Sharp estimates for the polynomial approximation in weighted Sobolev spaces (Q745266)

The authors obtain a sharp estimate of the form \[ \inf_{p \in \mathcal{P}_{n-1}} \| u-p \|_{V_{\omega_{\alpha}}^{s}(\Omega)} \leq C_{st} N^{-(t-s)} \| u \|_{V_{\omega_{\alpha}}^{t}(\Omega)}, \;\;0 \leq s \leq t \leq N, \] for the case in which \(\omega_{\alpha}\) is a Jacobi, Laguerre or Hermite weight function and \(\Omega\) is an interval, the half-line or the entire real line, respectively; \(\mathcal{P}_{n}\) denotes the set of all algebraic polynomials of order \(\leq n,\) and the space \(V_{\omega_{\alpha}}^{s}(\Omega)\) is defined as the set of all functions \(u\) for which \[ \| u \|^{2}_{V_{\omega_{\alpha}}^{s}(\Omega)} = \int_{\Omega} \, \sum_{j=0}^{s} \, \omega_{\alpha+j}(x) \Bigm| \frac{d^{j}u}{dx^{j}}(x) \Bigm|^{2} \, dx < \infty. \] Further, it is shown that the orthogonal polynomials corresponding to \(\omega_{\alpha}\) form an orthogonal basis in \(V_{\omega_{\alpha}}^{m}(\Omega)\) for each integer \(m \geq 0;\) it is found the constant \(C_{stN\alpha}\) in the sharp estimate \[ | u-u_{N-1} |_{V_{\omega_{\alpha}}^{s}(\Omega)} \leq C_{stN\alpha} | u |_{V_{\omega_{\alpha}}^{t}(\Omega)}, \;\;0 \leq s \leq t \leq N, \] in closed form, where \(u_{N-1}\) is the best approximation in \(\mathcal{P}_{N-1}\) to \(u \in V_{\omega_{\alpha}}^{s}(\Omega),\) and \(| u |_{V_{\omega_{\alpha}}^{s}(\Omega)}\) is the semi-norm defined by \[ | u |^{2}_{V_{\omega_{\alpha}}^{s}(\Omega)} = \int_{\Omega} \, \omega_{\alpha+s}(x) | D^{s}u(x) |^{2} \, dx; \] finally, an sharp Markov-Bernstein type estimate \[ | p_{N} |_{V_{\omega_{\alpha}}^{t}(\Omega)} \leq C_{stN\alpha}^{-1} | p_{N} |_{V_{\omega_{\alpha}}^{s}(\Omega)} \] is obtained, where \(t \geq s \geq 0,\) \(p_{N} \in \mathcal{P}_{N}\) and \(C_{stN\alpha}\) is the same constant as above.