Block Jacobi matrices and zeros of multivariate orthogonal polynomials
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- by Yuan Xu PDF
- Trans. Amer. Math. Soc. 342 (1994), 855-866 Request permission
Abstract:
A commuting family of symmetric matrices are called the block Jacobi matrices, if they are block tridiagonal. They are related to multivariate orthogonal polynomials. We study their eigenvalues and joint eigenvectors. The joint eigenvalues of the truncated block Jacobi matrices correspond to the common zeros of the multivariate orthogonal polynomials.References
-
N. I. Akhiezer and I. M. Glazman, Theory of linear operators in Hilbert space, Ungar, New York, 1961.
- T. S. Chihara, An introduction to orthogonal polynomials, Mathematics and its Applications, Vol. 13, Gordon and Breach Science Publishers, New York-London-Paris, 1978. MR 0481884
- J. Dombrowski, Orthogonal polynomials and functional analysis, Orthogonal polynomials (Columbus, OH, 1989) NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., vol. 294, Kluwer Acad. Publ., Dordrecht, 1990, pp. 147–161. MR 1100292, DOI 10.1007/978-94-009-0501-6_{7}
- Joanne Dombrowski, Tridiagonal matrix representations of cyclic selfadjoint operators, Pacific J. Math. 114 (1984), no. 2, 325–334. MR 757504
- Tom Koornwinder, Two-variable analogues of the classical orthogonal polynomials, Theory and application of special functions (Proc. Advanced Sem., Math. Res. Center, Univ. Wisconsin, Madison, Wis., 1975) Math. Res. Center, Univ. Wisconsin, Publ. No. 35, Academic Press, New York, 1975, pp. 435–495. MR 0402146 H. Möller, Polynomideale und Kubaturformeln, Thesis, Univ. of Dortmund, 1973.
- H. M. Möller, Kubaturformeln mit minimaler Knotenzahl, Numer. Math. 25 (1975/76), no. 2, 185–200. MR 405815, DOI 10.1007/BF01462272
- C. R. Morrow and T. N. L. Patterson, Construction of algebraic cubature rules using polynomial ideal theory, SIAM J. Numer. Anal. 15 (1978), no. 5, 953–976. MR 507557, DOI 10.1137/0715062 I. P. Mysovskikh, Numerical characteristics of orthogonal polynomials in two variables, Vestnik Leningrad Univ. Math. 3 (1976), 323-332.
- I. P. Mysovskikh, The approximation of multiple integrals by using interpolatory cubature formulae, Quantitative approximation (Proc. Internat. Sympos., Bonn, 1979) Academic Press, New York-London, 1980, pp. 217–243. MR 588184
- Frigyes Riesz and Béla Sz.-Nagy, Functional analysis, Frederick Ungar Publishing Co., New York, 1955. Translated by Leo F. Boron. MR 0071727
- Hans Joachim Schmid, On cubature formulae with a minimum number of knots, Numer. Math. 31 (1978/79), no. 3, 281–297. MR 514598, DOI 10.1007/BF01397880
- Hans Joachim Schmid, Interpolatorische Kubaturformeln, Dissertationes Math. (Rozprawy Mat.) 220 (1983), 122 (German). MR 735919 —, Minimal cubature formulae and matrix equation, preprint.
- Marshall Harvey Stone, Linear transformations in Hilbert space, American Mathematical Society Colloquium Publications, vol. 15, American Mathematical Society, Providence, RI, 1990. Reprint of the 1932 original. MR 1451877, DOI 10.1090/coll/015
- A. H. Stroud, Approximate calculation of multiple integrals, Prentice-Hall Series in Automatic Computation, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1971. MR 0327006 G. Szegö, Orthogonal polynomials, Amer. Math. Soc. Colloq. Publ., vol. 23, Amer. Math. Soc., Providence, R.I., 4th ed., 1975.
- Yuan Xu, On multivariate orthogonal polynomials, SIAM J. Math. Anal. 24 (1993), no. 3, 783–794. MR 1215438, DOI 10.1137/0524048
- Yuan Xu, Multivariate orthogonal polynomials and operator theory, Trans. Amer. Math. Soc. 343 (1994), no. 1, 193–202. MR 1169912, DOI 10.1090/S0002-9947-1994-1169912-X
- Yuan Xu, Gaussian cubature and bivariate polynomial interpolation, Math. Comp. 59 (1992), no. 200, 547–555. MR 1140649, DOI 10.1090/S0025-5718-1992-1140649-8
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 342 (1994), 855-866
- MSC: Primary 42C05; Secondary 65D99, 65F99
- DOI: https://doi.org/10.1090/S0002-9947-1994-1258289-7
- MathSciNet review: 1258289