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Ghost matrices and a characterization of symmetric Sobolev bilinear forms. (English) Zbl 1214.15016

Summary: We characterize symmetric Sobolev bilinear forms defined on \(\mathcal P\times \mathcal P\), where \(\mathcal P\) is the space of polynomials. More specifically we show that symmetric Sobolev bilinear forms, like symmetric matrices, can be re-written with a diagonal representation. As an application, we introduce the notion of a ghost matrix, extending some classic work of Stieltjes.

MSC:

15A63 Quadratic and bilinear forms, inner products
05A15 Exact enumeration problems, generating functions
35A25 Other special methods applied to PDEs

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[1] Álvarez de Morales, M.; Pérez, T. E.; Piñar, M. A.; Ronveaux, A., Orthogonal polynomials associated with a nondiagonal Sobolev inner product with polynomial coefficients, (Orthogonal Functions, Moment Theory, and Continued Fractions. Theory and Applications (Campinas, 1996). Orthogonal Functions, Moment Theory, and Continued Fractions. Theory and Applications (Campinas, 1996), Lecture Notes in Pure and Applied Mathematics, vol. 199 (1998), Marcel Dekker: Marcel Dekker New York), 343-358 · Zbl 0930.33005
[2] G.E. Andrews, L.L. Littlejohn, A combinatorial interpretation of the Legendre-Stirling numbers, Proc. Amer. Math. Soc., in press.; G.E. Andrews, L.L. Littlejohn, A combinatorial interpretation of the Legendre-Stirling numbers, Proc. Amer. Math. Soc., in press. · Zbl 1167.05002
[3] Barrios Rolania, D.; López-Lagomasino, G.; Pijeira Cabrera, H., The moment problem for a Sobolev inner product, J. Approx. Theory, 100, 2, 364-380 (1999) · Zbl 0980.44008
[4] J. Blankenagel, Anwendungenadjungierter Polynomoperatoren, Doctoral Dissertation, Universität Köln, 1971.; J. Blankenagel, Anwendungenadjungierter Polynomoperatoren, Doctoral Dissertation, Universität Köln, 1971.
[5] Boas, R. P., The Stieltjes moment problem for functions of bounded variation, Bull. Amer. Math. Soc., 45, 399-404 (1939) · Zbl 0021.30702
[6] Chihara, T. S., An Introduction to Orthogonal Polynomials (1978), Gordon and Breach Publishers: Gordon and Breach Publishers New York · Zbl 0389.33008
[7] Danese, A. E., Present status and current trends in the theory of orthogonal polynomials and special functions, Rend. Sem. Mat. Univers. Politec. Torino, 35, 1-20 (1976/1977)
[8] Duran, A. J., The Stieltjes moment problem for rapidly decreasing functions, Proc. Amer. Math. Soc., 107, 731-741 (1989) · Zbl 0676.44007
[9] Duran, A. J., Functions with given moments and weight functions for orthogonal polynomials, Rocky Mountain J. Math., 23, 87-104 (1993) · Zbl 0777.44003
[10] Gautschi, W., Orthogonal Polynomials: Computation and Approximation. Orthogonal Polynomials: Computation and Approximation, Numerical Mathematics and Scientific Computation Series (2004), Oxford University Press: Oxford University Press London · Zbl 1130.42300
[11] Han, S. S.; Kim, S. S.; Kwon, K. H., Orthogonalizing weights for Tchebycheff sets of polynomials, Bull. London Math. Soc., 24, 361-367 (1992) · Zbl 0768.33007
[12] Krall, H. L.; Frink, O., A new class of orthogonal polynomials: the Bessel polynomials, Trans. Amer. Math. Soc., 65, 100-115 (1949) · Zbl 0031.29701
[13] Kwon, K. H.; Littlejohn, L. L., Sobolev orthogonal polynomials and second-order differential equations, Rocky Mountain J. Math., 28, 2, 547-594 (1998) · Zbl 0930.33004
[14] Lay, David C., Linear Algebra and its Applications (2003), Addison-Wesley Publishing Co.: Addison-Wesley Publishing Co. New York · Zbl 0900.15001
[15] Littlejohn, L. L., On the classification of differential equations having orthogonal polynomial solutions, Ann. Mat. Pura Appl., 4, 35-53 (1984) · Zbl 0571.34003
[16] Marcellán, F.; Alfaro, M.; Rezola, M. L., Orthogonal polynomials on Sobolev spaces: old and new directions, J. Comput. Appl. Math., 49, 225-232 (1993) · Zbl 0790.42015
[17] Marcellán, F.; Szafraniec, F. H., The Sobolev-type moment problem, Proc. Amer. Math. Soc., 128, 2309-2317 (2000) · Zbl 0951.44002
[18] Marcellán, F.; Szafraniec, F. H., A matrix algorithm towards solving the moment problem of Sobolev type, Linear Algebra Appl., 331, 1-2, 155-164 (2001) · Zbl 0980.65049
[19] Piziak, R.; Odell, P. L., Matrix Theory: From Generalized Inverses to Jordan Form. Matrix Theory: From Generalized Inverses to Jordan Form, Pure and Applied Mathematics Series (2007), Chapman and Hall/CRC: Chapman and Hall/CRC Boca Raton · Zbl 1114.15002
[20] Schäfke, F. W.; Wolf, G., Einfache verallgemeinerte Klassische orthogonalpolynome, J. Reine Angew. Math., 262-263, 339-355 (1973) · Zbl 0272.33018
[21] Stieltjes, T. J., Recherches sur les fractions continues, Ann. Faculte Sci. Toulousse, 1 (1894), [T 1 122, 1 9 A 5-47] · JFM 25.0326.01
[22] Szegö, G., Orthogonal Polynomials, vol. 23 (1975), American Mathematical Society Colloquium Publications: American Mathematical Society Colloquium Publications Providence, Rhode Island · JFM 65.0278.03
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