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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.

We consider recurrence relations for the polynomials orthonormal with respect to the Sobolev-type inner product and generated by classical orthogonal polynomials, namely: Jacobi polynomials, Legendre polynomials, Chebyshev polynomials of the first and the second kind, Gegenbauer (ultraspherical) polynomials, Hermite polynomials.

The mechanical quadrature method, which has proven itself to efficiently solve singular integral equations of various types, is also quite effective in solving the hypersingular integral equation. A unified quadrature formula is obtained for a hypersingular integral with the weight function of the Jacobi polynomials. Formula is applicable for all admissible values, both real and complex, of the exponents of the weight function. The method has been tested to solve a simple model problem for an elastic half-plane containing an edge crack of finite length in the framework of the plane problem of the elasticity theory, the defining equation of which can be written in the form of a hypersingular integral equation for crack opening. The convergence of the method is shown, and the results of solving the problem are presented in the form of a graph and a table. [ABSTRACT FROM AUTHOR]

The authors analyze the properties of wavelets based on Jacobi polynomials. In particular, orthogonality conditions for these wavelets are considered, as well as minimization of the Riesz ratio. These problems lead to the solution of systems of nonlinear equations by the method proposed earlier by the authors. [ABSTRACT FROM AUTHOR]