#### For citation:

Guseinov I. G., Gadzhimirzaev R. M. Two-dimensional limit series in ultraspherical Jacobi polynomials and their approximative properties. *Izvestiya of Saratov University. Mathematics. Mechanics. Informatics*, 2021, vol. 21, iss. 4, pp. 422-433. DOI: 10.18500/1816-9791-2021-21-4-422-433, EDN: SDAGUK

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

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.

- Pashkovskiy S.
*Vychislitel’nye primeneniia mnogochlenov i riadov Chebysheva*[Numerical Applications of Polynomials and Tchebychev Series]. Moscow, Nauka, 1983. 384 p. (in Russian). - Malvar H. S.
*Signal Processing with Lapped Transforms.*Artech House, 1992. 380 p. - Trefethen L. N.
*Finite Difference and Spectral Methods for Ordinary and Partial Differential Equation.*Cornell University, 1996. 299 p. - Sharapudinov I. I.
*Mnogochleny, ortogonal’nye na setkah. Teoriya i prilozheniya*[Polynomials Orthogonal on Grids. Theory and Applications]. Makhachkala, Izdatelstvo Dagestan Pedag. Univ., 1997. 255 p. (in Russian). - Mukundan R., Ramakrishnan K. R.
*Moment Functions in Image Analysis. Theory and Applications.*Singapore, World Scientific, 1998. 164 p. - Dedus F. F., Mahortyh S. A., Ustinin M. N., Dedus A. F.
*Obobshchennyi spektral’no-analiticheskii metod obrabotki informatsionnykh massivov. Zadachi analiza izobrazhenii i raspoznavaniia obrazov*[Generalized Spectral and Analytic Method of Data Arrays Processing. Problems of Image Analysis and Pattern Recognition]. Moscow, Mashinostroenie, 1999. 356 p. (in Russian). - Trefethen L. N.
*Spectral Methods in Matlab.*Philadelphia, SIAM, 2000. 181 p. - Sharapudinov I. I. Approximation of discrete functions and Chebyshev polynomials orthogonal on the uniform grid.
*Mathematical Notes*, 2000, vol. 67. iss. 3, pp. 389–397. https://doi.org/10.1007/BF02676675 - Sharapudinov I. I. Limit ultraspherical series and their approximation properties.
*Mathematical Notes*, 2013, vol. 94. iss. 2, pp. 281–293. https://doi.org/10.1134/S0001434613070274 - Szego G.
*Orthogonal Polynomials.*AMS Colloq. Publ., vol. 23, 1939. 440 p. (Russ. ed.: Moscow, Fizmatgiz, 1962. 500 p.).

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