Polynomial approximation in convex domains
Let \(G\) be a compact convex domain in \(\mathbb{R}^{d}\) and \(\Gamma \subset \partial G\) denotes the set of its conic points (\(x \in \Gamma\) if there are \(d\) linearly independent supporting hyperplanes of \(G\) through \(x\)). For \(f \in C(G)\) and \(t>0,\) the \(k\)th order modulus of smoothness is given by \[ \omega_{k}(f;t) = \sup_{| h | \leq t} \, \sup_{x \in G_{kh}} \, | \Delta_{h}^{k} f(x) | , \] where \(G_{kh} = \{ x : [x,x+kh] \subset G \},\) \(h \in \mathbb{R}^{d}\) and \[ \Delta_{h}^{k} f(x) = \sum_{j=0}^{k} \, (-1)^{k-j} \binom{k}{j} f(x+jh). \] The main result of the paper is the following Theorem. Let \(f \in C^{l}(G)\) and \(\Gamma \subset \partial G,\) \(l \geq 0,\) \(k,n \in \mathbb{Z}_{+}\) satisfy the conditions \(k \geq 1,\) \(n \geq k+l-1\) and \(\operatorname{card} \Gamma < \infty.\) Then there is a linear operator \(R_{n} : C(G) \to \mathcal{P}_{n}\) such that \[ | f(x) - R_{n}f(x) | \leq c \, \Delta_{n}(x)^{l} \sup_{| \alpha | = l} \, \omega_{k}\left( D^{\alpha}f ; \Delta_{n}(x) \right), \quad x \in G; \] \(\mathcal{P}_{n}\) is the space of polynomials in \(x \in \mathbb{R}^{d}\) of degree \(n;\) \(c>0\) is a constant independent of \(x, f, n;\) \(\Delta_{n}(x) = \frac{\sqrt{\mathrm{dist} (x,\Gamma)}}{n+1} + \frac{1}{(n+1)^{2}}\) and \(| \alpha | = \sum_{i=1}^{d} \, \alpha_{i}\) for \(\alpha = (\alpha_{1}, \dots, \alpha_{d}) \in \mathbb{Z}^{d}_{+}.\)
scientific article; zbMATH DE number 6944961
scientific article