On the Convergence of Trigonometrical Interpolation at Equi-Distant Knots

Abstract

where f(x) is the function interpolated. Naturally the question arises, under what conditions the polynomial T5(x) converges to the function f(x) when n is increasing. Papers dealing with this problem generally make some assumptions concerning the nature of the function f(x) in the whole segment [0, 27r]. Such a treatment makes it possible to apply the results established in the theory of the best approximation of functions and leads to different tests of uniform convergence of T0 (x) towards f(x). These tests prove very similar to those of uniform convergence of Fourier-series. Yet there is also a second possibility of treatment which it is natural to call a local one. Namely, we may examine properties of f(x) near a fixed point X0 which will cause the convergence of Tn(xo) towards f(xo). I know of only one paper treating the problem in this way, namely the classic memoir by Vallee-Poussin.' Vallee-Poussin has proved that the equality