An implementation of Christoffel’s theorem in the theory of orthogonal polynomials

An algorithm for the construction of the polynomials associated with the weight function w(t)P(t) from those associated with w(t) is given for the case when P(t) is a polynomial which is nonnegative in the interval of orthogonality. The relation of the algorithm to the LR algorithm is also discussed. Introduction. In several problems of numerical analysis, particularly in the construction of Gaussian quadrature rules with preassigned nodes, the following problem arises. Given the orthogonal polynomials {pj(t)j associated with a weight function w(t) on the interval (a, b) and a polynomial P(t) of degree m which is nonnegative on the interval (a, b), construct the orthogonal polynomials { q j(t)} associated with the weight function P(t)w(t) on the same interval. A theorem of Christoffel [1] gives an explicit expression for the polynomial qQ(t) in the form Pn(t) Pn+ 1(t) ..P., .(t) Pn(zl) Pn+l(Zl) ..P.+. (z1) (l1) q.(t)P(t) = P.(Z2) P.+l(Z2) P.+. (Z2) Pn(Zm) Pn+i(Zm) ... Pn+m(Zm) where the Zk, k = I(I)m, are the roots of P(t). If some root, zi, is of multiplicity j, then the corresponding rows of (1) are replaced by the derivatives of order 0, 1, ... * j 1 of the polynomials pr(t), r n(l)n + m, at t = zi. For numerical calculations, Eq. (1) is very clumsy to use, even for simple evaluation of the polynomial qn(t) at a point, unless m is small. Often, the three-term recurrence relation (2) pj(t) = (tbj)pj_(t) -gjpj2(t), j = 1, 2, * with p0(t) = 1 and p1(t) = 0, is known because it is more convenient to obtain [4], [5] and to use [5], [6]. The main result of this paper is to prove a theorem, equivalent to Christoffel's, which states an explicit construction of the three-term recurrence relation (3) qj(t) = (t Bj)q1_1(t) Gjqq2(t), j = 1, 2, . . Received March 9, 1970, revised May 13, 1970. AMS 1969 subject classifications. Primary 6510; Secondary 6525, 6550, 3345.