An Introduction to Harmonic Analysis (second corrected - download pdf or read online

By Yitzhak Katznelson

ISBN-10: 0486633314

ISBN-13: 9780486633312

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Extra resources for An Introduction to Harmonic Analysis (second corrected edition)

Sample text

N=2 - - - converges lor all IE T. iogn 5. FOURIER SERIES OF SQUARE SUMMABLE FU NCTlONS In some respects the greatest success in representing functions by means of their Fourier series happens for square summable functions. The reason is that L2(T) is a Hilbert space. g) = 1 2n f - J(t) g(t) dt , and in this Hilbert space the exponentials form a complete orthogonal system. We start this section with a brief review of the basic properties of orthonormal and complete systems in ab~tract Hilbert space and conclude with the corresponding statements about Fourier series in Ll(T).

Bn. t2 is a Hilbert space. 5 is equivalent to the statement that the correspondence / ....... {f(1I)} is an isometry between L2(T) and f2. (II EXERCISES FOR SECTION 5 1. Let { 'Pn} n= 1 be an orthogonal system in a Hilbert space :R. ff. Show that

W E B. t Assume conversely that f ..... f~ is well defined, hence bounded, in 8. W = f~ - i(2"+1)t(e-1I2.. +1)'l)~ S! which means that II liB is bounded by twice the norm over B of the mapping f -T f~ . 8), I Sn liB = I S! liB, the theorem ~ follows. (T) norm. EXERCISES FOR SECTION I I. Show that the Lebesgue constants L" L" Hint: = 2 4/1t log n = liD" ilL' satisfy + 0(1). )I1 dt In ~-+112 2. L 200 ((t) = iKN/t) 1 does not converge in 1}(T). 3. Let {an} be an even sequence of positive numbers,convex on (0, 00) and vanishing at infinity (cf.

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

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