Ugulava, Duglas On some approximation properties of a generalized Fejér integral. (English) Zbl 1252.43003 Bull. Georgian Natl. Acad. Sci. (N.S.) 6, No. 1, 32-38 (2012). Let \(G\) be a locally compact abelian Hausdorff group, and \(\hat G\) be its dual, i.e., the set of all characters on \(G\). \(U_{\hat G}\) denotes the collection of all symmetric compact sets from \(\hat G\) which are closures of neighborhoods of the unity in \(\hat G\). The product of the sets \(K\) and \(T\) is given by \(KT=\left\{g:\,g=g_1g_2,\,g_1\in K,\,g_2\in T\right\}\). The characteristic function of a set \(K\) is denoted by \((1)_K\).The author considers the following generalized Fejér means for a function from \(L^p\). Let \(I\subset{\mathbb R}^+\) be an ordered unbounded set. Let us consider a generalized sequence of sets \(K_\alpha\in U_{\hat G}\), such that \(K_\alpha\subset K_\beta\) if \(\alpha<\beta\), \(\alpha,\beta\in I\), and \(\cup_{\alpha\in I} K_\alpha=\hat G\). Then, \[ \sigma_{K_\alpha}\left(f\right)\left(g\right)\equiv\left(f*V_{K_\alpha}\right)\left(g\right)= \int_G f\left(h\right)V_{K_\alpha}\left(h^{-1}g\right)\,dh, \] where \[ V_{K_\alpha}\left(g\right)= \left(\text{mes}\,K_\alpha\right)^{-1}\left(\left(\hat 1\right)_{K_\alpha}\left(g\right)\right)^2, \quad K_\alpha\in U_{\hat G}. \] The main results of the article are the following two statements:Theorem 1. Let \(\left\{K_\alpha\right\}\) be a sequence in \(U_{\hat G}\) satisfying \[ \lim_{\alpha\to\infty}\frac{\text{mes}\,\left(TK_\alpha\right)}{\text{mes}\,\left(K_\alpha\right)}=1, \] for all fixed \(T\in U_{\hat G}\), and \(S\subset G\) be a compact set. If a function \(f\in L^\infty\left(G\right)\) is continuous at a neighborhood of \(S\), then \(\sigma_{K_\alpha}\left(f\right)\) converges to \(f\) uniformly on \(S\) as \(\alpha\to\infty\).Theorem 2. Let \(f\in L^p\left(G\right)\), \(1\leq p\leq 2\), and a sequence of sets \(K_\alpha\in U_{\hat G}\) satisfy the condition \[ \limsup_{\alpha\in I}\left\{\text{mes}\,\left(K_\alpha\right)-\text{mes}\,\left(K_\alpha\cap\left(\chi K_\alpha\right)\right)\right\} \neq 0, \] for any fixed \(\chi\in\hat G\). Then the condition \[ \left\|f-\sigma_{K_\alpha}\left(f\right)\right\|_{L^p\left(G\right)}=o\left(\text{mes}\,\left(K_\alpha\right)\right),\quad \alpha\to\infty, \] implies \(f\left(g\right)=0\) a.e. on \(G\).Several important applications of these theorems with concrete groups \(G\) are also given. They lead to results on the convergence of several means of Fourier integrals.The article can be considered as a continuation of the author’s recent work [Georgian Math. J. 19, No. 1, 181–193 (2012; Zbl 1238.41021)]. It should be interesting for specialists in abstract harmonic analysis as well as in Fourier analysis. Reviewer: Alexander Tovstolis (Stillwater) Cited in 1 Document MSC: 43A55 Summability methods on groups, semigroups, etc. 43A15 \(L^p\)-spaces and other function spaces on groups, semigroups, etc. 43A25 Fourier and Fourier-Stieltjes transforms on locally compact and other abelian groups 43A40 Character groups and dual objects 43A70 Analysis on specific locally compact and other abelian groups Keywords:locally compact abelian Hausdorff group; characters; Fourier transform; generalized Fejér integral; convergence of Fourier integral’s means Citations:Zbl 1238.41021 PDFBibTeX XMLCite \textit{D. Ugulava}, Bull. Georgian Natl. Acad. Sci. (N.S.) 6, No. 1, 32--38 (2012; Zbl 1252.43003)