On approximation of functions of two variables by sums of the form \(\varphi (x) + \psi (y)\) in \(L_ p\) spaces \((1\leq p\leq\infty )\)
The author discusses some questions related to the problem of approximating functions \(f(x,y)\in L_p(G)\), \(G\subset\mathbb R^2\), by sums of the form \(f(x)+f(y) \in L_p(G)\). The following three possible versions of the statement of the problem are considered: (1) \(f(x)+g(y)\in L_p(G)\); (2) \(f(x),g(y)\in L_p(G)\); (3) \(f(x)\in L_p(X_G),\;g(y)\in L_p(Y_G)\), where \(X_G\) and \(Y_G\) are projections onto the \(x\)- and \(y\)-axes. According to these versions, the author considers the following three subspaces for \(1\leq p\leq\infty\): \(L^1_p=\{w\in L_p(G)\: w=f(x)+g(y), f,g\) are measurable and finite a.e.\}; \(L^2_p=\{w\in L_p(G)\: w=f(x)+g(y), f,g\in L_p(G)\}\); \(L^3_p=\{w\in L_p(G)\: w=f(x)+g(y), f\in L_p(X_G), g\in L_p(Y_G)\}\). A subspace \(M\) of a Banach space \(Z\) is called proximinal, if for each function \(f\in Z\), there is a best approximation in the subspace \(M\). The main results of the article are the following theorems. Theorem 1. Let \(G\) be an open set in \(\mathbb R^2\). Then the subspace \(L^2_{\infty}(G)\) is proximinal. Theorem 3. Let \(G\) be an open set in \(\mathbb R^2\). Then the subspace \(L^1_1(G)\) is proximinal. The author claims that the statement of Theorem 3 remains valid for the \(L_p(G)\) spaces with \(1<p<\infty\).
scientific article; zbMATH DE number 1046573