Discrete least squares and quadrature formulas

Author:
M. Wayne Wilson

Journal:
Math. Comp. **24** (1970), 271-282

MSC:
Primary 65.55

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275677-3

MathSciNet review:
0275677

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Abstract | References | Similar Articles | Additional Information

Abstract: The purpose of this paper is two-fold. Firstly, we explore some of the intimate connections between discrete least squares processes and quadratures. Secondly, we present an algorithm to construct Gauss-type integration formulas, and consider briefly the method proposed by Gautschi [2].

**[1]**P. J. Davis,*Interpolation and Approximation*, Blaisdell, Waltham, Mass., 1963. MR**28**#393. MR**0157156 (28:393)****[2]**W. Gautschi, "Construction of Gauss-Christoffel quadrature formulas,"*Math. Comp.*, v. 22, 1968, pp. 251-270. MR**37**#3755. MR**0228171 (37:3755)****[3]**W. Gautschi, "Algorithm, Gaussian quadrature formulas,"*Comm. ACM*, v. 11, 1968, pp. 432-436.**[4]**A. H. Stroud & D. Secrest,*Gaussian Quadrature Formulas*, Prentice-Hall, Englewood Cliffs, N.J., 1966.*MR***34**#2185. MR**0202312 (34:2185)****[5]**G. W. Struble, "Orthogonal polynomials: Variable-signed weight functions,"*Numer. Math.*v. 5, 1963, pp. 88-94. MR**27**#4342. MR**0154393 (27:4342)****[6]**G. Szegö,*Orthogonal Polynomials*, 2nd ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R. I., 1959. MR**21**#5029. MR**0106295 (21:5029)****[7]**A. E. Taylor,*Introduction to Functional Analysis*, Wiley, New York, 1958. MR**20**#5411. MR**0098966 (20:5411)****[8]**V. Tchakaloff, "Formules de cubature méchaniques à coefficients non négatifs,"*Bull. Sci. Math.*, (2)v. 81, 1957, pp. 123-134. MR**20**#1145. MR**0094632 (20:1145)****[9]**M. W. Wilson,*Geometrie Aspects of Quadratures with Non-Negative Weights*, Ph.D. Thesis, Brown University, Providence, R. I., 1969.**[10]**M. W. Wilson, "Necessary and sufficient conditions for equidistant quadrature formulas,"*SIAM J. Numer. Anal.*(To appear.) MR**0282530 (43:8241)****[11]**M. W. Wilson, "A general algorithm for nonnegative quadrature formulas,"*Math. Comp.*, v. 23, 1969, pp. 253-258. MR**0242374 (39:3705)**

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Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1970-0275677-3

Keywords:
Discrete inner product,
discrete least squares,
least squares approximations,
numerical integration,
Gaussian integration,
orthogonal polynomials,
construction of inner products,
construction of orthogonal polynomials

Article copyright:
© Copyright 1970
American Mathematical Society