APPROXIMATE CALCULATION OF SUMS I: BOUNDS FOR THE ZEROS OF GRAM POLYNOMIALS~*

摘要

Let N be a positive integer and x_j be N equidistant points. We propose an algorithmic approach for approximate calculation of sums of the form ∑_(j=1)~N F(x_j). The method is based on the Gaussian type quadrature formula for sums, where gn,k(N) are the zeros of the so-called Gram polynomials. This allows the calculation of sums with very large number of terms N to be reduced to sums with a much smaller number of summands n. The first task in constructing such a formula is to calculate its nodes gn,k(N). In this paper we obtain precise lower and upper bounds for gn,k(N). Numerical experiments show that the estimates for the zeros gn,k(N) are very sharp and that the proposed method for calculation of sums is efficient.