Generalized polynomial approximation providing the best reference (Q1387386)

Suppose that \(f\) is continuous on a compact set \(Q\subset\mathbb{R}^n\), \(\Delta\) is a compact set in \(\mathbb{R}^n\), \(0\in\Delta\). Suppose that \(t\in\mathbb{R}^n\) and \(\Delta_t:= t+\Delta\subset Q\). The function \[ \varphi(x)= \varphi_t(x)= f(t+ x),\quad x\in\Delta\tag{1} \] is said to be a fragment of the functions \(f\) related to the point \(t\) and compact \(\Delta\). In application, we often meet the problem in which the function \(f(x)\), \(x\in Q\) and \(\varphi(x)\), \(x\in\Delta\) are given, and it is required to find the position \(t\) of the fragment function \(\varphi\), that is to find a value \(t\) for which the condition (1) is satisfied. This problem is called ``reference problem''. For the solution of the reference problem it is natural to take one of the values of \(T= T(t)\) that provides the lower bound \[ \inf\{\| f(x+ T)- \varphi_t(x)\|_\Delta: \Delta_t\subset Q\}, \] where \(\|\cdot\|_\Delta\) is some norm (for example \(\| f\|= \max\{| f(x)|:x\in \Delta\}\). The Euclidean distance \(| t- T(t)|\) is called ``reference error''. In this paper, the author investigates the extremum problem of finding a polynomial approximation for a function of several variables so that this approximation by a fragment of the function provides the best reference. He obtains an upper bound for the modulus of informativity \[ J_\Delta(r, f)= \sup_{Y,T} \{| Y|:\| f(x)- f(x+ T)\|_{\Delta_T}\leq r,\;\Delta_T\subset Q\} \] if \(f(x)\) is a polynomial or rational function.