Read e-book online An Introduction to Harmonic Analysis (second corrected PDF

By Yitzhak Katznelson

ISBN-10: 0486633314

ISBN-13: 9780486633312

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Additional info for An Introduction to Harmonic Analysis (second corrected edition)

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3 We turn now to some simple results about the order of magnitude of Fourier coefficients of functions satisfying various smoothness conditions. Theorem: If f E L'(T) is absolutely continuous, then j(n) . = o(l/n). 6 we have fen) /'mann-Lebesgue lemma 1'(n) ~ O. = (1/in)l'(n) and by the Rie- .... oo. 4). '~. 5 Theorem: Inl' Iff is of bounded variation on T, then 25 I. Fourier Series on T < Var(f) IJ( n) I = 2xl n I. (t)dt I 11(n) I = = -~ II 2mn J e-in1df(t) I ~ Var(f~ 2x In I .... t + y) - f(t)l.

4). '~. 5 Theorem: Inl' Iff is of bounded variation on T, then 25 I. Fourier Series on T < Var(f) IJ( n) I = 2xl n I. (t)dt I 11(n) I = = -~ II 2mn J e-in1df(t) I ~ Var(f~ 2x In I .... t + y) - f(t)l. 6) Q(f, h) We clearly have n(f, h) ~ = I f(t + h) - j(n) = ~ 2x IILI . w(f, h). Theorem: For n =F 0, IJ(n) I ~ PROOF: f(t) ~ n(f,,:J. 7. IffELiplT),thenj(lI) = O(n-«). '(T) then I 11(n)lq < OCJ. The case p = 2 will be proved in the following section. The case 1 < p < 2 will be proved in chapter IV.

In the second half of this section we show how various smoothness conditions affect the size of the Fourier coefficients. The effect of square integrability will be discussed in the following section. 1. Theorem: Let {an}~; -co be an even sequellce of nOl/negative numbers tending to zero at infinity. 1) Then there exists a /lonnegative fut/ctioJl IE Ll(T) such that J(II) = an' 1. 1) implies that (an - an+ I) is monotonically decreasing with n, hence lim n(a n 0, - an + 1) = a o - aN - = ,,-+00 and consequently N L n(a n - I + a n + 1 -2a n ) N(a N - aN+ I) n=1 converges to ao as N -).

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An Introduction to Harmonic Analysis (second corrected edition) by Yitzhak Katznelson


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