Some extremal problems for algebraic polynomials in loaded spaces

B. P. Osilenker

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 2 (2010), pp. 53-65

Résumé

Let $$ \Pi _N^{(r)}(x)=\sum_{k=N-r+1}^Na_k^0x^k+\sum_{j=0}^{N-r}a_jx^j \quad(a_N^{(0)}>0) $$ be an algebraic polynomial with fixed coefficients $a_k^{(0)}$. For the $l$th derivative of the mentioned polynomial we solve the following extremal problems: in a loaded Jacobi space with the inner product $$ \langle f,g\rangle=\frac{\Gamma(\alpha+\beta+2)}{2^{\alpha+\beta+1}\Gamma(\alpha+1)\Gamma(\beta+1)}\int_{-1}^1fg(1-x)^\alpha(1+x)^\beta\,dx+Lf(1)g(1)+Mf(-1)g(-1), $$ $(L,M\ge0)$, find $\inf\langle D^l[\Pi_N^{(r)}(x)],D^l[\Pi_N^{(r)}(x)]\rangle$ ($D=\frac d{dx}$, $0\le l\le N-r$) and calculate extremal polynomials.

Détail

Keywords: extremal problem, loaded spaces, loaded orthogonal polynomials, algebraic polynomials, classical Jacobi polynomials.