Localizing the Lozinski-Harshiladze theorem on projections into the space of trigonometric polynomials

Abstract

w 1. We think of periodic continuous functions as functions of the complex variable z = e ~x on the unit circle with Fourier series f(z)~ ~ akz k k ~ ~ and norm ]]f]l = Max ]f(z)l. By a projection T. we mean a linear operator that makes correspond to every such function a trigonometric polynomial of order at most n (i.e. an f with vanishing Fourier coefficients for [k[>n) but leaves each trigonometric polynomial of order =<n invariant, f-~ ~ a k z k is such an operator and the theorem referred to in k= --n