where q is a positive integer and f(x) is a real function which, together with its first 2p derivatives, is continuous in the interval [0, 1]. The method involves calculating trapezoidal rule approximationsto 1 f(t) dt with different mesh ratios and combining these results. In the case that f(x) is a periodic function with period 1, this method reduces to a finite form of a method due to Goldberg and Varga [2]. 2. Relations between various operators. It is convenient to introduce condensed notation for various quantities which are used in this paper. The following definitions are mainly notational in character and express familiar functionals in terms of linear operator notation. The Fourier coefficients may be defined as follows. DEFINITION 2.1.