Constrained approximation with Jacobi weights (Q2786455)

In this paper the authors continue their research on constrained polynomial approximation of functions, a subject on which they already have several known contributions. Here they get exact results on the degree of \(k\) -monotone \((k=1,2)\) polynomial approximation with the Jacobi weighted norm of \(L_{w_{\alpha ,\beta }}^{p}\), \(w_{\alpha ,\beta }\left( x\right) =\left( 1+x^{\alpha }\right) \left( 1-x^{\beta }\right) \), \(\alpha , \beta >-1/p\) if \( 1\leq p<\infty \) or \(\alpha ,\beta \geq 0\) if \(p=\infty \). To this aim the authors introduce appropriate modifications of other known moduli of smoothness, as well as novel techniques. They apply the new results to the study of certain related constrained spline approximation problems. The paper is clearly written and the proofs are complete and rigorous.