Product-integration rules and their convergence

Abstract

Abstract.An algorithm, based on the use of orthogonal polynomials, for product-integration is outlined. A general discussion on the convergence of such quadrature rules for finite intervals is then given. The paper concludes with five examples for each of which sufficient conditions for convergence of the quadrature rule are given.

Product integration over infinite intervals. I. Rules based on the zeros of Hermite polynomials

“…Because this holds for any polynomial of degree < n -1, the desired result follows on taking p"_, to be the polynomial of best approximation to/in the sense of the norm ||-1| r D 5. Practical Construction of the Rules.…”

“…Transformations of this kind are discussed in [5]. The rule so obtained can be viewed as a product-integration rule for the original interval, based not on polynomials but instead on the functions obtained from the polynomials by the transformation from the finite to the infinite interval.…”

“…Now we write (3)(4)(5)(6)(7) so that k(x) = e~xlK(x), k*(x) = e~x2K*(x), \k*-k\\p+)= \\K*-K\\p-\ It follows from the assumption (2.1) that K G Lp~\ Let us now require that K* be a polynomial. Since the polynomials are dense in the space L<p) (see [13, Lemma 2]), we can choose K* so that the first two terms on the right-hand side of (3.6) are each less than e, where e > 0 is a given arbitrary number.…”

“…Practical Construction of the Rules. The construction is a special case of that given by [5], [8], [15] for product integration based on the zeros of orthogonal polynomials.…”

“…Introduction. In this paper we propose a product-integration method, in the sense of [2], [5], [6], [18]- [22], [25], for evaluating an integral of the form /oo k(x)f(x)dx, -00 where / is a smooth function and A: is a Lebesgue integrable function. Additional conditions on k and/will be imposed later.…”