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By Yitzhak Katznelson

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Deduce that an(J1,r:) ~ 0 if /l is positive. 3. Show that a trigonometric series L ane int such that L~Naneint ~ 0 for all Nand f ET is a Fourier-Sticltjes series of a positive measure. 4. L 11(n)\q l/q ) 5 Ilflium L lalll P < cc, a" and Ilg IILq(T) ~ (L lanIP)l/p. A~suming this, show that if {an} is a numerical sequence satisfying there exists a function gE Lq(T) such that g(n) = 5. The elements of the dual space of Cm(T) are called distributions of order m on T. 2"'(T) = (Cm(T»* the space of distributions of order m on T.

Then 30 An Introduction to Harmonic Analysis 2 J(n)/nt IV (b) / = lim ill the L2(T) norm. N-co -N (c) Given any sequence {a n}:'= _ co of complex numbers satisfying =j(n). IIa l2 < 00. gE L2(T). (lI)g(n). II We denote by t 2 the space of sequences {an}:: co such that an 12 < 00. 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.

3) lim 2 ')' (1 N-oo and since n~ l~n] ~ n ~- - ) 101 = i(F(O) - teO»~ N +1 n 0, the theorem follows. An Introduction to Harmonic Analysis 24 I Corollary: If an> 0, anln =00, then L~la .. sinnt is not a Fourier series. Hence there exist trigonometric series with coefficients tending to zero which are not Fourier series. 1, the senes Ln=2 1-= L/n/;;'22-) ogn is a ogn - -'-I . senes . , 2 Its . conjugate . lnl~2 e IS not. 3 We turn now to some simple results about the order of magnitude of Fourier coefficients of functions satisfying various smoothness conditions.

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