On some approximation properties of a generalized Fejér integral (Q2911312)

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 \((1)_K\).NEWLINENEWLINEThe 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, NEWLINE\[NEWLINE \sigma_{K_\alpha}\left(f\right)\left(g\right)\equiv\left(f*V_{K_\alpha}\right)\left(g\right)= \int_G f\left(h\right)V_{K_\alpha}\left(h^{-1}g\right)\,dh, NEWLINE\]NEWLINE where NEWLINE\[NEWLINE V_{K_\alpha}\left(g\right)= \left(\text{mes}\,K_\alpha\right)^{-1}\left(\left(\hat 1\right)_{K_\alpha}\left(g\right)\right)^2, \quad K_\alpha\in U_{\hat G}. NEWLINE\]NEWLINE The main results of the article are the following two statements:NEWLINENEWLINETheorem 1. Let \(\left\{K_\alpha\right\}\) be a sequence in \(U_{\hat G}\) satisfying NEWLINE\[NEWLINE \lim_{\alpha\to\infty}\frac{\text{mes}\,\left(TK_\alpha\right)}{\text{mes}\,\left(K_\alpha\right)}=1, NEWLINE\]NEWLINE 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 \(\alpha\to\infty\).NEWLINENEWLINETheorem 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 NEWLINE\[NEWLINE \limsup_{\alpha\in I}\left\{\text{mes}\,\left(K_\alpha\right)-\text{mes}\,\left(K_\alpha\cap\left(\chi K_\alpha\right)\right)\right\} \neq 0, NEWLINE\]NEWLINE for any fixed \(\chi\in\hat G\). Then the condition NEWLINE\[NEWLINE \left\|f-\sigma_{K_\alpha}\left(f\right)\right\|_{L^p\left(G\right)}=o\left(\text{mes}\,\left(K_\alpha\right)\right),\quad \alpha\to\infty, NEWLINE\]NEWLINE implies \(f\left(g\right)=0\) a.e. on \(G\).NEWLINENEWLINESeveral 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.NEWLINENEWLINEThe 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.