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Analytic approximation of transmutation operators and related systems of functions

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Abstract

In Kravchenko and Torba (J Comput Appl Math 275:1–26, 2015) a method for approximate solution of Sturm–Liouville equations and related spectral problems was presented based on the construction of the Delsarte transmutation operators. The problem of numerical approximation of solutions and eigendata was reduced to approximation of a primitive of the potential by a finite linear combination of certain specially constructed functions obtained from the generalized wave polynomials introduced in Khmelnytskaya et al. (J Math Anal Appl 399:191–212, 2013) and Kravchenko and Torba (Complex Anal Oper Theory 9:379–429, 2015). The method allows one to compute both lower and higher eigendata with an extreme accuracy. Since the solution of the approximation problem is the main step in the application of the method, the properties of the system of functions involved are of primary interest. In Kravchenko and Torba (J Comput Appl Math 275:1–26, 2015) two basic properties were established: the completeness in appropriate functional spaces and the linear independence. In this paper we present a considerably more complete study of the systems of functions. We establish their relation with another linear differential second-order equation, find out certain operations (in a sense, generalized derivatives and antiderivatives) which allow us to generate the next such function from a previous one. We obtain the uniqueness of the coefficients of expansions in terms of such functions and a corresponding generalized Taylor theorem, as well as formulas for exact expansion coefficients involving the operations mentioned above. We also construct the invertible integral operators transforming powers of the independent variable into the functions under consideration and establish their commutation relations with differential operators. We present some error bounds for the solution of the approximation problem depending on the smoothness of the potential and show that these error bounds are close to optimal in order. Also, we provide a rigorous justification of the alternative formulation of the proposed method allowing one to make use of the known initial values of the solutions at the left endpoint of the spectral problem.

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Authors and Affiliations

  1. Departamento de Matemáticas, CINVESTAV del IPN, Unidad Querétaro, Libramiento Norponiente No. 2000, Fracc. Real de Juriquilla, C.P. 76230, Querétaro, QRO, Mexico

    Vladislav V. Kravchenko & Sergii M. Torba

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  1. Vladislav V. Kravchenko
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  2. Sergii M. Torba
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Correspondence to Vladislav V. Kravchenko.

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To 60th birthday anniversary of Prof. Dr. Sergei Grudsky.

Research was supported by CONACYT, Mexico via the Projects 166141 and 222478.

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Kravchenko, V.V., Torba, S.M. Analytic approximation of transmutation operators and related systems of functions. Bol. Soc. Mat. Mex. 22, 389–429 (2016). https://doi.org/10.1007/s40590-016-0103-0

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