Approximate calculation of sums. I: Bounds for the zeros of Gram polynomials (Q2927837)

The authors interest is to approximate sums of the form \(\sum_{j=1}^{N}F(j)\) for large \(N\). The method bases on a Gaussian type quadrature formula. The idea is to get NEWLINE\[NEWLINE \sum_{j=1}^{N}F(x_j)\rightarrow \int_{-1}^{+1}F(x)dx\approx \sum_{k=1}^{n} a_k F(x_k), NEWLINE\]NEWLINE where \(x_j=-1+\frac{2j-1}{N}, \;\;1\leq j\leq N, \;\;n\ll N \).NEWLINENEWLINEFor this end, they consider the Gram polynomials as well as the corresponding inner product. The explicit formula of the above polynomials, shows that their zeros converge to the zeros of Legendre polynomials as \(N \rightarrow \infty\). The authors used then, the quadratic decomposition of orthogonal polynomials to give lower and upper bounds for the zeros of Gram polynomials.NEWLINENEWLINESome illustrative examples show how this approximation of sums over 1000 and 10000 works in some situations.