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Mat. Sb., 2003, Volume 194, Number 3, Pages 115–148 (Mi msb723)  

This article is cited in 19 scientific papers (total in 19 papers)

Mixed series in ultraspherical polynomials and their approximation properties

I. I. Sharapudinov

Daghestan Scientific Centre of the Russian Academy of Sciences

Abstract: New (mixed) series in ultraspherical polynomials $P_n^{\alpha,\alpha}(x)$ are introduced. The basic difference between a mixed series in the polynomials $P_n^{\alpha,\alpha}(x)$ and a Fourier series in the same polynomials is as follows: a mixed series contains terms of the form $\dfrac{2^rf_{r,k}^\alpha}{(k+2\alpha)^{[r]}}P_{k+r}^{\alpha-r,\alpha-r}(x)$, where $1\leqslant r$ is an integer and $f_{r,k}^\alpha$ is the $k$ th Fourier coefficient of the derivative $f^{(r)}(x)$ with respect to the ultraspherical polynomials $P_k^{\alpha,\alpha}(x)$. It is shown that the partial sums ${\mathscr Y}_{n+2r}^\alpha(f,x)$ of a mixed series in the polynomial $P_k^{\alpha,\alpha}(x)$ contrast favourably with Fourier sums $S_n^\alpha(f,x)$ in the same polynomials as regards their approximation properties in classes of differentiable and analytic functions, and also in classes of functions of variable smoothness. In particular, the ${\mathscr Y}_{n+2r}^\alpha(f,x)$ can be used for the simultaneous approximation of a function $f(x)$ and its derivatives of orders up to $(r- 1)$, whereas the $S_n^\alpha(f,x)$ are not suitable for this purpose.

DOI: https://doi.org/10.4213/sm723

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English version:
Sbornik: Mathematics, 2003, 194:3, 423–456

Bibliographic databases:

UDC: 517.5
MSC: 41A58, 42C10
Received: 25.10.2001 and 12.11.2002

Citation: I. I. Sharapudinov, “Mixed series in ultraspherical polynomials and their approximation properties”, Mat. Sb., 194:3 (2003), 115–148; Sb. Math., 194:3 (2003), 423–456

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. I. I. Sharapudinov, “Mixed Series of Chebyshev Polynomials Orthogonal on a Uniform Grid”, Math. Notes, 78:3 (2005), 403–423  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    2. I. I. Sharapudinov, “Approximation properties of mixed series in terms of Legendre polynomials on the classes $W^r$”, Sb. Math., 197:3 (2006), 433–452  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    3. I. I. Sharapudinov, “Approximation Properties of the Vallée-Poussin Means of Partial Sums of a Mixed Series of Legendre Polynomials”, Math. Notes, 84:3 (2008), 417–434  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    4. I. I. Sharapudinov, G. N. Muratova, “Nekotorye svoistva $r$-kratno integrirovannykh ryadov po sisteme Khaara”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 9:1 (2009), 68–76  mathnet  elib
    5. I. I. Sharapudinov, T. I. Sharapudinov, “Mixed Series of Jacobi and Chebyshev Polynomials and Their Discretization”, Math. Notes, 88:1 (2010), 112–139  mathnet  crossref  crossref  mathscinet  isi  elib
    6. I. I. Sharapudinov, “Approximating smooth functions using algebraic-trigonometric polynomials”, Sb. Math., 201:11 (2010), 1689–1713  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    7. I. I. Sharapudinov, M. G. Magomed-Kasumov, S. R. Magomedov, “Polinomy, ortogonalnye po Sobolevu, assotsiirovannye s polinomami Chebysheva pervogo roda”, Dagestanskie elektronnye matematicheskie izvestiya, 2015, no. 4, 1–14  mathnet  crossref  elib
    8. I. I. Sharapudinov, Z. D. Gadzhieva, “Polinomy, ortogonalnye po Sobolevu, porozhdennye mnogochlenami Meiksnera”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:3 (2016), 310–321  mathnet  crossref  mathscinet  elib
    9. I. I. Sharapudinov, “Asimptoticheskie svoistva polinomov, ortogonalnykh po Sobolevu, porozhdennykh polinomami Yakobi”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 6, 1–24  mathnet  crossref  elib
    10. I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Sistemy funktsii, ortogonalnykh otnositelno skalyarnykh proizvedenii tipa Soboleva s diskretnymi massami, porozhdennykh klassicheskimi ortogonalnymi sistemami”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 6, 31–60  mathnet  crossref  elib
    11. I. I. Sharapudinov, “Approximation Properties of Fourier Series of Sobolev Orthogonal Polynomials with Jacobi Weight and Discrete Masses”, Math. Notes, 101:4 (2017), 718–734  mathnet  crossref  crossref  mathscinet  isi  elib
    12. I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Raznostnye uravneniya i polinomy, ortogonalnye po Sobolevu, porozhdennye mnogochlenami Meiksnera”, Vladikavk. matem. zhurn., 19:2 (2017), 58–72  mathnet
    13. I. I. Sharapudinov, S. R. Magomedov, “Systems of functions orthogonal in the sense of Sobolev associated with Haar functions and the Cauchy problem for ODEs”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 1–15  mathnet  crossref
    14. I. I. Sharapudinov, Z. D. Gadzhieva, R. M. Gadzhimirzaev, “Sobolev orthogonal functions on the grid, generated by discrete orthogonal functions and the Cauchy problem for the difference equation”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 29–39  mathnet  crossref
    15. I. I. Sharapudinov, “O priblizhenii resheniya zadachi Koshi dlya nelineinykh sistem ODU posredstvom ryadov Fure po funktsiyam, ortogonalnym po Sobolevu”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 66–76  mathnet  crossref
    16. M. S. Sultanakhmedov, “Cauchy problem for the difference equation and Sobolev orthogonal functions on the finite grid, generated by discrete orthogonal functions”, Dagestanskie elektronnye matematicheskie izvestiya, 2017, no. 7, 77–85  mathnet  crossref
    17. I. I. Sharapudinov, “Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums”, Sb. Math., 209:9 (2018), 1390–1417  mathnet  crossref  crossref  adsnasa  isi  elib
    18. Sharapudinov I.I., “Sobolev Orthogonal Polynomials Associated With Chebyshev Polynomials of the First Kind and the Cauchy Problem For Ordinary Differential Equations”, Differ. Equ., 54:12 (2018), 1602–1619  crossref  mathscinet  isi  scopus
    19. I. I. Sharapudinov, “Sobolev-orthogonal systems of functions and the Cauchy problem for ODEs”, Izv. Math., 83:2 (2019), 391–412  mathnet  crossref  crossref  adsnasa  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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