Geodesic
Convergence of the Fourier Series in Meixner–Sobolev
Matematičeskie zametki, Tome 115 (2024) no. 3, pp. 330-347 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We study the convergence of Fourier series in the system of polynomials $\{m_{n,N}^{\alpha,r}(x)\}$ orthonormal in the sense of Sobolev and generated by the system of modified Meixner polynomials. In particular, we show that the Fourier series of $f\in W^r_{l^p_{\rho_N}(\Omega_\delta)}$ in this system converges to $f$ pointwise on the grid $\Omega_\delta$ as $p\geqslant 2$. In addition, we study the approximation properties of partial sums of Fourier series in the system $\{m_{n,N}^{0,r}(x)\}$.
Keywords: Sobolev-type inner product, Fourier series, approximative property
Mots-clés : Meixner polynomial, Lebesgue function. 
@article{MZM_2024_115_3_a1,
     author = {R. M. Gadzhimirzaev},
     title = {Convergence of the {Fourier} {Series} in {Meixner{\textendash}Sobolev}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {330--347},
     year = {2024},
     volume = {115},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a1/}
}
TY  - JOUR
AU  - R. M. Gadzhimirzaev
TI  - Convergence of the Fourier Series in Meixner–Sobolev
JO  - Matematičeskie zametki
PY  - 2024
SP  - 330
EP  - 347
VL  - 115
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a1/
LA  - ru
ID  - MZM_2024_115_3_a1
ER  - 
%0 Journal Article
%A R. M. Gadzhimirzaev
%T Convergence of the Fourier Series in Meixner–Sobolev
%J Matematičeskie zametki
%D 2024
%P 330-347
%V 115
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a1/
%G ru
%F MZM_2024_115_3_a1
R. M. Gadzhimirzaev. Convergence of the Fourier Series in Meixner–Sobolev. Matematičeskie zametki, Tome 115 (2024) no. 3, pp. 330-347. http://geodesic.mathdoc.fr/item/MZM_2024_115_3_a1/

[1] P. Barry, P. M. Rajković, M. D. Petković, “An application of Sobolev orthogonal polynomials to the computation of a special Hankel determinant”, Approximation and Computation, Springer Optim. Appl., 42, Springer, New York, 2011, 53–60 | DOI | MR

[2] F. Marcellán, Y. Xu, “On Sobolev orthogonal polynomials”, Expo. Math., 33:3 (2015), 308–352 | DOI | MR

[3] I. I. Sharapudinov, “Ortogonalnye po Sobolevu sistemy funktsii i nekotorye ikh prilozheniya”, UMN, 74:4 (448) (2019), 87–164 | DOI | MR | Zbl

[4] 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 | MR

[5] I. Area, E. Goboy, F. Marcellán, “Inner products involving differences: the Meixner–Sobolev polynomials”, J. Differ. Equations Appl., 6:1 (2000), 1–31 | DOI | MR

[6] S. F. Khwaja, A. B. Olde-Daalhuis, “Uniform asymptotic approximations for the Meixner–Sobolev polynomials”, Anal. Appl. (Singap.), 10:3 (2012), 345–361 | DOI | MR

[7] H. Bavinck, H. V. Haeringen, “Difference equations for generalized Meixner polynomials”, J. Math. Anal. Appl., 184:3 (1994), 453–463 | DOI | MR

[8] H. Bavinck, R. Koekoek, “Difference operators with Sobolev type Meixner polynomials as eigenfunctions”, Comput. Math. Appl., 36:10–12 (1998), 163–177 | MR

[9] 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 | DOI | MR

[10] I. I. Sharapudinov, I. G. Guseinov, “Polinomy, ortogonalnye po Sobolevu, porozhdennye polinomami Sharle”, Izv. Sarat. un-ta. Nov. ser. Ser.: Matematika. Mekhanika. Informatika, 18:2 (2018), 196–205 | DOI | MR

[11] J. Moreno-Balcázar, “$\delta$-Meixner–Sobolev orthogonal polynomials: Mehler–Heine type formula and zeros”, J. Comput. Appl. Math., 284 (2015), 228–234 | DOI | MR

[12] R. S. Costas-Santos, A. Soria-Lorente, J.-M. Vilaire, “On polynomials orthogonal with respect to an inner product involving higher-order differences: the Meixner case”, Mathematics, 10 (2022), 1952 | DOI

[13] R. M. Gadzhimirzaev, “Approximative properties of Fourier–Meixner sums”, Probl. Anal. Issues Anal., 7 (25):1 (2018), 23-40 | DOI | MR

[14] R. M. Gadzhimirzaev, “Otsenka funktsii Lebega summ Fure po modifitsirovannym polinomam Meiksnera”, Matem. zametki, 106:4 (2019), 519–530 | DOI | MR

[15] I. I. Sharapudinov, Mnogochleny, ortogonalnye na setkakh, Izd-vo Dag. gos. ped. un-ta, Makhachkala, 1997