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By Yitzhak Katznelson

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Hint: The distance between two orthogonal vectors of norm 1 is ")2. 3. Yt'is complete if, and only if, the set of finite linear combinations of {'Pn} is dense in £' . 4. ' lib; apply the Cauchy-Schwarz inequality to the last identity, 5. Assume IEU(T) and /(n) = aClnl-k). r t. Hint: IJ(O)/ :::; 1I/1Ic.. I. Fourier Series on T 31 6. ABSOLUTELY CONVERGENT FOURIER SERIES We shall study absolutely convergent Fourier series in some detail later on: here we mention only some elementary facts.

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.

14) holds for every Borel set E. 42 An Introduction to Harmonic Analysis being discrete. lEM(T). l#)({O}). l * p#)({O}) = o. The discrete part of a measure J1 can be "recovered" from its Fourier-Stieltjes series. l({r}) = lim N-+Cf) PROOF: · () Th e f unctIOns TN t = I '!.... ? {t(n)e +::;v iii - 1 lnt • t D () 1 1 L... ; e- int eint 2"N-l N t--r ~ iN + + -N are bounded by 1 and tend to zero uniformly outside any neighborhood of t = r. 15) lim = O. l5). Corollary (Wiener): Let /1 E M(T).

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