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By Yitzhak Katznelson

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3). = 0,1, ... , and ~ Bernstein's theorem is sharp; there exist functions in LiPl/2(T) the Fourier series of which does not converge absolutely. A classical example is the Hardy-Littlewood series 00 ein log n n= I n • L ___ e'nt 1, p. 197). 6. (see [28], Vol. I. 3 can be relaxed if we assume that f is of bounded variation. Theorem (Zygmund): Let f be of bounded variation sume f E LipiT) for some IX > O. Then f E A(T). 011 T and as- We refer to [28], Vol. 1, p. 241, for the proof. 5 Remark: There is a change of scene in this section compared with the rest of the chapter.

All that we have to do is show that the operator I -+ I is closed, that is, that if linif. = I and lim!.. g = I. (j) = lim - j sgn (j)l. -+00 1-+ P = tl(O) + ·HI + if) '" L j(j) eiJt o = lU) . II. 'the Convergence of Fourier Series 49 is a well-defined. bounded linear operator on B. Conversely. if the mapping f -TP is well-defined in a space B. then B admits conjuga. tion since 1 = - i(2P - f - 1(0». Theorem: Let B be a homogeneous Banach space on T and assume that for feB and for all n. 7) Then B admits conjugation if, and only if.

E i • t. If/eL1(T) and if the series conjugate to Il(n) einr is the Fourier series of some function g e Ll(T) , we call g the conjugate lunction ofI and denote i! by J This definition is adequate for the purposes of this section; however, it does not define I for all Ie Ll(T) and we shall extend it later. DEFINITION: for every IE B, A space of functions B c Ll(T) admits conjugation if is defined and belongs to B. I If B is a homogeneous Banach space which admits conjugation, then the mapping I -+ lis a bounded linear operator on B.

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