Estimates of weights in Gauss-type quadrature

Abstract

f(x) z P(x)f(x,) P(xi) = 0, (1) ~~~~P'(xi)(x - xX) PI(xi) 560, i = 1 2, ,n, algebraically valid for polynomials f of degree v < n, the degree of P, has a rather limited direct use in polynomial approximation theory. Combined with various restrictions on P to be in a basis of a set of polynomials with suitable properties, it becomes more useful. Let P*(x) be of degree n + 1, so that P*(x) = axPl(x) - bP(x) - cP*(x) for constants a, b, and c, P* representing a polynomial of degree v < n. We set K(x, t) = K(t, x) P*(x)P(t) - P*(t)P(X) a polynomial of degree n in x for each t, so that K(x, t) = aP(x)P(t) + cK*(x, t), K* being defined in terms of P and P* exactly as K is determined by P* and P. In particular, K(x, x) = P(x)P* (x)- P*(x)P'(x); and (1) is modified to become