A result of P?lya states that every sequence of quadrature formulas Qn(f)
with n nodes and positive Cotes numbers converges to the integral I(f) of
a continuous function f provided Qn(f) = I(f) for a space of algebraic
polynomials of certain degree that depends on n. The classical case when the
algebraic degree of precision is the highest possible is well-known and the
quadrature formulas are the Gaussian ones whose nodes coincide with the
zeros of the corresponding orthogonal polynomials and the Cotes (Christoffel)
numbers are expressed in terms of the so-called kernel polynomials. In many
cases it is reasonable to relax the requirement for the highest possible
degree of precision in order to gain the possibility to either approximate
integrals of more specific continuous functions that contain a polynomial
factor or to include additional fixed nodes. The construction of such
quadrature processes is related to quasi-orthogonal polynomials. Given a
sequence {Pn}n?0 of monic orthogonal polynomials and a fixed integer k, we
establish necessary and sufficient conditions so that the quasi-orthogonal
polynomials {Qn}n?0 defined by Qn(x) = Pn(x) + ?k-1,i=1 bi,nPn-i(x), n ?
0, with bi,n ? R, and bk-1,n ? 0 for n ? k-1, also constitute a sequence
of orthogonal polynomials. Therefore we solve the inverse problem for
linearly related orthogonal polynomials. The characterization turns out to
be equivalent to some nice recurrence formulas for the coefficients bi,n. We
employ these results to establish explicit relations between various types
of quadrature rules from the above relations. A number of illustrative
examples are provided.
On the Darboux transformations and sequences of p-orthogonal polynomials