Download e-book for iPad: An Introduction to Harmonic Analysis (second corrected by Yitzhak Katznelson

By Yitzhak Katznelson

ISBN-10: 0486633314

ISBN-13: 9780486633312

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

Example text

R K/ nl- I • Prove that for all nand l sinjtl ~ 1rr t, + I. 4. i(n) = 0 (the Riemann-Lebesgue lemma). In this section we discuss the following three questions: (a) Can the Riemann-Lebesgue lemma be improved to provide a certain rate of vanishing of J( n) as 1n 1-+ OC!? 1). (b) In view of the negative answer to (a), is it true that any sequence {all} which tends to zero as In 1-+ OC! is the sequence of Fourier coefficients of some f E LI(T)? 2). (c) How are properties like bounded ness, continuity, smoothness, etc.

Absolutely convergent sequences) and their" Fourier transforms" which happen to be continuous functions on T. 7 with the roles of T and Z reversed. EXERCISES FOR SECTION 6 l. • , let f,. E A(T) and 1If,. IIA(T) ~ I. Assume that f" converge to I uniformly on T. Show that IEA(T) and II/IIA(T) ~ l. 2. Show that the conditions in exercise I are not sufficient to imply lim,,_oo III- f,. IIA(1) = 0; however, if we add the assumption that II I IIA(T) = lim 3. ~(T) ~ O. ~ Il define Ail) = { 1-a- 11 1 1 for It!

N = max (0, 2N + 1 j It follows that -I I). 5. l E M(T). There exists a unique linear operator on B having the following properties: Jl I JlII (i) (ii) ~ II Jt IIM(T) A Jlf(n) • = fi(n)f(n) for all fE B. (n)e , that is, Jl is completely determined on the polynomials in B. If Jl is bounded it is completely determined. 8) since Jl would then have a unique extension of norm ~"It IIM(T) to all of B. IlM(T)IIJ liB, and since I1J = lim" ... eM(T). Then {{i(n)](n)} is the sequence oj Fourier coefficients oj some Junction in B.

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


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