Geodesic
An Extremal Problem for Algebraic Polynomials in the Symmetric Discrete Gegenbauer–Sobolev Space
Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 411-425

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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.
Mots-clés : algebraic polynomial
Keywords: discrete Gegenbauer–Sobolev space, Gegenbauer probability measure, extremal problem, Hilbert space, Gram–Schmidt orthogonalization. 
B. P. Osilenker. An Extremal Problem for Algebraic Polynomials in the Symmetric Discrete Gegenbauer–Sobolev Space. Matematičeskie zametki, Tome 82 (2007) no. 3, pp. 411-425. http://geodesic.mathdoc.fr/item/MZM_2007_82_3_a9/
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