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 ∑2rk=1akPn+k(x) added; here Pm(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 WrHμL2 and Aq(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.
Citation:
I. I. Sharapudinov, “Approximation Properties of the Operators Yn+2r(f) and of Their Discrete Analogs”, Mat. Zametki, 72:5 (2002), 765–795; Math. Notes, 72:5 (2002), 705–732
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\by I.~I.~Sharapudinov
\paper Approximation Properties of the Operators $\mathscr Y_{n+2r}(f)$ and of Their Discrete Analogs
\jour Mat. Zametki
\yr 2002
\vol 72
\issue 5
\pages 765--795
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\transl
\jour Math. Notes
\yr 2002
\vol 72
\issue 5
\pages 705--732
\crossref{https://doi.org/10.1023/A:1021421425474}
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Linking options:
https://www.mathnet.ru/eng/mzm466
https://doi.org/10.4213/mzm466
https://www.mathnet.ru/eng/mzm/v72/i5/p765
This publication is cited in the following 23 articles:
I. I. Sharapudinov, “Sobolev-orthogonal systems of functions and some of their applications”, Russian Math. Surveys, 74:4 (2019), 659–733
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
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
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
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
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
I. I. Sharapudinov, “Special series in Laguerre polynomials and their approximation properties”, Siberian Math. J., 58:2 (2017), 338–362
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
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
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
I. I. Sharapudinov, “Asimptoticheskie svoistva polinomov, ortogonalnykh po Sobolevu, porozhdennykh polinomami Yakobi”, Dagestanskie elektronnye matematicheskie izvestiya, 2016, no. 6, 1–24
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
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
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
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
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
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
I. I. Sharapudinov, T. I. Sharapudinov, “Mixed Series of Jacobi and Chebyshev Polynomials and Their Discretization”, Math. Notes, 88:1 (2010), 112–139
I. I. Sharapudinov, “Approximating smooth functions using algebraic-trigonometric polynomials”, Sb. Math., 201:11 (2010), 1689–1713
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
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