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Topics In Math

by: Otilia Murray I

Topics In Math MAT 280

Otilia Murray I
GPA 3.88

Albert Schwarz

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Albert Schwarz
Class Notes
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This 2 page Class Notes was uploaded by Otilia Murray I on Tuesday September 8, 2015. The Class Notes belongs to MAT 280 at University of California - Davis taught by Albert Schwarz in Fall. Since its upload, it has received 18 views. For similar materials see /class/187389/mat-280-university-of-california-davis in Mathematics (M) at University of California - Davis.

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Date Created: 09/08/15
MAT 280 Applied amp Computational Harmonic Analysis Supplementary Notes 11 by Naoki Saito The Generalized Functions o The generalized functions have more singular behavior than functions thus the name generalized functions and are always de ned as linear functionals on the dual space Thus before we discuss the generalized functions we need to know the following De nition Let 3C be a vector space over say 1 A linear map from SC to C is called a linear functional on 3C lf 3C is a normed vector space then the space LOC C of bounded linear functionals on 3C is called the dual space and denoted by 39 or 35 Examples The dual of LPUR 1 lt p lt 00 is DIOR where 11 14 1 These numbers are called conjugate exponents In particular L2 is self dual Similarly the dual of the sequence space KHZ is 12quot Holder s Inequality Let p and q are conjugate exponents Then for any f 6 LP 9 6 L97 we have Hng1 3 HfHPHgHQ As you can see the CauchySchwarz inequality is a special version of this with p q 12 The proof is a great exercise The Riesz Representation Theorem Suppose p and q are conjugate exponents with 1 lt p lt 00 Then for each linear functional tp 6 LP there exists 9 E Lq such that f due for all f 6 LP In other words is isometrically isomorphic to L I o The more singular the class of the generalized functions the more regular its dual We now de ne the Schwartz class 8 E FOUR SupmeR nk3 f lt 007for any lot 6 N which are very smooth and decay faster than any polynomial at in nity ie a very nice class of functions An example The Gaussian g1c e r Then we consider the dual 8 You can imagine that members of this class can be very singular or spiky This dual space is called the tempered distributions Being as a linear functional each member of 8 acts on the Schwartz functions More precisely if F E 8 and 4 E 8 then the value of F at 4 F is a linear map from 8 to Cl is denoted as F7 4 f dun An example the Dirac delta function 6a E 8 is de ned as 64 In other words 0C 6z1c4gt1c dun 40 oc For any F E 8 and any 4 E 8 we can de ne the following operations Differentiation 3 F74gt 1 39F3 4gt This can be shown by integration by parts An example 6 4 6 4 4gt 0 Convolution F gt1 4gt1c Ruthi where y 4gty An example 6 gt1 Fourier transform F7 4 177 3 w w An example F 6 then 64 6 4 This essentially shows that Mg 3 1 Using the translation operator we can also have Squot6 a e gwi and 339e2 im 6 a 0 De nition A tempered distribution F on IR is called periodic with period A if F 7 739 mp F 7 4 for all f E 8 A sequence of tempered distributions FM is said to converge temperately to a tempered distribution F if Fmtf gt Ffgt as 77 gt 00 for all f E 8 See that all these operations and de nitions are now moved to the nice spouses of F Theorem If F is a periodic tempered distribution then F can be expanded in a temperately con vergent Fourier series skeQWimA ie 1774 heQWik 4 for an 6 E 8 Moreover the coef cients 04k satisfy 04k 3 C1 for some C N Z 0 Conversely if 0 is any sequence satisfying this estimate the series 04k 6279mm converges temperately to a periodic tempered distribution De ne the Shah function or comb function III10 Mm The facts about this function 1 Since this is a periodic tempered distribution we can expand it into the temperately convergent Fourier series III zz 6279mm Note that 04k 3 for all k E Z 2 lmMO lmAg itfo 56 Using the Shah function and its Fourier transform we can see that the Fourier transform of the Fourier series of a periodic function on A27 142 as follows GO GO ft N gakemmm 2 Wag 1 ie line spectrum discrete As you can see as A gets large we are doing the ner sampling in the frequency domain ie f e L2A2 142 L f e was gt1 convolution multiplication Home L 1A11111A it L it Periodization with period A Discretization with rate 1 and scaling with factor 1 I Periodization of a function with compact support ltgt Discretization in frequency domain with amplitude rescaling For the details of the facts in these notes see 1 Chap 9 2 Chap 9 3 Chap 1 References 1 G B FOLLAND Fourier Analysis and Its Applications Wadsworth amp BrooksCole 1992 2 7 Real Analysis Modern Techniques and Their Applications John Wiley amp Sons Inc 2nd ed 1999 3 E M STEIN AND G WEISS Introduction to Fourier Analysis on Euclidean Spaces Princeton Univ Press 1971


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