On a two-dimensional analogue of the orthogonal Jacobi polynomials of a discrete variable

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.