×

Some biorthogonal polynomials arising in numerical analysis and approximation theory. (English) Zbl 1483.42016

The aim of the paper under review is the study of some biorthogonal polynomials that arise in certain topics of Numerical Analysis and Approximation Theory such as Numerical Integration and Convergence Acceleration.
Throughout their work the authors discuss the most general form of biorthogonality, that is, involving two families of polynomials orthogonal with respect to certain measures.They provide a survey of these polynomials, biorthogonal with respect different measures such as the logarithm or exponentials.
The authors also discuss the positivity of the weights in the interpolatory quadrature formulas generated by these biorthogonal polynomials. Finally they show the application of the potential theory, powerful tool in so many problems involving polynomials, to the study of some topics of biorthogonal polynomials such as aymptotics and zero distributions.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
11C08 Polynomials in number theory
30C10 Polynomials and rational functions of one complex variable
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Brezinski, C., Biorthogonality and its Applications to Numerical Analysis (1992), Marcel Dekker: Marcel Dekker New York · Zbl 0757.41001
[2] Iserles, A.; Nørsett, S., On the theory of biorthogonal polynomials, Trans. Amer. Math. Soc., 306, 455-474 (1988) · Zbl 0662.42017
[3] Kuijlaars, A.; McLaughlin, K. T-R., A Riemann-Hilbert problem for biorthogonal polynomials, J. Comput. Appl. Math., 178, 313-320 (2005) · Zbl 1096.42009
[4] Bertola, M.; Gekhtman, M.; Szmigielski, J., Cauchy biorthogonal polynomials, J. Approx. Theory, 162, 832-867 (2010) · Zbl 1202.33012
[5] L.G. Gonzalez Ricardo, G. Lopez Lagomasino, Strong asymptotic of Cauchy biorthogonal polynomials and orthogonal polynomials with varying measure, manuscript. · Zbl 1510.42035
[6] Iserles, A.; Nørsett, S. P., Bi-orthogonality and zeros of transformed polynomials, J. Comput. Appl. Math., 19, 39-45 (1987) · Zbl 0636.42022
[7] Iserles, A.; Nørsett, S. P.; Saff, E. B., On transformations and zeros of polynomials, Rocky Mountain J. Math., 21, 331-357 (1991) · Zbl 0754.26007
[8] Peherstorfer, F., Characterization of positive quadrature formulas, SIAM J. Math. Anal., 12, 935-942 (1981) · Zbl 0481.41025
[9] Peherstorfer, F., Characterization of quadrature formulas II, SIAM J. Math. Anal., 15, 1021-1030 (1984) · Zbl 0596.41044
[10] Peherstorfer, F., Positive quadrature formulas III: asymptotics of weights, Math. Comp., 77, 2241-2259 (2008) · Zbl 1198.65051
[11] Schmid, H. J., A note on positive quadrature rules, Rocky Mountain J. Math., 19, 395-404 (1989) · Zbl 0691.41032
[12] Sidi, A., Convergence properties of some nonlinear sequence transformations, Math. Comp., 33, 315-326 (1979) · Zbl 0401.41020
[13] Sidi, A., Analysis of convergence of the T-transformation for power series, Math. Comp., 35, 833-850 (1980) · Zbl 0441.40004
[14] Levin, D., Development of non-linear transformations for improving convergence of sequences, Int. J. Comput. Math., B3, 371-388 (1973) · Zbl 0274.65004
[15] Sidi, A., Practical Extrapolation Methods (2003), Cambridge University Press: Cambridge University Press Cambridge · Zbl 1041.65001
[16] Sidi, A., Numerical quadrature and nonlinear sequence transformations: unified rules for efficient computation of integrals with algebraic and logarithmic endpoint singularities, Math. Comp., 35, 851-874 (1980) · Zbl 0441.41021
[17] Sidi, A.; Lubinsky, D. S., On the zeros of some polynomials that arise in numerical quadrature and convergence acceleration, SIAM J. Numer. Anal., 20, 589-598 (1983) · Zbl 0521.65002
[18] Lubinsky, D. S.; Sidi, A., Strong asymptotics for polynomials biorthogonal to powers of \(\log x\), Analysis, 14, 341-379 (1994) · Zbl 0814.30022
[19] Elbert, C., Strong asymptotics of the generating polynomials of the Stirling numbers of the second kind, J. Approx. Theory, 109, 198-217 (2001) · Zbl 0977.05003
[20] Zhao, Yu-Qiu, A uniform asymptotic expansion of the single variable Bell polynomials, J. Comput. Appl. Math., 150, 329-355 (2003) · Zbl 1025.41020
[21] Lubinsky, D. S.; Sidi, A., Positive interpolatory quadrature rules generated by some biorthogonal polynomials, Math. Comp., 79, 845-855 (2010) · Zbl 1202.41029
[22] Lubinsky, D. S.; Stahl, H., Some explicit biorthogonal polynomials, (Chui, C. K.; Neamtu, M.; Schumaker, L. L., Approximation Theory XI (2005), Nashboro Press: Nashboro Press Brentwood, TN), 279-285 · Zbl 1069.42016
[23] Borodin, A., Biorthogonal ensembles, Nuclear Phys. B, 536, 704-732 (1999) · Zbl 0948.82018
[24] Claeys, T.; Romano, T., Biorthogonal ensembles with two-particle interactions, Nonlinearity, 27, 2419-2444 (2014) · Zbl 1314.60024
[25] Kuijlaars, A.; Molag, L. D., The local universality of Muttalib-Borodin biorthogonal ensembles with parameter \(\theta = \frac{ 1}{ 2} \), Nonlinearity, 32, 3023-3081 (2019) · Zbl 1427.60013
[26] D. Wang, L. Zhang, A vector Riemann-Hilbert approach to the Muttalib-Borodin ensembles, arXiv:2103.10327 [math.PR].
[27] VanAssche, W.; Fano, G.; Ortolani, F., Asymptotic behavior of the coefficients of some sequences of polynomials, SIAM J. Math. Anal., 18, 1597-1615 (1987) · Zbl 0622.33007
[28] Lubinsky, D. S.; Soran, I., Weights whose biorthogonal polynomials admit a Rodrigues formula, J. Math. Anal. Appl., 324, 805-819 (2006) · Zbl 1106.33010
[29] Claeys, T.; Wang, D., Random matrices with equispaced external source, Comm. Math. Phys., 328, 1023-1077 (2014) · Zbl 1291.15085
[30] Sidi, A., Numerical quadrature for some infinite range integrals, Math. Comp., 38, 127-142 (1982) · Zbl 0506.41034
[31] Sidi, A., Converging factors for some asymptotic moment series that arise in numerical quadrature, J. Aust. Math. Soc. B, 24, 223-233 (1982) · Zbl 0508.65010
[32] Sidi, A.; Lubinsky, D. S., Biorthogonal polynomials and numerical integration formulas for infinite intervals, J. Numer. Anal. Ind. Appl. Math., 2, 209-226 (2007) · Zbl 1221.65081
[33] Sidi, A., Problems 5-8, (Brass, H.; Hämmerlin, G., Numerical Integration III (1988), Birkhäuser: Birkhäuser Berlin), 321-325
[34] Sidi, A., Biorthogonal polynomials and numerical quadrature formulas for some finite-range integrals with symmetric weight functions, J. Comput. Appl. Math., 272, 221-238 (2014) · Zbl 1298.65043
[35] Lubinsky, D. S.; Sidi, A., Zero distribution of composite polynomials and polynomials biorthogonal to exponentials, Constr. Approx., 28, 343-371 (2008) · Zbl 1181.42027
[36] Bertola, M., The matrix models and biorthogonal polynomials, (The Oxford Handbook of Random Matrix Theory (2011), Oxford University Press: Oxford University Press Oxford), 310-328 · Zbl 1237.81111
[37] Lubinsky, D. S.; Sidi, A., Polynomials biorthogonal to dilations of measures and their asymptotics, J. Math. Anal. Appl., 397, 91-108 (2013) · Zbl 1253.42021
[38] Andrievskii, V. V.; Blatt, H. P., Discrepancy of Signed Measures and Polynomial Approximation (2002), Springer: Springer New York · Zbl 0995.30001
[39] Ransford, T., Potential Theory in the Complex Plane (1995), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0828.31001
[40] Saff, E. B.; Totik, V., Logarithmic Potentials with External Fields (1996), Springer: Springer New York
[41] Stahl, H.; Totik, V., General Orthogonal Polynomials (1992), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0791.33009
[42] Landkof, N. S., Foundations of Modern Potential Theory (1972), Springer · Zbl 0253.31001
[43] Lubinsky, D. S.; Sidi, A.; Stahl, H., Asymptotic zero distribution of biorthogonal polynomials, J. Approx. Theory, 190, 26-49 (2015) · Zbl 1310.42014
[44] Bloom, T.; Levenberg, N.; Totik, V.; Wielonsky, F., Modified logarithmic potential theory and applications, Int. Math. Res. Not. IMRN, 4, 1116-1154 (2017) · Zbl 1405.31006
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.