Sobolev orthogonal polynomials generated by Jacobi and Legendre polynomials, and special series with the sticking property for their partial sums

The paper is concerned with polynomials $p^{\alpha,\beta}_{r,k}(x)$, $k=0,1,\dots$, orthonormal with respect to the Sobolev-type inner product
$$ \langle f,g\rangle =\sum_{\nu=0}^{r-1}f^{(\nu)}(-1)g^{(\nu)}(-1)+\int_{-1}^{1}f^{(r)}(t)g^{(r)}(t)(1-t)^\alpha(1+t)^\beta\, dt\,, $$
where $r$ is an arbitrary natural number. Fourier series in the polynomials $p_{r,k}(x)=p^{0,0}_{r,k}(x)$ and some generalizations of them are introduced. Partial sums of such series are shown to retain certain important properties of the partial sums of Fourier series in the polynomials $p_{r,k}(x)$, in particular, the property of $r$-fold coincidence (sticking) of the original function $f(x)$ and the partial sums of the Fourier series in the polynomials $p_{r,k}(x)$ at the points $-1$ and $1$. The main emphasis is put on problems of approximating smooth and analytic functions by partial sums of such generalized series, which are special series in ultraspherical Jacobi polynomials, whose partial sums have the sticking property at the points $-1$ and $1$.
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