Shape preserving approximation on the real line with exponential weights
The paper deals with weighted polynomial approximation on the real line. In the 1950's, Achieser, Mergelyan and Pollard solved the well-known `Bernstein's weighted problem' posed by \textit{S. Bernstein} in 1924 [Bull. Math. Soc. France 52, 399--410 (1924; JFM 50.0195.01)]. As stated by \textit{H. Pollard} [Proc. Am. Math. Soc. 4, 869--875 (1953; Zbl 0057.29802)] it was written as follows. Here for \(W: \mathbb{R}\rightarrow (0,1]\), \(C_W\) denotes the space of all \(f\in C(\mathbb{R})\) such that \(\lim _{x\rightarrow \pm \infty}f(x)W(x)=0\) and \(\|f\|_W:=\sup _{x\in \mathbb{R}}|f(x)W(x)|\). Theorem. Suppose \(x^n\in C_{W}\), for an non-negative integer \(n\). Then, in favor that \(\{x^n\}_{n=0}^{\infty}\) is fundamental in \(C_{W}\) it is necessary and sufficient that {\parindent=8mm \begin{itemize}\item[(i)] \(\int _{-\infty}^{+\infty}\frac{\log (1/W(t))}{1+t^2}\,dt=\infty\); \item[(ii)] there exists a sequence of polynomials \(\{P_n\}\) such that for each \(x\): \[ \lim _{n\rightarrow \infty}P_n(x)W(x)=1\quad\text{ while }\quad\sup _{n\geq 1}\|P_n\|_W<\infty. \] \end{itemize}} In particular, the so-called Freud's weight \(W_{\alpha }(x):=e^{-|x|^{\alpha }}\) satisfies conditions (i) and (ii) if \(\alpha \geq 1\), and therefore the linear span of \(x^n\), \(n\in \mathbb{Z}_+\) is dense in \(C_{W_{\alpha }}\) if \(\alpha \geq 1\). The main goal of the paper is to investigate `Bernstein's weighted problem' in Shape Preserving Approximation using Freud's weight. For \(f\in C_W\), let \[ E_n(f,W):=\inf _{P\in \Pi_n}\|f-P\|_W \quad \quad E_n^k(f,W):=\inf _{P\in \Pi_n^k}\|f-P\|_W, \] where \(\Pi _n\) and \(\Pi _n^k\) denote the sets of all algebraic and all \(k\)-monotone algebraic polynomials of degree \(\leq n\), respectively (\(f:\mathbb{R}\rightarrow \mathbb{R}\) is said to be \(k\)-monotone if \([x_0,x_1,\dots x_k,f]\geq 0\) for any collection of \(k+1\) distinct points \(x_0,x_1,\dots , x_k\)). The main result of the paper reads as follows: Theorem 1. Let \(W_{\alpha }(x):=e^{-|x|^{alpha }}\), \(\alpha \geq 1\), and suppose that \(f\in C_{W_{\alpha }}\) is \(k\)-monotone. Then \[ \lim _{n\rightarrow \infty }E_n^k(f,W_{\alpha })=0. \]
scientific article; zbMATH DE number 5558974
scientific article