On trigonometric interpolation

Abstract

which in case of derivatives take the place of (1). (ArP(t)/(At)r is the rth divided difference formed with the constant increment At =2wx/(2n+1)), and we shall show that here, as in Marcinkiewicz's case, the exponent 2 can be replaced by any p ?1. As an application, we shall give bounds for the p-norms of the derivatives of interpolating polynomials in terms of the corresponding quantities for the interpolated function. This, in turn, allows us to derive some information as to how the order of approximation yielded by trigonometric interpolation depends upon the integrability properties of the derivatives of the interpolated function. To simplify the notation, we shall introduce, in connection with any system of m equidistant points modulo 27r, tl, t2, . . ., tmi the step functions wm(t) defined as follows: