The Banach algebra \(A^*\) and its properties
The authors consider the Beurling algebra \[ A^*(R)= \Biggl\{f:|f|_{A^*(R)}= |\widehat f|_{L^*(R)}= \int^\infty_0 \{\text{ess sup}|\widehat f(u)|: x\leq|u|<\infty\} dx<\infty\Biggr\} \] and its analog \(A^*(T)\), where \(R= (-\infty,\infty)\), \(T=[-\pi, \pi)\) and \(\widehat f\) is the Fourier transform of \(f\). They investigate properties of \(A^*(T)\) as an algebra, describe its dual space and establish analogs of Herz and Wiener-Ditkin theorems for it. It is shown, in particular, that \(A^*(T)\) is a regular Banach algebra with trivial radical and its space of maximal ideals coincides with \(T\). Several inclusion results comparing the algebra \(A^*(T)\) with certain Banach spaces of smooth functions are given. The following result detects a close connection of \(A^*(T)\) with problems of summability of the Fourier series at Lebesgue points. Theorem. The linear means \[ (f* d\mu)_N(x)= \int_R f\Biggl({x- u\over N}\Biggr) d\mu(u) \] converge to \(f(x)\) as \(N\to\infty\) at all the Lebesgue points of each \(f\in L_1(T)\) if and only if the measure \(\mu\) is absolutely continuous with respect to Lebesgue measure and \(\mu'\in L^*(R)\).
scientific article; zbMATH DE number 1019260