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Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics

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Ukrainian Mathematical Journal Aims and scope

Let (u, v) be a pair of quasidefinite and symmetric linear functionals with {Pn}n≥0 and {Qn}n≥0 as respective sequences of monic orthogonal polynomial (SMOP). We define a sequence of monic polynomials {Rn}n≥0 as follows:

\(\begin{array}{cc}\frac{{P}_{n+2}^{\mathrm{^{\prime}}}\left(x\right)}{n+2}+{b}_{n}\frac{{P}_{n}^{\mathrm{^{\prime}}}\left(x\right)}{n}-{Q}_{n+1}\left(x\right)={d}_{n-1}\left(x\right),& n\ge 1.\end{array}\)

We present necessary and sufficient conditions for {Rn}n≥0 to be orthogonal with respect to a quasidefinite linear functional w. In addition, we consider the case where {Pn}n≥0 and {Qn}n≥0 are monic Chebyshev polynomials of the first and second kinds, respectively, and study the relative outer asymptotics of Sobolev polynomials orthogonal with respect to the Sobolev inner product

\(\langle p,q\rangle s=\underset{-1}{\overset{1}{\int }}pq{\left(1-{x}^{2}\right)}^{-1/2}dx+{\uplambda }_{1}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}}{\left(1-{x}^{2}\right)}^{1/2}dx+{\uplambda }_{2}\underset{-1}{\overset{1}{\int }}{p}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}{q}^{\mathrm{^{\prime}}\mathrm{^{\prime}}}d\mu \left(x\right),\)

where μ is a positive Borel measure associated with w and λ1, λ2 > 0; λ2 is a linear polynomial of λ1.

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References

  1. M. Abramowitz and I. A. Stegun (eds.), Handbook of Mathematical Functions, 10th edn., Dover, New York (1972).

    Google Scholar 

  2. M. Alfaro, F. Marcellán, and M. L. Rezola, “Orthogonal polynomials on Sobolev spaces: old and new directions,” J. Comput. Appl. Math., 48, 113–131 (1993).

    Article  MathSciNet  Google Scholar 

  3. M. Alfaro, A. Peña, J. Petronilho, and M. L. Rezola, “On linearly related orthogonal polynomials and their functionals,” J. Math. Anal. Appl., 287, 307–319 (2003).

    Article  MathSciNet  Google Scholar 

  4. A. Ali Kheli, A. Belkebir, and M. N. Bouras, “On a new combination of orthogonal polynomials sequences,” Vladikavkaz Math. J., 24, No. 3, 5–20 (2022).

  5. D. Barrios and G. López Lagomasino, “Asymptotic behavior of solutions of general three term recurrence relations,” Adv. Comput. Math., 26, 9–37 (2007).

  6. T. S. Chihara, An Introduction to Orthogonal Polynomials, Gordon & Breach, New York (1978).

    Google Scholar 

  7. E. B. Christoffel, “Über die Gaussische Quadratur und eine Verallgemeinerung derselben,” J. Reine Angew. Math., 55, 61–82 (1858).

    MathSciNet  Google Scholar 

  8. A. Delgado, Ortogonalidad No Estándar: Problemas Directos e Inversos [in Spanish], Ph. D. Thesis, Univ. Carlos III de Madrid, Spain (2006).

  9. A. Delgado and F. Marcellán, “On an extension of symmetric coherent pairs of orthogonal polynomials,” J. Comput. Appl. Math., 178, 155–168 (2005).

    Article  MathSciNet  Google Scholar 

  10. H. Due nas, F. Marcellán, and A. Molano, “An inverse problem associated with (1, 1)-symmetric coherent linear functionals,” Integral Transforms Spectr. Funct., 30, No. 7, 526–546 (2019).

  11. H. Dueñas, F. Marcellán, and A. Molano, “On symmetric (1, 1)-coherent pairs and Sobolev orthogonal polynomials: an algorithm to compute Fourier coefficients,” Rev. Colombiana Mat., 53, No. 2, 139–164 (2019).

    Article  MathSciNet  Google Scholar 

  12. Ya. L. Geronimus, “On the polynomials orthogonal with respect to a given number sequence,” Zap. Mat. Otdel. Kharkov. Univ. i NII Mat. i Mekh., 17, 3–18 (1940).

  13. Ya. L. Geronimus, “On the polynomials orthogonal with respect to a given number sequence and a theorem by W. Hahn,” lzv. Akad. Nauk SSSR, 4, No. 2, 215–228 (1940).

  14. M. E. H. Ismail and D. R. Masson, “Generalized orthogonality and continued fractions,” J. Approx. Theory, 83, 1–40 (1995).

    Article  MathSciNet  Google Scholar 

  15. A. Iserles, P. E. Koch, S. P. Nørsett, and J. M. Sanz-Serna, “On polynomials orthogonal with respect to certain Sobolev inner products,” J. Approx. Theory, 65, 151–175 (1991).

    Article  MathSciNet  Google Scholar 

  16. M. N. de Jesús, F. Marcellán, J. Petronilho, and N. Pinzón-Cortés, “(M,N)-coherent pairs of order (m, k) and Sobolev orthogonal polynomials,” J. Comput. Appl. Math., 256, 16–35 (2014).

    Article  MathSciNet  Google Scholar 

  17. M. N. de Jesús and J. Petronilho, “On linearly related sequences of derivatives of orthogonal polynomials,” J. Math. Anal. Appl., 347, 482–492 (2008).

    Article  MathSciNet  Google Scholar 

  18. D. C. Lewis, “Polynomial least square approximations,” Amer. J. Math., 69, 273–278 (1947).

    Article  MathSciNet  Google Scholar 

  19. D. S. Lubinsky, “Asymptotics of orthogonal polynomials: some old, some new, some identities,” Acta Appl. Math., 61, 207–256 (2000).

    Article  MathSciNet  Google Scholar 

  20. F. Marcellán and J. J. Moreno Balcázar, “Asymptotics and zeros of Sobolev orthogonal polynomials on unbounded supports,” Acta Appl. Math., 94, No. 2, 163–192 (2006).

  21. F. Marcellán and S. Varma, “On an inverse problem for a linear combination of orthogonal polynomials,” J. Difference Equat. Appl., 20, 570–585 (2014).

    Article  MathSciNet  Google Scholar 

  22. F. Marcellán and Y. Xu, “On Sobolev orthogonal polynomials,” Expo. Math., 33, No. 3, 308–352 (2015).

    Article  MathSciNet  Google Scholar 

  23. H. G. Meijer, “A short history of orthogonal polynomials in a Sobolev space I. The non-discrete case,” Nieuw Arch. Wisk. (4), 14, 93–113 (1996).

    MathSciNet  Google Scholar 

  24. J. Petronilho, “On the linear functionals associated to linearly related sequences of orthogonal polynomials,” J. Math. Anal. Appl., 315, 379–393 (2006).

    Article  MathSciNet  Google Scholar 

  25. G. Szegő, “Orthogonal polynomials,” Amer. Math. Soc. Colloq. Publ., 23, 4th edn., American Mathematical Society, Providence, RI (1975).

  26. V. B. Uvarov, “Relation between polynomials orthogonal with different weights,” Dokl. Akad. Nauk SSSR, 126, 33–36 (1959).

    MathSciNet  Google Scholar 

  27. V. B. Uvarov, “The connection between systems of polynomials that are orthogonal with respect to different distribution functions,” Z. Vychisl. Mat. Mat. Fiz., 9, 1253–1262 (1969); English translation: Comput. Math. Math. Phys., 9, 25–36 (1969).

  28. A. Zhedanov, “Rational spectral transformations and orthogonal polynomials,” J. Comput. Appl. Math., 85, 67–86 (1997).

    Article  MathSciNet  Google Scholar 

Download references

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Authors and Affiliations

  1. Escuela de Matemáticas y Estadística, Universidad Pedagógica y Tecnológica de Colombia, Duitama, Colombia

    Luis Alejandro Molano Molano

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  1. Luis Alejandro Molano Molano
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Correspondence to Luis Alejandro Molano Molano.

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Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 75, No. 10, pp. 1411–1428, October, 2023. Ukrainian DOI: https://doi.org/10.37863/umzh.v75i10.7293.

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Molano, L. Inverse Problems, Sobolev–Chebyshev Polynomials, and Asymptotics. Ukr Math J 75, 1601–1620 (2024). https://doi.org/10.1007/s11253-024-02281-3

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  • DOI: https://doi.org/10.1007/s11253-024-02281-3

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