Extension of functions with \(\omega\)-rapid polynomial approximation
For a weight function \(\omega: [0, \infty[\to [0, \infty[\) we denote by \({\mathcal E}_{(\omega)}(\mathbb{R}^ N)\) the class of all \(\omega\)- ultradifferentiable functions of Beurling-type on \(\mathbb{R}^ N\). Each element in \({\mathcal E}_{(\omega)}(\mathbb{R}^ N)\) is a function with \(\omega\)-rapid polynomial approximation on each compact set in \(K\subset \mathbb{R}^ N\), whenever \(\omega\) is a strong weight function, i.e.: \[ \sup_{l\in \mathbb{N}} \inf_{p\in {\mathcal P}^ N_ l} \| f- p\|_ K e^{B\omega(l)}< \infty,\qquad\text{for all } B\geq 1, \] where \({\mathcal P}^ N_ l\) denotes the space of all polynomials in \(N\) variables of degree \(\leq l\) and \(\|\cdot\|_ K\) denotes the sup-norm on \(K\). In the present note there is given a family of weight functions \(\omega\) such that for each function \(f\) with \(\omega\)-rapid polynomial approximation defined on a compact set \(K\) satisfying Markov's inequality for some \(M\) and \(\alpha\): \[ \| p^{(\beta)}\|_ K\leq M l^{\alpha| \beta|} \| p\|_ K,\;p\in {\mathcal P}^ N_ l,\;l\in \mathbb{N},\;\beta\in \mathbb{N}^ N_ 0, \] can be extended to an \(\omega\)-ultradifferentiable function on \(\mathbb{R}^ N\). However, this is not true for the small Gevrey-classes \(\Gamma^{(d)}\).
ultradifferentiable functions of Beurling-type
\(\omega\)-rapid polynomial approximation
scientific article; zbMATH DE number 782145