A characterization of positive quadrature formulae

Author:
Yuan Xu

Journal:
Math. Comp. **62** (1994), 703-718

MSC:
Primary 41A55; Secondary 65D32

MathSciNet review:
1223234

Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: A positive quadrature formula with *n* nodes which is exact for polynomials of degree , is based on the zeros of certain quasi-orthogonal polynomials of degree *n*. We show that the quasi-orthogonal polynomials that lead to the positive quadrature formulae can all be expressed as characteristic polynomials of a symmetric tridiagonal matrix with positive subdiagonal entries. As a consequence, for a fixed *n*, every positive quadrature formula is a Gaussian quadrature formula for some nonnegative measure.

**[1]**T. S. Chihara,*An introduction to orthogonal polynomials*, Gordon and Breach Science Publishers, New York-London-Paris, 1978. Mathematics and its Applications, Vol. 13. MR**0481884****[2]**Erik A. van Doorn,*Representations and bounds for zeros of orthogonal polynomials and eigenvalues of sign-symmetric tri-diagonal matrices*, J. Approx. Theory**51**(1987), no. 3, 254–266. MR**913621**, 10.1016/0021-9045(87)90038-4**[3]**G. Freud,*Orthogonal polynomials*, Pergamon Press, Oxford, 1971.**[4]**Walter Gautschi,*Advances in Chebyshev quadrature*, Numerical analysis (Proc. 6th Biennial Dundee Conf., Univ. Dundee, Dundee, 1975) Springer, Berlin, 1976, pp. 100–121. Lecture Notes in Math., Vol. 506. MR**0468117****[5]**J. Gilewicz and E. Leopold,*On the sharpness of results in the theory of location of zeros of polynomials defined by three-term recurrence relations*, Orthogonal polynomials and applications (Bar-le-Duc, 1984) Lecture Notes in Math., vol. 1171, Springer, Berlin, 1985, pp. 259–266. MR**838992**, 10.1007/BFb0076552**[6]**C. A. Micchelli and T. J. Rivlin,*Numerical integration rules near Gaussian quadrature*, Israel J. Math.**16**(1973), 287–299. MR**0366003****[7]**Franz Peherstorfer,*Characterization of positive quadrature formulas*, SIAM J. Math. Anal.**12**(1981), no. 6, 935–942. MR**635246**, 10.1137/0512079**[8]**Franz Peherstorfer,*Characterization of quadrature formula. II*, SIAM J. Math. Anal.**15**(1984), no. 5, 1021–1030. MR**755862**, 10.1137/0515079**[9]**H. J. Schmid,*A note on positive quadrature rules*, Rocky Mountain J. Math.**19**(1989), no. 1, 395–404. Constructive Function Theory—86 Conference (Edmonton, AB, 1986). MR**1016190**, 10.1216/RMJ-1989-19-1-395**[10]**G. Szegő,*Orthogonal polynomials*, 4th ed., Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, RI, 1975.**[11]**Yuan Xu,*Weight functions for Chebyshev quadrature*, Math. Comp.**53**(1989), no. 187, 297–302. MR**970703**, 10.1090/S0025-5718-1989-0970703-2**[12]**Yuan Xu,*Quasi-orthogonal polynomials, quadrature, and interpolation*, J. Math. Anal. Appl.**182**(1994), no. 3, 779–799. MR**1272153**, 10.1006/jmaa.1994.1121

Retrieve articles in *Mathematics of Computation*
with MSC:
41A55,
65D32

Retrieve articles in all journals with MSC: 41A55, 65D32

Additional Information

DOI:
https://doi.org/10.1090/S0025-5718-1994-1223234-0

Keywords:
Quadrature formula,
quasi-orthogonal polynomial,
tridiagonal matrix

Article copyright:
© Copyright 1994
American Mathematical Society