Wiener type theorems for Jacobi series with nonnegative coefficients (Q2880659)

Let \(\alpha\geq\beta\geq -1/2\) be fixed, \(\mathbb P\) be the class of all functions \(f\in L^1\) such that the corresponding Jacobi coefficients \(\hat{f}(k)=\hat{f}(\alpha,\beta;k)\) are nonnegative for all \(k\in\mathbb N_0.\) If \(Y\) is any subspace of \(L^1,\) then \(Y_{loc}\) is the class of all \(f\in L^1\) with the following property: there exists a nondegenerate interval \(I\subset[-1,1]\) with \(1\in I\) such that for any \(\phi\in C^{\infty}\) supported on \(I,\) \(f\phi\in Y.\) A subspace \(X\subset L^1\) is called solid if \(f,g\in L^1,\) \(|\hat{f}(k)|\leq \hat{g}(k)\) for all \(k\in\mathbb N_0\) and any \(g\in X\) imply that \(f\in X.\)NEWLINENEWLINEFirstly the authors prove that for any solid space \(X,\) NEWLINE\[NEWLINEX_{loc}\cap\mathbb P=X\cap\mathbb PNEWLINE\]NEWLINE (an analogue of Wiener's theorem for Jacobi expansions). Next they prove that the Jacobi expansion of a function in \(\mathbb P\) that is locally essentially bounded near 1 is absolutely and uniformly convergent on \([-1,1].\) A refinement of this result is given where Besov spaces and their local analogues are considered rather than mere continuity.