The optimal set of the modulus of continuity in the sharp Jackson inequality in the space \(L_ 2\).
To a function \(f\in L_2[-\pi,\pi]\) and a compact set \(Q\subset [-\pi,\pi]\) is denoted the supremum \(w(f,\varphi)=\sup_{t\in Q}\| f(\cdot,+t)-f(\cdot)\|)_{L_2[-\pi,\pi]}\) which is an analog of the modules of continuity. If is denoted by \(K(n,Q)\) the best constant in Jackson's inequality between the best approximation of the function \(f\) by trigonometric polynomials of degree \((n-1)\) in the space \(L_2[-\pi,\pi]\) and the modulus of continuity \(w(f,Q)\) on the strength of a result of Yudin, it is shown that if the measure of the set \(Q\) is less than \(\underline\pi\) then \(K(n,Q)>\frac1{\sqrt 2}\). It was known from results of Chernykh that \(K(n,Q)\geq \frac1{\sqrt2}\) and \(K[n,[0,\frac\pi n]]=\frac1{\sqrt2}\). The strict inequality proved that \(K(n,Q)\geq \frac1{\sqrt2}\) and \(K[n,[0,\frac\pi n]]=\frac1{\sqrt2}\). The strict inequality proved in this paper means that \(K(n,Q)\) as function of the measure of \(Q\) have a strict maximum when the measure of \(Q\) is \(\frac \pi n\) in \([0,\underline\pi]\).
approximation by trigonometric polynomials
scientific article; zbMATH DE number 2177674