Geodesic
On a two-dimensional analogue of the orthogonal Jacobi polynomials of a discrete variable
Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 3, pp. 277-284 
F. M. Korkmasov. On a two-dimensional analogue of the orthogonal Jacobi polynomials of a discrete variable. Sibirskij žurnal vyčislitelʹnoj matematiki, Tome 10 (2007) no. 3, pp. 277-284. http://geodesic.mathdoc.fr/item/SJVM_2007_10_3_a4/
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Voir la notice de l'article provenant de la source Math-Net.Ru

It is shown that if $P_i^{\alpha,\beta}(x)$ ($\alpha,\beta>-1$, $i=0,1,2,\dots$) are classical Jacobi polynomials, the system of polynomials of two variables $\{\Psi_{mn}^{\alpha,\beta}(x,y)\}_{m,n=0}^r=\{P_m^{\alpha,\beta}(x)P_n^{\alpha,\beta}(y)\}_{m,n=0} ^r$ ($r=m+n\leq N-1$) is an orthogonal system on the grid $\Omega_{N\times N}=\{(x_i,y_i)\}_{i,j=0}^N\subset[-1,1]^2$, where $x_i$, and $y_j$ are zeros of the Jacobi polynomial $P_N^{\alpha,\beta}(x)$. Given an arbitrary continuous function $f(x,y)$ on the square $[-1,1]^2$, we construct two-dimensional discrete partial Fourier–Jacobi sums of the rectangular type $S_{m,n,N}^{\alpha,\beta}(f;x,y)$ over the orthonormal system $\{\widehat\Psi_{mn}^{\alpha,\beta}(x,y)\}_{m,n=0}^r$. Estimates of the Lebesgue function $L_{m,n,N}^{\alpha,\beta}(f;x,y)$ for the discrete Fourier–Jacobi sums $S_{m,n,N}^{\alpha,\beta}(f;x,y)$ depending on the position of a point $(x,y)$ on the square $[-1,1]^2$ are obtained.Besides, an application of the orthogonal Jacobi polynomials of a discrete variable $\Psi_{mn}^{\alpha,\beta}(x,y)$ to some applied problems of geophysics is considered.

[1] Pashkovskii S., Vychislitelnye primeneniya mnogochlenov i ryadov Chebysheva, Nauka, M., 1983 | MR

[2] Solodovnikov V. V., Dmitriev A. N., Egupov N. D., Spektralnye metody rascheta i proektirovaniya sistem upravleniya, Mashinostroenie, M., 1986 | Zbl

[3] Korkmasov F. M., “Approksimativnye svoistva srednikh Valle-Pussena dlya diskretnykh summ Fure–Yakobi”, Sib. matem. zhurn., 45:2 (2004), 334–355 | MR | Zbl

[4] Korkmasov F. M., “Approksimativnye svoistva diskretnykh summ Fure–Yakobi”, Zhurn. “Vestnik molodykh uchenykh”. Seriya “Prikladnaya matematika i mekhanika”, 2005, no. 1, 15–25

[5] Suetin P. K., Ortogonalnye mnogochleny po dvum peremennym, Nauka, M., 1988 | MR | Zbl

[6] Yanushauskas A. I., Kratnye trigonometricheskie ryady, Nauka, Novosibirsk, 1986 | MR | Zbl

[7] Daugavet I. K., “O postoyannykh Lebega dlya dvoinykh ryadov Fure”, Metody vychislenii, Vyp. 6, Izd-vo Leningr. un-ta, L., 1970, 8–13 | MR

[8] Kuznetsova O. I., “Ob asimptoticheskom povedenii konstant Lebega dlya posledovatelnosti treugolnykh chastnykh summ dvoinykh ryadov Fure”, Sib. matem. zhurn., 18:3 (1977), 629–636 | MR | Zbl

[9] Baiborodov S. P., “Konstanty Lebega mnogogrannikov”, Matem. zametki, 32:6 (1982), 817–822 | MR

[10] Podkorytov A. N., “Poryadok rosta konstant Lebega summ Fure po poliedram”, Vestnik LGU. Matematika, mekhanika, astronomiya, 1982, no. 7, 110–111 | MR | Zbl

[11] Agakhanov S. A., Natanson G. I., “Funktsiya Lebega summ Fure–Yakobi”, Vestnik LGU. Seriya matematika, 1968, no. 1, 11–23 | MR | Zbl

[12] Badkov V. M., “Otsenki funktsii Lebega i ostatka ryada Fure–Yakobi”, Sib. matem. zhurn., 9:6 (1968), 1263–1283 | MR | Zbl