An Introduction to Harmonic Analysis (second corrected by Yitzhak Katznelson

By Yitzhak Katznelson

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Convergence seems to be closely related to the existence and properties of the so-called conjugate function. In this chapter we give only a temporary incomplete definition of the conjugate function. A proper definition and the study of the basic properties of conjugation are to be found in chapter ITI. ]. 1) Sif) = Sn(f, t) = ~pace on T. As usual we write 2:" l(j) iiI. 2) lim "-00 1/ SnCf) - f liB = O. Our purpose in this section is to characterize the spaces B which have this property. 46 II. The Convergence of Fourier Series 47 We have introduced the operators Sn:f -+ Sin in chapter I.

The mapping J -+ {j(n) }nez of A(T) into [I (the Banach SPace of absolutely convergent sequences) is clearly inl linear and one-to-one. 'ooane converges uniformly on T and, denoting its sum by g, we have an = g(n). It follows that the mapping above is an isomorphism of A(T) onto [I. 1) 00 L IJ(n)l· ~ -00 With this norm A{T) is a Banach space isometric to it is an algebra. Lemma: (I; we now claim Assume that J,gEA(T). 2 Not every continuous function on T has an absolutely convergent Fourier series, and those that have cannott be characterized by smoothness conditions (see exercise 5 of this section).

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. 1. We denote by A{T) the space of (continuous) functions on T having an absolutely convergent Fourier series, that is, the functions J for which L:::'",1J(II) I < 00.

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