On a theorem of Feichtinger and Weisz

Abstract. The so-called \(\theta\)-summation is well-known in the theory of approximation. A remarkable result gives a necessary and sufficient condition for uniformly or \(L^1\)-norm convergence of \(\theta\)-means if \(\theta\) has compact support. This condition is nothing else but the integrability of the (trigonometric) Fourier transform of \(\theta\). Later this theorem was improved by Feichtinger and Weisz showing the same result for \(\theta\)'s belonging to a suitable Wiener algebra \(W(C,{\ell}_1)\). If \(\theta\) is compactly supported then \(\theta\in W(C,{\ell}_1)\) holds evidently but there are functions \(\theta\in W(C,{\ell}_1)\) with unbounded support. In this work we extend the statement of Feichtinger and Weisz. To this end a new space \(S(C,{\ell}_1)\) of functions will be constructed for which we prove the validity of the integrability condition. A simple consideration leads to the proper inclusion \(W(C,{\ell}_1)\subset S(C,{\ell}_1)\).