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Local estimates for Jacobi polynomials. (English) Zbl 1134.33006

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: \(\int_{U_n(x)}| p_n^{(\alpha ,\beta)}(t)| ^2\omega_k^{(\alpha,\beta)}(t)\,dt\leq\frac{C(\alpha,\beta)}{n}\).

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
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