On Birkhoff Quadrature Formulas

In an earlier work the author has obtained new quadrature formulas (see (1.3)) based on function values and second derivatives on the zeros of nn(i) as defined by (1.2). The proof given earlier was quite long. The object of this paper is to provide a proof of this quadrature formula which is extremely simple and indeed does not even require the use of fundamental polynomials of (0,2) interpolation. Introduction. In [7] the author obtained some new quadrature formulas based on the zeros (1.1) 1 = Xin > X2n > ■ ■ ■ > Xn-i,n > Xn,n = -1 of (1.2) Wnix) = i\-X2)P'n_iix), where Pn_i(x) denotes the Legendre polynomials of degree < n—1. More precisely it was proved that the quadrature formula 2(2n 3) y^1 fjXkn) + nin 2)(2n 1) ¿ (P„_i(a:fcn))2 1 n^il-x2kn)f"jxkn) + nin l)(n 2)(2n 1) ¿J (P„-i(xfcn))2 is exact for all polynomials / of degree at most 2n 1. The interesting feature of this quadrature formula is that it is based on function values and second derivatives on the zeros of n„(x) as defined by (1.2). Moreover the above quadrature formula provides the solution of the open problems XXXVI, XXXVII, XXXVIII and XXXIX raised by P. Turan [6]. Proof of this quadrature formula was obtained by integrating the fundamental polynomials of (0,2) interpolation investigated earlier by J. Balazs and P. Turan [1]. (By the problem of (0,2) interpolation we mean that the values and second derivatives of the interpolatory polynomials are prescribed at the given nodes. For many contributions on Birkhoff quadrature formulas we refer the reader to the interesting monograph of Lorentz et al. [4].) As noted in my paper [7] the integration of the fundamental polynomials of (0,2) interpolation is complicated and runs in many pages. The main object of this paper is to provide a proof of this quadrature formula which is extremely simple and indeed does not even require the use of fundamental polynomials of (0,2) Received by the editors April 2, 1985. 1980 Mathematics Subject Classification. Primary 41A05; Secondary 41A55. ©1986 American Mathematical Society 0002-9939/86 $1.00 + $.25 per page