Felten, Michael Local estimates for Jacobi polynomials. (English) Zbl 1134.33006 JIPAM, J. Inequal. Pure Appl. Math. 8, No. 1, Paper No. 3, 7 p. (2007). Let \(\omega_k^{(\alpha ,\beta)}(x):=(\sqrt{1-x}+\frac{1}{n})^{2\alpha}(\sqrt{1+x}+\frac{1}{n})^{2\beta}\), \(x\in [-1,1]\) defines the so-called modified Jacobi weights. The author proves that the orthogonal Jacobi polynomials \(p_n^{(\alpha ,\beta)}\), \((\alpha ,\beta\geq -\frac{1}{2})\) are bounded on some set of subintervals \(U_n(x)\) of \([-1,1]\) by a constant divided by a function similar to \(\omega_k^{(\alpha,\beta)}\). He shows that his local estimate leads to the following estimate: \(\int_{U_n(x)}| p_n^{(\alpha ,\beta)}(t)| ^2\omega_k^{(\alpha,\beta)}(t)\,dt\leq\frac{C(\alpha,\beta)}{n}\). Reviewer: Jacek Gilewicz (Marseille) Cited in 2 Documents MSC: 33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis Keywords:Jacobi polynomials; Jacobi weights; local estimates PDFBibTeX XMLCite \textit{M. Felten}, JIPAM, J. Inequal. Pure Appl. Math. 8, No. 1, Paper No. 3, 7 p. (2007; Zbl 1134.33006) Full Text: EuDML EMIS