A conjecture of M. Golomb on optimal and nearly-optimal linear approximation

Abstract

In 1964, M. Golomb, in his survey paper on optimal and nearly-optimal linear approximation, presented at the General Motors Conference [3], called attention to an unsolved problem. It is the purpose of this note to solve this problem and at the same time to give a certain extension of the Harsiladze-Lozinskii theorem. The authors are indebted to Professor P. L. Butzer for many helpful discussions and for a critical reading of the manuscript. Let C2n be the space of continuous 27r-periodic functions with Cebysev norm, Un the class of trigonometric polynomials of degree <n,and En[f] = inf{ \\fp\\\p G Un} the error of best approximation of an ƒ G C2ix by elements of IIW for an n G P ={0, 1, 2, • • • } . A sequence {Un}n(E? of bounded linear operators on C27r into C27T is called asymptotically optimal [3] for a given subset Y C C2lT if