An estimate for coefficients of polynomials in \(L^2\) norm. II. (English) Zbl 0863.26012
[For part I see the first author and A. Guessab, Proc. Am. Math. Soc. 120, 165-171 (1994; Zbl 0797.26010)].
For the class \(\mathcal P_n\) of algebraic polynomials \(P(x)=\sum_{k=0}^na_kx^k\) of degree at most \(n\), endowed with the norm \(|P|_{d\sigma} = \bigl(\int_{\mathbb{R}}|P(x)|^2 d\sigma(x)\bigr)^{1/2}\), where \(d\sigma(x)\) is a nonnegative measure on \({\mathbb{R}}\), the authors determine the best constants \(C_{k,n}\) so that \(|a_k|\leq C_{n,k}(d\sigma)|P|_{d\sigma}\) (\(k=0,...,n\)), when \(P\in {\mathcal P}_n\), and such that \(P(\eta_k)=0\), \(k=1,...,m\) and \(\eta_k\in {\mathbb{C}}\).
For the class \(\mathcal P_n\) of algebraic polynomials \(P(x)=\sum_{k=0}^na_kx^k\) of degree at most \(n\), endowed with the norm \(|P|_{d\sigma} = \bigl(\int_{\mathbb{R}}|P(x)|^2 d\sigma(x)\bigr)^{1/2}\), where \(d\sigma(x)\) is a nonnegative measure on \({\mathbb{R}}\), the authors determine the best constants \(C_{k,n}\) so that \(|a_k|\leq C_{n,k}(d\sigma)|P|_{d\sigma}\) (\(k=0,...,n\)), when \(P\in {\mathcal P}_n\), and such that \(P(\eta_k)=0\), \(k=1,...,m\) and \(\eta_k\in {\mathbb{C}}\).
Reviewer: L.M.Kocić (Niš)
MSC:
| 26C05 | Real polynomials: analytic properties, etc. |
| 26D05 | Inequalities for trigonometric functions and polynomials |
| 33C65 | Appell, Horn and Lauricella functions |
| 41A44 | Best constants in approximation theory |
| 30C15 | Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) |
