On Birkhoff quadrature formulas

Author:
A. K. Varma

Journal:
Proc. Amer. Math. Soc. **97** (1986), 38-40

MSC:
Primary 41A55

MathSciNet review:
831383

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Abstract: 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 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.

**[1]**J. Balázs and P. Turán,*Notes on interpolation. II. Explicit formulae*, Acta Math. Acad. Sci. Hungar.**8**(1957), 201–215. MR**0088590****[2]**-,*Notes on interpolation*. III, Acta Math. Hungar.**9**(1958), 195-213.**[3]**Vladimir Ivanovich Krylov,*Approximate calculation of integrals*, Translated by Arthur H. Stroud, The Macmillan Co., New York-London, 1962, 1962. MR**0144464****[4]**George G. Lorentz, Kurt Jetter, and Sherman D. Riemenschneider,*Birkhoff interpolation*, Encyclopedia of Mathematics and its Applications, vol. 19, Addison-Wesley Publishing Co., Reading, Mass., 1983. MR**680938****[5]**Paul Nevai and A. K. Varma,*A new quadrature formula associated with the ultraspherical polynomials*, J. Approx. Theory**50**(1987), no. 2, 133–140. MR**888295**, 10.1016/0021-9045(87)90004-9**[6]**P. Turán,*On some open problems of approximation theory*, J. Approx. Theory**29**(1980), no. 1, 23–85. P. Turán memorial volume; Translated from the Hungarian by P. Szüsz. MR**595512**, 10.1016/0021-9045(80)90138-0**[7]**A. K. Varma,*On some open problems of P. Turán concerning Birkhoff interpolation*, Trans. Amer. Math. Soc.**274**(1982), no. 2, 797–808. MR**675080**, 10.1090/S0002-9947-1982-0675080-2

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DOI:
https://doi.org/10.1090/S0002-9939-1986-0831383-9

Article copyright:
© Copyright 1986
American Mathematical Society