Chistyakova, Nataly; Logvinenko, Vladimir Polynomial approximation on varying sets. (English) Zbl 0858.41003 J. Approximation Theory 86, No. 2, 144-178 (1996). The authors consider functions \(f:\mathbb{R}^d\to\mathbb{C}\) such that for a given sequence \(\{\gamma_m\}^\infty_{m=1}\), \(\gamma_m>0\), and for every \(\xi\in\mathbb{R}\) there exist polynomials \(P_m(x)=P_m(x;\xi)\), \(\deg P_m\leq m\), \(m=1,2,\dots\), which satisfy the following conditions \[ \sup\{|f(x)-P_m(x;\xi)|: |x-\xi|\leq\gamma_m\}\leq Ce^{-m},\;C=C_f. \] Smoothness, quasianalytic and analytic properties of \(f\) in terms of the sequences \(\{\gamma_m\}\) are investigated in the paper. The authors use these properties to obtain estimates (Cartwright-type theorems) on \(\mathbb{R}^d=\text{Re }\mathbb{C}^d\) for entire functions of exponential type bounded on some discrete subset of \(\mathbb{R}^d\). They construct a weight function \(\varphi:\mathbb{R}^d\to\mathbb{R}\), \(d>1\), such that algebraic polynomials are dense in \(C^0_{\varphi|A}(A)\) for every affine subspace \(A\subset\mathbb{R}^d\), \(A\neq\mathbb{R}^d\), but not dense in the space \(C^0_\varphi(\mathbb{R}^d)\). Reviewer: L.I.Ronkin (Khar’kov) MSC: 41A10 Approximation by polynomials 41A63 Multidimensional problems 32A15 Entire functions of several complex variables Keywords:polynomial approximation; quasianalyticity; entire functions of exponential type PDFBibTeX XMLCite \textit{N. Chistyakova} and \textit{V. Logvinenko}, J. Approx. Theory 86, No. 2, 144--178 (1996; Zbl 0858.41003) Full Text: DOI