On the nonclassical approximation method for periodic functions by trigonometric polynomials
A periodic function is approximated in the uniform norm by the linear polynomial means of its Fourier series. The means in question are generated by a multiplier function \(\varphi(x)\) supported in \([-\pi,\pi]\) and such that not only \(\varphi(0)=1\) but, contrary to the classical situation, there are other points at which \(\varphi\) takes the value \(1\). Three theorems are proven in the paper. They have something in common, more precisely, in each of them a necessary and sufficient condition for the rate of approximation to be equivalent either to a certain modulus of smoothness or to a \(K\)-functional is given in terms of the belonging of the ratio of \(1-\varphi(x)\) and a certain function, related to the points where \(\varphi\) is equal to \(1\), to the Wiener algebra of absolutely convergent Fourier integrals. Unfortunately, the paper is not well enough translated, with numerous deviations from standard language and even statements that are misleading in translation.
scientific article; zbMATH DE number 6152303