Approximation Properties of Fourier Series of Sobolev Orthogonal Polynomials with Jacobi Weight and Discrete Masses

We study Fourier series of Jacobi polynomials $P_k^{\alpha-r,-r}(x)$, $k=r,r+1,\dots$, orthogonal with respect to the Sobolev-type inner product of the following form:
$$ \langle f,g\rangle=\sum_{\nu=0}^{r-1} f^{(\nu)}(-1)g^{(\nu)}(-1) +\int_{-1}^1f^{(r)}(t)g^{(r)}(t)(1-t)^\alpha\,dt. $$
It is shown that such series are a particular case of mixed series of Jacobi polynomials $P_k^{\alpha,\beta}(x)$, $k=0,1,\dots$, considered earlier by the author. We study the convergence of mixed series of general Jacobi polynomials and their approximation properties. The results obtained are applied to the study of the approximation properties of Fourier series of Sobolev orthogonal Jacobi polynomials $P_k^{\alpha-r,-r}(x)$.