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Mat. Zametki, 2002, Volume 72, Issue 5, Pages 765–795 (Mi mz466)  

This article is cited in 22 scientific papers (total in 22 papers)

Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs

I. I. Sharapudinov

Daghestan State Pedagogical University

Abstract: This paper is devoted to the study of the approximation properties of linear operators which are partial Fourier–Legendre sums of order $n$ with $2r$ terms of the form $\sum _{k=1}^{2r}a_kP_{n+k}(x)$ added; here $P_m(x)$ denotes the Legendre polynomial. Due to this addition, the linear operators interpolate functions and their derivatives at the endpoints of the closed interval $[-1,1]$, which, in fact, for $r=1$ allows us to significantly improve the approximation properties of partial Fourier–Legendre sums. It is proved that these operators realize order-best uniform algebraic approximation of the classes of functions $W_rH_{L_2}^\mu $ and $A_q(B)$. With the aim of the computational realization of these operators, we construct their discrete analogs by means of Chebyshev polynomials, orthogonal on a uniform grid, also possessing nice approximation properties.

DOI: https://doi.org/10.4213/mzm466

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English version:
Mathematical Notes, 2002, 72:5, 705–732

Bibliographic databases:

UDC: 517.518.8
Received: 20.04.2001

Citation: I. I. Sharapudinov, “Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs”, Mat. Zametki, 72:5 (2002), 765–795; Math. Notes, 72:5 (2002), 705–732

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. S. Sultanakhmedov, T. N. Shakh-Emirov, T. I. Sharapudinov, M. G. Magomed-Kasumov, G. G. Akniev, R. M. Gadzhimirzaev, “Ob identifikatsii parametrov lineinykh sistem na osnove polinomov Chebysheva pervogo roda i polinomov Chebysheva, ortogonalnykh na ravnomernoi setke”, Dagestanskie elektronnye matematicheskie izvestiya, 2014, no. 2, 1–32  mathnet  crossref  elib
    8. 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
    9. T. I. Sharapudinov, “Diskretnye polinomy, ortogonalnye po Sobolevu, assotsiirovannye s polinomami Chebysheva, ortogonalnymi na ravnomernoi setke”, Dagestanskie elektronnye matematicheskie izvestiya, 2015, no. 4, 15–20  mathnet  crossref  elib
    10. I. I. Sharapudinov, “Nekotorye spetsialnye ryady po obschim polinomam Lagerra i ryady Fure po polinomam Lagerra, ortogonalnym po Sobolevu”, Dagestanskie elektronnye matematicheskie izvestiya, 2015, no. 4, 31–73  mathnet  crossref  elib
    11. I. I. Sharapudinov, T. I. Sharapudinov, “Ob odnovremennom priblizhenii funktsii i ikh proizvodnykh posredstvom polinomov Chebysheva, ortogonalnykh na ravnomernoi setke”, Dagestanskie elektronnye matematicheskie izvestiya, 2015, no. 4, 74–117  mathnet  crossref  elib
    12. 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
    13. R. M. Gadzhimirzaev, “Ryady Fure po polinomam Meiksnera, ortogonalnym po Sobolevu”, Izv. Sarat. un-ta. Nov. ser. Ser. Matematika. Mekhanika. Informatika, 16:4 (2016), 388–395  mathnet  crossref  mathscinet  elib
    14. I. I. Sharapudinov, T. I. Sharapudinov, “Polynomials, orthogonal on Sobolev, derived by the Chebyshev polynomials, orthogonal on the uniform net”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 5, 56–75  mathnet  crossref
    15. I. I. Sharapudinov, “Asimptoticheskie svoistva polinomov, ortogonalnykh po Sobolevu, porozhdennykh polinomami Yakobi”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 6, 1–24  mathnet  crossref  elib
    16. 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
    17. 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
    18. 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
    19. I. I. Sharapudinov, “Special series in Laguerre polynomials and their approximation properties”, Siberian Math. J., 58:2 (2017), 338–362  mathnet  crossref  crossref  isi  elib  elib
    20. Sharapudinov I.I. 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. Equ., 54:1 (2018), 49–66  crossref  mathscinet  zmath  isi  scopus  scopus
    21. 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
    22. 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
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