An Extremal Problem for Algebraic Polynomials in the Symmetric Discrete Gegenbauer–Sobolev Space

We study discrete Sobolev spaces with symmetric inner product
$$ \langle f,g\rangle_\alpha =\int_{-1}^1fg\,d\mu_\alpha+M[f(1)g(1)+f(-1)g(-1)]+K[f'(1)g'(1)+f'(-1)g'(-1)], $$
where $M\ge0$, $K\ge0$, and
$$ d\mu_\alpha(x) =\frac{\Gamma(2\alpha+2)} {2^{2\alpha+1}\Gamma^2(\alpha+1)}\,(1-x^2)^\alpha\,dx,\qquad \alpha>-1, $$
is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate
$$ \inf_{a_0,a_1,\dots,a_{N-r}}\biggl\{ \langle P^{(r)}_N,P^{(r)}_N\rangle_\alpha,1\le r\le N-1,P^{(r)}_N(x) =\sum_{j=N-r+1}^{N}a^0_j x^j+\sum_{j=0}^{N-r}a_j x^j\biggr\}, $$
where the $a^0_j$, $j=N-r+1,N-r+2,\dots,N-1,N$, $a^0_N>0$, are fixed numbers, and find the extremal polynomial.