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.