Inegrl Trnsform Definiions Funcion Spce funcion spce A funcion spce is liner spce of funcions defined on he sme domins & rnges. Liner Mpping liner mpping Le VF, WF e liner spces over he field F. A mpping f : V W, x f x is liner. x y f x y f x f y x, y V &, F Inegrl Trnsform inegrl rnsform def Le e he funcion spce over liner spce W, he funcion spce over liner spce V. = or The inegrl rnsform from o y he kernel K is liner mpping: : f F f where f & F such h FΩ f ; Ω V d KΩ; f where V & Ω W FΩ is clled he inegrl rnsform of funcion f y he kernel KΩ;. The inverse rnsform from o y he kernel H is liner mpping: : F f F where f & F such h f FΩ; W dω H; Ω FΩ where V & Ω W f is clled he inverse rnsform of FΩ y he kernel H; Ω. Lineriy implies: f g f g F G F G f, g &, Fourier ( idempoen ) Kernel V W & KΩ; H; Ω noe: he kernel of he fourier rnsform is no fourier kernel.
2 InegrlTrnsform.n Convoluion ( Flung / Folding ) convoluion def Le e he funcion spce over liner spce V; = or Le f, g The convoluion ( flung ) f g of f nd g in V is defined s f g C V dτ f Τ gτ where C is some consn. V Fourier Trnsform Kernel KΩ, C e i Ω H, Ω C ' e i Ω C C ' 2 Definiion fourier rnsform def FΩ f ; Ω C d e i Ω f where f is piecewise coninuous, differenile, soluely inegrle ( d f exiss ). Fourier Inegrl Theorem The of his heorem is rher involved. To egin, we oin he Riemnn-Leesgue Lemm for finie inervls; hen exend i o infinie ones. This eges he Loclizion Lemm, s for finie, hen infinie, inervls. Finlly, we rrive he Fourier Inegrl Theorem. Riemnn-Leesgue Lemm Riemnn-Leesgue lemm Le f e piecewise coninuous for d f sin d f cos The elow ssumes f o e coninuous in,. The cse for f piece-wise coninuous is proved y pplying he echnique o ech coninuous segmen individullly. d f sin Τ where sinτ sinτ dτ f Τ sin Τ
InegrlTrnsform.n 2 d f sin d f sin d f sin d f sin d f sin d f sin d f f sin d f sin d f sin d f sin I I 2 I 3 where I d f sin I 2 d f f sin I 3 d f sin Since f is piecewise coninuous, i is ounded, ie. M f M. Using d g d g, sin x we hve I M d M I 3 M Now, y he men vlue heorem, f f f ' s f f N I 2 N where s where N mx f ' Thus 2 d f sin 2 M N Hence: d f sin Proof for he cos cse proceeds in he sme mnner. Wriing e i cos i sin, we hve d f e i
4 InegrlTrnsform.n Corollry Riemnn-Leesgue Corollry Le. f e piecewise coninuous for 2. f is soluely inegrle. ie. d d f sin f exiss. d f cos d f sin d f sin d f sin f is soluely inegrle d f d f sin lim d f sin d f sin lim d f d f sin Loclizion Lemm loclizion lemm Le f ' e piecewise coninuous for d f sin Null 2 f The elow ssumes f ' o e coninuous. The cse for f ' piece-wise coninuous is proved y pplying he echnique o ech coninuous segmen individullly. Le where d f sin f ' is coninuous I 2 I I 2 I f d sin I 2 d f f sin f f is coninuous in, Now: d sin Hence: u du sin u u sin d f 2 du sin u u f 2
InegrlTrnsform.n Corollry Le. f ' e piecewise coninuous for 2. f is soluely inegrle. ie. d d f sin d f x sin 2 f 2 f x f exiss. Proof for he s pr is nlogous o h used in he Riemnn-Leesgue corollry. 2nd pr is oined y simple chnge of vrile. Corollry 2 Le. f ' e piecewise coninuous for 2. f is soluely inegrle. ie. d f sin Null d 2 f f f exiss. d f sin d f sin 2 f Fourier Inegrl Theorem fourier inegrl hm Le. f ' e piecewise coninuous for 2. f is soluely inegrle. ie. d f exiss. 2 f dω dτ f Τ cos Ω Τ f 2 f 2 If f is coninuous. f f 2 dω dω dω dτ f Τ cos Ω Τ dτ e i Ω Τ f Τ dτ e i Ω Τ f Τ
6 InegrlTrnsform.n Now: Using f 2 f dτ f Τ Τ Τ dτ f Τ sin sin Τ Τ ( ) sin Τ dω cos Ω Τ Τ f 2 f dω dτ f Τ cos Ω Τ Inverse Trnsform f FΩ; C ' dω e i Ω FΩ CC ' 2 FIT f 2 dω C ' C ' dτ e i Ω Τ f Τ dω e i Ω C dτ e i Ω Τ f Τ dω e i Ω FΩ Dirc Del Funcion 2 dω e i Ω FIT f 2 f dω Τ dτ e i Ω Τ f Τ dτ f Τ 2 dτ f Τ Τ 2 dω e i Ω Τ dω e i Ω Τ Trnsform of Derivives f n ; Ω i Ω n FΩ
InegrlTrnsform.n Le FΩ f ; Ω C F Ω d f d ; Ω C C f e i Ω d e i Ω f d e i Ω d f d i Ω C i Ω FΩ where f since i is soluely inegrle. The generl cse cn e proved y inducion. d e i Ω f Convoluion ( Flung ) Theorem dτ gτ f Τ dω e i Ω GΩ F Ω where F, G re fourier rnsforms of f, g, respecively Prsevl Relion d g f C ' C dω G Ω F Ω Using f C ' GΩ C dω e i Ω FΩ d e i Ω g d g f C ' C ' C ' C d g dω FΩ dω FΩ GΩ dω e i Ω F Ω d g e i Ω Lplce Trnsform Kernel Ks, e s H, s 2 i e s x, Re s s Definiion Given funcion f of rel vrile. Is Lplce rnsform Fs is defined, if he inegrl exiss, s Fs f ; s d e s f d e s f Θ
8 InegrlTrnsform.n If s, e s f M Fs exiss s s f is hen sid o e of exponenil order. For, e s f f Fs does no exiss if f n n Inverse Trnsform f Fs; Γ i 2 i Γ i ds e s Fs FIT f 2 dω dτ e i Ω Τ f Τ Mellin Trnsform Kernel Ks, s H, s 2 i s x, Re s s Hnkel Trnsform Kernel K n k, r r J n k r H n r, k k J n k r k, r, n inegers References I.N.Sneddon, "The Use of Inegrl Trnsforms", McGrw Hill (72) G.Arfken, "Mhemicl Mehods for Physiciss", 3rd ed., Chp 5. F.W.Byron Jr., R.W.Fuller, "Mhemics of Clssicl & Qunum Physics", Vol II, Addison Wesley, (69,7)