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Sobolev Orthogonal Polynomials Associated with Chebyshev Polynomials of the First Kind and the Cauchy Problem for Ordinary Differential Equations

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Abstract

We consider the polynomials Tr,n(x) (n = 0, 1,…) generated by Chebyshev polynomials Tn(x) and forming a Sobolev orthonormal system with respect to the inner product

$$\langle f,g\rangle = \sum\limits_{\nu = 0}^{r - 1} {{f^{(\nu)}}} ( - 1){g^{(\nu)}}(-1) + \int\limits_{-1}^1 {{f^{(r)}}} (x){g^{(r)}}(x)\mu (x)dx,$$

, where μ(x) = 2π−1(1 − x2)−1/2. It is shown that the Fourier sums in the polynomials Tr,n(x) (n = 0, 1,…) give a convenient and efficient tool for approximately solving the Cauchy problem for ordinary differential equations.

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References

  1. Kwon, K.H. and Littlejohn, L.L., The orthogonality of the Laguerre polynomials {L(-k) n (x)} for positive integers k, Ann. Numer. Anal., 1995, no. 2, pp. 289–303.

    MathSciNet  MATH  Google Scholar 

  2. Kwon, K.H. and Littlejohn, L.L., Sobolev orthogonal polynomials and second-order differential equations, Rocky Mountain J. Math., 1998, vol.28, pp. 547–594.

    Article  MathSciNet  MATH  Google Scholar 

  3. Marcellan, F., Alfaro, M., and Rezola, M.L., Orthogonal polynomials on Sobolev spaces: old and new directions, J. Comput. Appl. Math., 1993, vol. !48, pp. 113–131.

    Article  MathSciNet  MATH  Google Scholar 

  4. Iserles, A., Koch, P.E., Norsett, S.P., and Sanz-Serna, J.M., On polynomials orthogonal with respect to certain Sobolev inner products, J. Approx. Theory, 1991, vol.65, pp. 151–175.

    Article  MathSciNet  MATH  Google Scholar 

  5. Meijer, H.G., Laguerre polynomials generalized to a certain discrete Sobolev inner product space, J. Approx. Theory, 1993, vol.73, pp. 1–16.

    Article  MathSciNet  MATH  Google Scholar 

  6. Lopez, G., Marcellan, F., and Vanassche, W., Relative asymptotics for polynomials orthogonal with respect to a discrete Sobolev inner product, Constr. Approx., 1995, vol.11, no. 1, pp. 107–137.

    Article  MathSciNet  MATH  Google Scholar 

  7. Marcellan, F. and Xu, Y., On Sobolev orthogonal polynomials, Expo. Math., 2015, vol.33, no. 3, pp. 308–352.

    Article  MathSciNet  MATH  Google Scholar 

  8. Sharapudinov, I.I., Several special series in general Laguerre polynomials and Fourier series in Laguerre Sobolev-orthogonal polynomials, Dagestan Elektron. Mat. Izv., 2015, no. 4, pp. 31–73.

    Google Scholar 

  9. Sharapudinov, I.I., Special series in Laguerre polynomials and their approximation properties, Sib. Math. J., 2017, vol.58, no. 2, pp. 338–362.

    Article  MathSciNet  MATH  Google Scholar 

  10. Sharapudinov, I.I., Sobolev-orthogonal systems of functions associated with an orthogonal system, Izv. Math., 2018, vol.82, no. 1, pp. 212–244.

    Article  MathSciNet  MATH  Google Scholar 

  11. Sharapudinov, I.I. and Magomed-Kasumov, M.G., On representation of a solution to the Cauchy problem by a Fourier series in Sobolev-orthogonal polynomials generated by Laguerre polynomials, Differ. Equations, 2018, vol.54, no. 1, pp. 49–66.

    Article  MathSciNet  MATH  Google Scholar 

  12. Trefethen, L.N., Spectral Methods in Matlab, Fhiladelphia: SIAM, 2000.

    Book  MATH  Google Scholar 

  13. Trefethen, L.N., Finite Difference and Spectral Methods for Ordinary and Partial Differential Equations, Ithaca: Cornell Univ., 1996.

    Google Scholar 

  14. Solodovnikov, V.V., Dmitriev, A.N., and Egupov, N.D., Spektral’nye metody rascheta i proektirovaniya sistem upravleniya (Spectral Methods for Calculation and Design of Control Systems), Moscow: Mashinostroenie, 1986.

    MATH  Google Scholar 

  15. Pashkovskii, S., Vychislitel’nye primeneniya mnogochlenov i ryadov Chebysheva (Computational Applications of Chebyshev Polynomials and Series), Moscow: Nauka, 1983.

    Google Scholar 

  16. Arushanyan, O.B., Volchenskova, N.I., and Zaletkin, S.F., Approximate solution of ordinary differential equations using the Chebyshev series, Sib. Elektron. Mat. Izv., 1983, no. 7, pp. 122–131.

    MATH  Google Scholar 

  17. Arushanyan, O.B., Volchenskova, N.I., and Zaletkin, S.F., Method for solving the Cauchy problems for ordinary differential equations using the Chebyshev series, Vychisl. Metody Program., 2013, vol.14, no. 2, pp. 203–214.

    MATH  Google Scholar 

  18. Arushanyan, O.B., Volchenskova, N.I., and Zaletkin, S.F., Application of Chebyshev series to integrate ordinary differential equations, Sib. Elektron. Mat. Izv., 2014, no. 11, pp. 517–531.

    Google Scholar 

  19. Lukomskii, D.S. and Terekhin, P.A., Application of the Haar system to solve the Cauchy problem for a first-order linear differential equation numerically, in Proc. 18th Intern. Saratov Winter School “Contemporary Problems of Theory of Functions and Its Applications”, Saratov, 2016, pp. 171–173.

    Google Scholar 

  20. Magomed-Kasumov, M.G., Approximate solution of ordinary differential equations by using mixed series in the Haar system, in Proc. 18th Intern. Saratov Winter School “Contemporary Problems of Theory of Functions and Its Applications”, Saratov, 2016, pp. 176–178.

    Google Scholar 

  21. Sharapudinov, I.I. and Magomed-Kasumov, M.G., Numerical method for solving the Cauchy problem for systems of ordinary differential equations by using a Sobolev orthogonal system generated by a cosine system, Dagestan Elektron. Mat. Izv., 2017, no. 8, pp. 53–60.

    Google Scholar 

  22. Sharapudinov, I.I., Approximation of functions of variable smoothness by Fourier–Legendre sums, Sb. Math., 2000, vol.191, no. 5, pp. 759–777.

    Article  MathSciNet  MATH  Google Scholar 

  23. Sharapudinov, I.I., Approximation properties of the operators Yn+2r(f) and of their discrete analogs, Math. Notes, 2002, vol.72, no. 5, pp. 705–732.

    Article  MathSciNet  MATH  Google Scholar 

  24. Sharapudinov, I.I., Smeshannye ryady po ortogonal’nym polinomam (Mixed Series in Orthogonal Polynomial), Makhachkala: DNTs RAN, 2004.

    Google Scholar 

  25. Sharapudinov, I.I., Mixed series of Chebyshev polynomials orthogonal on a uniform grid, Math. Notes, 2005, vol.78, no. 3, pp. 403–423.

    Article  MathSciNet  MATH  Google Scholar 

  26. Sharapudinov, I.I., Approximation properties of mixed series in terms of Legendre polynomials on the classes Wr, Sb. Math., 2006, vol.197, no. 3, pp. 433–452.

    Article  MathSciNet  MATH  Google Scholar 

  27. Sharapudinov, I.I., Approximation properties of the Vallée–Poussin means of partial sums of a mixed series of Legendre polynomials, Math. Notes, 2008, vol.84, no. 3, pp. 417–434.

    Article  MathSciNet  MATH  Google Scholar 

  28. Sharapudinov, I.I., Mixed series in ultraspherical polynomials and their approximation properties, Sb. Math., 2003, vol.194, no. 3, pp. 423–456.

    Article  MathSciNet  MATH  Google Scholar 

  29. Sharapudinov, I.I. and Sharapudinov, T.I., Mixed series of Jacobi and Chebyshev polynomials and their discretization, Math. Notes, 2010, vol.88, no. 1, pp. 112–139.

    Article  MathSciNet  MATH  Google Scholar 

  30. Szego, G., Orthogonal Polynomials, Amer. Math. Soc., 1939. Translated under the title Ortogonal’nye mnogochleny, Moscow: Gos. Izd. Fiz. Mat. Lit., 1962.

    Google Scholar 

  31. Timan, A.F., Teoriya priblizheniya funktsii deistvitel’nogo peremennogo (Theory of Approximation of Functions of a Real Variable), Moscow: Fizmatgiz, 1960.

    Google Scholar 

  32. Akhiezer, N.I., Lektsii po teorii approksimatsii (Lectures in the Theory of Approximation), Moscow: Nauka, 1965.

    Google Scholar 

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Authors and Affiliations

  1. Dagestan Scientific Center of the Russian Academy of Sciences, Makhachkala, 367032, Russia

    I. I. Sharapudinov

  2. Southern Mathematical Institute of the Vladikavkaz Scientific Center of the Russian Academy of Sciences, Vladikavkaz, 362027, Russia

    I. I. Sharapudinov

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  1. I. I. Sharapudinov
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Correspondence to I. I. Sharapudinov.

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Original Russian Text © I.I. Sharapudinov, 2018, published in Differentsial’nye Uravneniya, 2018, Vol. 54, No. 12, pp. 1645–1662.

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Sharapudinov, I.I. Sobolev Orthogonal Polynomials Associated with Chebyshev Polynomials of the First Kind and the Cauchy Problem for Ordinary Differential Equations. Diff Equat 54, 1602–1619 (2018). https://doi.org/10.1134/S0012266118120078

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