On the degree of approximation of a function by the partial sums of its Fourier series

Abstract

Abstract. When / is a lit periodic function with rth order fractional derivative, r > 0, of /»-bounded variation, Golubov has obtained estimates of the degree of approximation of/, in the L* norm, q > p, by the partial sums of its Fourier series. Here we consider the analogous problem for functions whose fractional derivatives are of ^-bounded variation and obtain estimates of the degree of approximation.in an Orlicz space norm. In a similar manner we shall extend various results that he obtained on degree of approximation in the sup norm. §1Let/be a real function of period 2tt. If the rth fractional derivative, r > 0, is of bounded/?-variation, Golubov [3] has obtained estimates of the degree of approximation of/, in the sup and Lq norms, q > /?, by the partial sums of the Fourier series offWe consider the analogous problem for functions whose fractional derivatives are of ^-bounded variation. In 2.1 we extend Salem's theorem of uniform convergence of Fourier series. In 2.2 we obtain estimates on the degree of approximation in an Orlicz space norm.In this chapter we shall review the background material which we shall require. In § 1 we concern ourselves with Orlicz spaces, §2 will present basic information on cp-bounded variation, and §3 contains the relevant information on the fractional integral and derivative.