Reconstruction of operators on classes of functions with constraints in integral norms (Q1386608)

The author considers the following problem of reconstruction of operators on classes of functions: Let an operator \(U\) which should be defined act from the set \(M\) into the normed space \(Z\) and the information \(\mathcal{P}\) about elements from \(M\) is known, \(\{S\}\) is the set of all univalent mappings from \(\mathcal{P}(M)\) into \(Z\), which are called methods of reconstruction of the operator \(U\) by the information \(\mathcal{P}\). The reconstruction error is defined by the equality \[ R(M,U,\mathcal{P},S)=\sup (\sup \left\| U(x)-S(y)\right\| :y\in \mathcal{P} (x):x\in M) \] The problem is to find the quantity \( E(M;U;\mathcal{P})=\inf (R(M,U,\mathcal{P},S):S\in \{S\}) \) and a method \(S^{\ast }\in \{S\}\) that realizes the infimum. The mapping \(S^{\ast }\) is called an optimal method of reconstruction of \(U\) on \(\;M\) by an information \(\mathcal{P}.\) The author solves some concrete problem of reconstruction of integral operators on a class of functions.