On some Sobolev spaces with matrix weights and classical type Sobolev orthogonal polynomials

References

• H. Azad, A. Laradji, and M.T. Mustafa, Polynomial solutions of differential equations, Adv. Differ. Equ. 58(2011) (2011), p. 12.
   Google Scholar
• J.J. Duistermaat and F.A. Grünbaum, Differential equations in the spectral parameter, Comm. Math. Phys. 103(2) (1986), pp. 177–240.
   Web of Science ®Google Scholar
• A.J. Durán, A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theor. 74(1) (1993), pp. 83–109.
   Web of Science ®Google Scholar
• A.J. Durán and M.D. de la Iglesia, Differential equations for discrete Jacobi–Sobolev orthogonal polynomials, J. Spectr. Theor. 8(1) (2018), pp. 191–234.
   Web of Science ®Google Scholar
• A. Erdélyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Higher Transcendental Functions. Vol. III. Based, In Part, on Notes Left by Harry Bateman, McGraw-Hill Book Company, Inc., New York–Toronto–London, 1955.
   Google Scholar
• W.N. Everitt, K.H. Kwon, L.L. Littlejohn, and R. Wellman, Orthogonal polynomial solutions of linear ordinary differential equations, Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999), J. Comput. Appl. Math. 133(1–2) (2001), pp. 85–109.
   Google Scholar
• E. Horozov, Automorphisms of algebras and Bochner's property for vector orthogonal polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 12 (2016), Paper No. 050, 14 pp.
   Google Scholar
• E. Horozov, Vector orthogonal polynomials with Bochner's property, Constr. Approx. 48(2) (2018), pp. 201–234.
   Web of Science ®Google Scholar
• E. Horozov, d-orthogonal analogs of classical orthogonal polynomials, SIGMA Symmetry Integrability Geom. Methods Appl. 14 (2018), Paper No. 063, 27 pp.
   Google Scholar
• H.K. Kim, K.H. Kwon, L.L. Littlejohn, and G.J. Yoon, Diagonalizability and symmetrizability of Sobolev-type bilinear forms: a combinatorial approach, Linear Algebra Appl. 460 (2014), pp. 111–124.
   Web of Science ®Google Scholar
• L. Klotz, A matrix generalization of a theorem of Szeg, Anal. Math. 18(1) (1992), pp. 63–72.
   Google Scholar
• R. Koekoek, P.A. Lesky, and R.F. Swarttouw, Hypergeometric Orthogonal Polynomials and their q-Analogues, With a Foreword by Tom H. Koornwinder, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2010.
   Google Scholar
• K.H. Kwon, L.L. Littlejohn, and G.J. Yoon, Ghost matrices and a characterization of symmetric Sobolev bilinear forms, Linear Algebra Appl. 431(1-2) (2009), pp. 104–119.
   Web of Science ®Google Scholar
• F. Marcellán, M. Alfaro, and M.L. Rezola, Orthogonal polynomials on Sobolev spaces: old and new directions, Proceedings of the Seventh Spanish Symposium on Orthogonal Polynomials and Applications (VII SPOA) (Granada, 1991), J. Comput. Appl. Math. 48(1–2) (1993), pp. 113–131.
   Google Scholar
• F. Marcellán and J.J. Moreno Balcázar, Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports, Acta Appl. Math. 94(2) (2006), pp. 163–192.
   Web of Science ®Google Scholar
• F. Marcellán and Y. Xu, On Sobolev orthogonal polynomials, Expo. Math. 33(3) (2015), pp. 308–352.
   Web of Science ®Google Scholar
• M. Marden, Geometry of Polynomials, 2nd ed., Mathematical Surveys, Vol. 3, American Mathematical Society, Providence, R.I., 1966.
   Google Scholar
• A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, With an appendix by M. V. Keldysh. Translations of Mathematical Monographs, 71. American Mathematical Society, Providence, RI, 1988.
   Google Scholar
• A. Martínez-Finkelshtein, Asymptotic properties of Sobolev orthogonal polynomials, Proceedings of the VIIIth Symposium on Orthogonal Polynomials and their Applications (Seville, 1997), J. Comput. Appl. Math. 99(1–2) (1998), pp. 491–510
   Google Scholar
• A. Martínez-Finkelshtein, Analytic aspects of Sobolev orthogonal polynomials revisited, Numerical analysis 2000, Vol. V, quadrature and orthogonal polynomials, J. Comput. Appl. Math. 127(1–2) (2001), pp. 255–266
   Google Scholar
• H.G. Meijer, A short history of orthogonal polynomials in a Sobolev space. I. The non-discrete case, 31st Dutch Mathematical Conference (Groningen, 1995), Nieuw Arch. Wisk. (4), 14(1) (1996), pp. 93–112.
   Google Scholar
• L. Rodman, An Introduction to Operator Polynomials, Operator Theory: Advances and Applications, Vol. 38, Birkhäuser Verlag, Basel, 1989.
   Google Scholar
• M. Rosenberg, The square-integrability of matrix-valued functions with respect to a non-negative Hermitian measure, Duke Math. J. 31 (1964), pp. 291–298.
   Web of Science ®Google Scholar
• V. Spiridonov and A. Zhedanov, Classical biorthogonal rational functions on elliptic grids, C. R. Math. Acad. Sci. Soc. R. Can. 22(2) (2000), pp. 70–76.
   Google Scholar
• G. Szegö, Orthogonal Polynomials, 4th ed., American Mathematical Society, Colloquium Publications, Vol. XXIII, American Mathematical Society, Providence, R.I., 1975
   Google Scholar
• A. Weron, On characterizations of interpolable and minimal stationary processes, Studia Math. 49 (1973/74), pp. 165–183.
   Web of Science ®Google Scholar
• Y. Xu, Approximation by polynomials in Sobolev spaces with Jacobi weight, J. Fourier Anal. Appl. 24(6) (2018), pp. 1438–1459.
   Web of Science ®Google Scholar
• S.M. Zagorodnyuk, On some classical type Sobolev orthogonal polynomials, J. Approx. Theor. 250 (2020), pp. 105337.
   Web of Science ®Google Scholar
• S.M. Zagorodnyuk, On a family of hypergeometric Sobolev orthogonal polynomials on the unit circle, Constr. Math. Anal. 3(2) (2020), pp. 84–75.
   Google Scholar