A Sharp Error Bound in Terms of an Averaged Modulus of Smoothness for Fourier Lagrange Coefficients

摘要

This paper discusses the approximation of Fourier coefficients by Fourier Lagrange coefficients. It gives an error bound in terms of an averaged modulus of smoothness. The sharpness of this estimate is shown as an application of a quantitative resonance principle by utilizing the aliasing phenomenon that occurs in the context of discrete Fourier transformation. The scenario is used to compare the averaged modulus with classical uniform and integral moduli of smoothness.