An Introduction to Harmonic Analysis (second corrected by Yitzhak Katznelson

By Yitzhak Katznelson

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Lemma: Let {M, • IISN-s~112 02 N = ~2 a"1',, I M+l L f{ = 10 , 1 2-+ 0 as M-+oo. }(' be a Hilbert space. }('. ). ) - 2: o"(tp,,,f) +L la,,12 = Ilfl2- L la,,12.... }('. }(' write aa. 3) The family {1'.. } in the statement of Bessel's inequality need not be finite nor even countable.

As in the case of summability, problems of convergence "in norms" are usually easier than those of pointwise convergence. summable functions was pIoved only recently (L. Carleson 1965). 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.

Yf' we say that fE ff is orthogolla I to E iff is orthogonal to every element of E. A set E c £ is ortliogona I if any two vectors in E are orthogonal to each other. Y{' is an orthonormal system if it is orthogonal and the norm of each vector in E is one. that is. if. whenever f. gEE.

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An Introduction to Harmonic Analysis (second corrected by Yitzhak Katznelson
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