A Cohen Type Inequality for Fourier Expansions of Orthogonal Polynomials with a Nondiscrete Jacobi-Sobolev Inner Product
<p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-128746-i1.gif"/></inline-formula> denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product <inline-formula> <graphic file="1029-242X-2010-12874...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
SpringerOpen
2010-01-01
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| Series: | Journal of Inequalities and Applications |
| Online Access: | http://www.journalofinequalitiesandapplications.com/content/2010/128746 |
| Summary: | <p/> <p>Let <inline-formula> <graphic file="1029-242X-2010-128746-i1.gif"/></inline-formula> denote the sequence of polynomials orthogonal with respect to the non-discrete Sobolev inner product <inline-formula> <graphic file="1029-242X-2010-128746-i2.gif"/></inline-formula>, where <inline-formula> <graphic file="1029-242X-2010-128746-i3.gif"/></inline-formula> and <inline-formula> <graphic file="1029-242X-2010-128746-i4.gif"/></inline-formula> with <inline-formula> <graphic file="1029-242X-2010-128746-i5.gif"/></inline-formula>, <inline-formula> <graphic file="1029-242X-2010-128746-i6.gif"/></inline-formula>. In this paper, we prove a Cohen type inequality for the Fourier expansion in terms of the orthogonal polynomials <inline-formula> <graphic file="1029-242X-2010-128746-i7.gif"/></inline-formula> Necessary conditions for the norm convergence of such a Fourier expansion are given. Finally, the failure of almost everywhere convergence of the Fourier expansion of a function in terms of the orthogonal polynomials associated with the above Sobolev inner product is proved.</p> |
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| ISSN: | 1025-5834 1029-242X |