Polynomial approximation on varying sets (Q1925038)

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 \(C^0_\varphi(\mathbb{R}^d)\).