Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations) |
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| Authors: | M Felten |
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| Institution: | (1) Faculty of Mathematics and Informatics, University of Hagen, 58084 Hagen, Germany |
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| Abstract: | The paper is concerned with bounds for integrals of the type , involving Jacobi polynomials p
n
(α,β)
and Jacobi weights w
(a,b)
depending on α,β, a, b > −1, where the subsets U
k
(x) ⊂ −1, 1] located around x and are given by ![$$
U_k (x) = \left {x - \tfrac{{\phi _k (x)}}
{k}, x + \tfrac{{\phi _k (x)}}
{k}} \right] \cap - 1, 1]
$$](/content/y16118177v84j716/10474_2007_6208_Article_IEq1.gif) with  . The functions to be integrated will also be of the type  on the domain −1,1] t/ U
k
(x). This approach uses estimates of Jacobi polynomials modified Jacobi weights initiated by Totik and Lubinsky in 1]. Various
bounds for integrals involving Jacobi weights will be derived. The results of the present paper form the basis of the proof
of the uniform boundedness of (C, 1) means of Jacobi expansions in weighted sup norms in 3].
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| Keywords: | and phrases" target="_blank"> and phrases Jacobi polynomials Jacobi weights local estimates bounds for integrals |
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