On a conjecture on the degree of approximation of Fourier series

Abstract

Conjecture. Let m be an arbitrary positive integer. If f (x) has ruth derivative f (") (x) on [- ~t, ~], and the Fourier series off (') (x) converges, then the degree of approximation of the Fourier series off (x) to f (x) is equal to z~(l/n'). It has been proved in [2] that the conjecture is true for even positive integer m. In this note, we prove that the conjecture is true for arbitrary positive integer m if the Fourier series of f(') (x) converges uniformly on [- ~, ~]. Theorem. Let m be an arbitrary positive integer. If f (x) has mth derivative f(m)(x) on [- n, 7r], and the Fourier series of f("~)(x) converges uniformly on [- ~, ~], then the degree of approximation of the Fourier series of f (x) to f (x) is uniformly equal to ~(1/n'). P r o o f. We need only prove that the theorem is true for m = 1. For arbitrary m, the theorem can be obtained by induction.