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

Example text

9) tend to zero. 4 Theorem (Principle of localization): Let feLI(T) alld assume that f vanishes ill an open interval I. Theil Sn(!. t) converge to zero for tel, and the convergence is uniform on closed subintervals of I. 3. If 10 is a closed subinterval of I, the . f(t - to)cos t/2 . functIOns 'P'o(l) = . 9) corresponding to f(t - to), to e To, tend to zero uniformly. ~ The principle of localization is often stated as follows: let f. g e Ll(T) and assume that f(t) = get) in some neighborhood of a point to.

3 yields Theorem (Dints te~t): Let fE LI(T). If I f(t + to) - f(to) I •( _ I t , Idt < 00 then -EXERCISES FOR SECTION 2 1. Show that if a sequence of continuous functions on some interval is unbounded on a dense subset of the interval, then it is bounded only on a set of the first category. 3» converges only on a set of the first category. 2. Show that for every given (denumerable) sequence {tn} there exists a continuous function whose Fourier series diverges at every tn' 3; Let g be the 2rr-periodic function defined by: g(O) = 0, g(t) = t - 1r for 0 < I < 21r.

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

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