Polynomial approximation in weighted Sobolev space (Q2720946)

This paper deals with the following inequality in the weighted Sobolev space \( W^{m,p}(\Omega, \sigma) \): NEWLINE\[NEWLINE\inf_{Q \in {\mathbf K}} ||f - Q||_{W^{m,p}(\Omega,\sigma)} \leq c(\Omega) \sum_{j} ||P_{j}(D) f||_{L_{p_{j}}(\Omega,\sigma)}, NEWLINE\]NEWLINE where \( ||.||_{W^{m,p}(\Omega,\sigma)} \) denotes the norm in the weighted Sobolev space \( W^{m,p}(\Omega,\sigma) \), \( \Omega \) being a bounded domain in \( {\mathbb R}^{n} \), \( n \geq 3 \), \( \{P_{j} \}^{k}_{j=1} \) is a set of homogeneous polynomials having no common complex zero and \( {\mathbf K} \) is the intersection of the kernels of the operators \( P_{j}(D) \). Possible applications in estimating the error of the finite element method are mentioned.