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Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2021, Volume 21, Issue 4, Pages 422–433
DOI: https://doi.org/10.18500/1816-9791-2021-21-4-422-433
(Mi isu907)
 

Scientific Part
Mathematics

Two-dimensional limit series in ultraspherical Jacobi polynomials and their approximative properties

I. G. Guseinov, R. M. Gadzhimirzaev

Dagestan Federal Research Center of RAS, 45 M. Gadzhieva St., Makhachkala 367025, Russia
References:
Abstract: Let $C[-1,1]$ be the space of functions continuous on the segment $[-1,1]$, $C[-1,1]^2$ be the space of functions continuous on the square $[-1,1]^2$. We denote by $P_n^\alpha(x)$ the ultraspherical Jacobi polynomials. Earlier, for function $f$ from the space $C[-1,1]$ limit series were constructed by the system of polynomials $P_n^\alpha(x)$ and the approximative properties of their partial sums were investigated. In particular, an upper bound for the corresponding Lebesgue function was obtained. Moreover, it was shown that the partial sums of the limit series, in contrast to the Fourier – Jacobi sums, coincide with the original function at the points $\pm1$. In this paper, for function $f(x, y)$ from the space $C[-1,1]^2$, we construct two-dimensional limit series by the system of ultraspherical Jacobi polynomials $P_n^\alpha(x)P_m^\beta(y)$ orthogonal on $[-1,1]^2$ with respect to the Jacobi-type weight-function. It is shown that the partial sum of the two-dimensional limit series coincides with $f(x, y)$ on the set $\{(-1,-1), (-1,1), (1, -1), (1,1)\}$ and is a projection on the subspace of algebraic polynomials $P(x,y)$. Using these properties, the approximative properties of the partial sums of the two-dimensional limit series are investigated. In particular, the behavior of the corresponding two-dimensional Lebesgue function is studied.
Key words: Jacobi polynomials, Fourier series, limit series, Lebesgue function, approximation properties.
Received: 25.05.2021
Accepted: 14.09.2021
Bibliographic databases:
Document Type: Article
UDC: 517.521.5
Language: Russian
Citation: I. G. Guseinov, R. M. Gadzhimirzaev, “Two-dimensional limit series in ultraspherical Jacobi polynomials and their approximative properties”, Izv. Saratov Univ. Math. Mech. Inform., 21:4 (2021), 422–433
Citation in format AMSBIB
\Bibitem{GusGad21}
\by I.~G.~Guseinov, R.~M.~Gadzhimirzaev
\paper Two-dimensional limit series in ultraspherical Jacobi polynomials and their approximative properties
\jour Izv. Saratov Univ. Math. Mech. Inform.
\yr 2021
\vol 21
\issue 4
\pages 422--433
\mathnet{http://mi.mathnet.ru/isu907}
\crossref{https://doi.org/10.18500/1816-9791-2021-21-4-422-433}
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    Известия Саратовского университета. Новая серия. Серия Математика. Механика. Информатика
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