Operator orthogonal polynomials are considered whose arguments are generators of strongly continuous semigroups of transformations of class C0 acting in a Banach space. Earlier such polynomials were considered by the authors in the case of the Chebyshev polynomials of the first and second kind. In this paper, more general classes of operator orthogonal polynomials are considered, which include the Jacobi and Aptekarev polynomials. Integral representations of operator fractional-rational functions and also of Bessel operator-valued functions of an imaginary argument are presented.
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References
A. I. Aptekarev, “Asymptotics of orthogonal polynomials in a neighborhood of endpoints of the interval of orthogonality,” Mat. Sb., 183, 43–62 (1992).
K. Iosida, Functional Analysis [Russian translation], Moscow (1967).
V. A. Kostin, “On the uniformly correct solvability of boundary-value problems for abstract equations with the Keldysh–Feller operator,” Differ. Uravn., 7, 1419–1425 (1984).
V. A. Kostin and M. N. Nebol’sina, “Chebyshev C 0-operator orthogonal polynomials and their representations,” Zap. Nauchn. Semin. POMI, 376, 64–88 (2010).
M. A. Krasnosel’skii, P. P. Zabreiko, E. I. Pustyl’nik, and P. E. Sobolevskii, Integral Operators in Spaces of Summable Functions [in Russian], Moscow (1966).
S. G. Krein, Linear Differential Equations in Banach Space [in Russian], Moscow (1967).
S. G. Krein and M. I. Khazan, “Differential equations in a Banach space,” Itogi Nauki Tekhn., 21, 130–264 (1983).
V. P. Maslov, Operational Methods [in Russian], Moscow (1973).
M. N. Nebol’sina, “Chebyshev orthogonal polynomials and the Neumann boundary-value problem,” Differ. Uravn., 46, 449–450 (2010).
G. Szegö, Orthogonal Polynomials [Russian translation], Moscow (1962).
P. K. Suetin, Classical Orthogonal Polynomials [in Russian], Moscow (1979).
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Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 456, 2017, pp. 125–134.
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Kostin, V.A., Nebol’sina, M.N. To the Theory of C0-Operator Orthogonal Polynomials. J Math Sci 234, 350–356 (2018). https://doi.org/10.1007/s10958-018-4011-x
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DOI: https://doi.org/10.1007/s10958-018-4011-x