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An extremal problem for algebraic polynomials in the symmetric discrete Gegenbauer–Sobolev space. (English. Russian original) Zbl 1165.46014

Math. Notes 82, No. 3, 366-379 (2007); translation from Mat. Zametki 82, No. 3, 411-425 (2007).
Summary: We study discrete Sobolev spaces with symmetric inner product \[ \left\langle{f,g}\right\rangle_\alpha=\int_{-1}^1{fg\,d\mu_\alpha}+M[f(1)g(1)+f(-1)g(-1)]+K[f'(1)g'(1)+f'(-1)g'(-1)], \]
where \(M \geq 0\), \(k \geq 0\), and
\[ d\mu_\alpha(x)=\frac{\Gamma(2\alpha+2)}{2^{2\alpha+1}\Gamma^2(\alpha+1)}(1-x^2)^\alpha\,dx,\quad\alpha>-1, \]
is the Gegenbauer probability measure. We obtain the solution of the following extremal problem: Calculate
\[ \inf\limits_{a_0 ,a_1,\dots,a_{N-r}}\left\{{\langle P_N^{(r)},P_N^{(r)}\rangle_\alpha,1\leq r\leq N-1,P_N^{(r)}(x)=\sum\limits_{j=N-r+1}^N{a_j^0 x^j}+\sum\limits_{j = 0}^{N-r}{a_jx^j}}\right\}, \]
where the \(a_{j}^{0}\), \(j=N-r+1,N-r+2,\dots,N-1,N\), \(a_{N}^{0}>0\) are fixed numbers, and find the extremal polynomial.

MSC:

46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables
33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
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