α + e α L lim w Hs 5.33 x + x cos, x 3, x 4 Xs Lxs 5.34 Lx + Lx Lcos s s Xs s Xs s + s + Xs s s + + 3s + 4 s Xs s + + 3s + 4 s + d s ds s s s + d x L s ds s + 3L + s + 4L + s + sin + 3 cos + 4 sin x + Xs 5.35 xτ sin τ dτ Xs Lx L + L xτ sin τ dτ + LxsLsin s s s + Xs s + ; Xs s + s 4 s + s 4 L Xs x + 4x + 5x δ, x, x 3 88

89 s Xs s 3 + 4sXs + 5Xs Lδ e s Xs 3 s + 4s + 5Xs 3 + e s Xs 3 s e s s x 3e sin + e sin U x + 4x + 8x δ, x, x s X sx x + 4sX x + 8X s X s + 4sX 4 + 8X s + 4s + 8X s + 7 X s + 7 s + 4s + 8 s + s s + + x L s + s L s + + e cos + 5 sin x e cos + 5 sin 5.8 : x e cos + 5 sin ; x e cos + 5 sin + e sin + 5 cos e 3 cos 7 sin ; x e 3 cos 7 sin + e 6 sin 4 cos e cos + 8 sin x + 4x + 8x x + 4x + 8x δ x, x 3 x, x

90 5.37 { x + y cos, x, y x + y s sxs + Y s s, Xs + sy s + Xs Lxs, Y s Lys x L s + Xs s +, Y s s s + s sin, y L s cos + 9

91 6 6., Wiersrass 6. [a, b] [c, d] fx, y C - d b b f fx, ydx x, ydx 6. dy y a a a b a < λ, λ < b λ a f b x, y y dx + f x, y y dx 6. λ λ b, λ a, y [c, d] 6. b c a f x, ydxdy y b a f x, ydydx c y b a fx, fx, cdx b a, b y a f y x, dx d d b fx, y + y dx fx, y dx fx, y + y fx, y y f y f y 6. λ > λ : λ ε + λ x, y + θ y f y x, y + θ y f y x >λ f y a f x, y y dx < ε, x, y + θ y f y f y x, y [ λ, λ] δ > : y y < δ ε + λε, y < δ d fx, ydx lim dy y fx, dx f x, ydx y b a f x, ydx y x, ydx < θ < x, y dx x, y dx y [c, d] fx, dx b a fx, cdx f y x, y f x, y y < ε fx, y + y dx fx, y dx y f x, ydx y : 9

92 6. Wiersrass [, ] fx ε > px max{ fx px : x [, ]} < ε 6.3 P [, ] [, ] C[, ] [, ] P [, ] C[, ] T n x cosncos x n Čebyšev cos nθ + i sin nθ cos θ + i sin θ n n cos nθ R nc k i k sin k θ cos n k θ k x cos θ [ n ] m [ n ] m T n x cosncos x nc m m sin m θ cos n m θ nc m m cos θ m θ cos n m θ [ n ] m n T n x n nc m x n m x m f C[, ] [ π, π] fx gθ fcos θ, π < θ π cos θ gθ g b n cos nθ, n N {} σ n θ n d n cos nθ k d n C.36 σ n θ gθ ε > n gθ d k cos kθ < ε k θ [ π, π] n θ Cos x n fx d k cos kcos x < ε x [, ] n px px k n fx d k T k x < ε x [, ] k n d k T k x k 9

93 As, Bs a n, b n c n CW T f a, b δx , 7 D n x d n T dx fa fa f g , 66, 77 Ffs, ˆfs F fs, ˇfs f # Ĝ Γx Hiω Hs hx K n x L a, b Lfs, F s , 7 L F s , 7 SR p α,β P n ψ j,k SR σ n x S n x T f U < s, > B Bessel Bromwich Bromwich C Cauchy-Schwarz , 34 convoluion C r D D lamber Duhamel , 87 F Fubbini L Lerch P Parseval Plancherel R RLC W Weiersrass Wiersrass Wilbraham-Gibbs n n Tempered disribuion Γ Gibbs , , ,

94 FFT , , , , , , , , , , , 66, Čebyšev Tes funcion space , , , , hea kernel Inversion Formula ,