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Riesz-Jacobi Transforms as Principal Value Integrals
Uppsala University, Disciplinary Domain of Science and Technology, Mathematics and Computer Science, Department of Mathematics, Analysis and Probability Theory.
Polish Acad Sci, Inst Math, Sniadeckich 8, PL-00656 Warsaw, Poland..
Polish Acad Sci, Inst Math, Sniadeckich 8, PL-00656 Warsaw, Poland..
2016 (English)In: Journal of Fourier Analysis and Applications, ISSN 1069-5869, E-ISSN 1531-5851, Vol. 22, no 3, 493-541 p.Article in journal (Refereed) PublishedText
Abstract [en]

We establish an integral representation for the Riesz transforms naturally associated with classical Jacobi expansions. We prove that the Riesz-Jacobi transforms of odd orders express as principal value integrals against kernels having non-integrable singularities on the diagonal. On the other hand, we show that the Riesz-Jacobi transforms of even orders are not singular operators. In fact they are given as usual integrals against integrable kernels plus or minus, depending on the order, the identity operator. Our analysis indicates that similar results, existing in the literature and corresponding to several other settings related to classical discrete and continuous orthogonal expansions, should be reinvestigated so as to be refined and in some cases also corrected.

Place, publisher, year, edition, pages
2016. Vol. 22, no 3, 493-541 p.
Keyword [en]
Jacobi expansion, Jacobi operator, Riesz transform, Integral representation, Principal value integral
National Category
Mathematics
Identifiers
URN: urn:nbn:se:uu:diva-298069DOI: 10.1007/s00041-015-9430-1ISI: 000376246800001OAI: oai:DiVA.org:uu-298069DiVA: diva2:944861
Available from: 2016-06-30 Created: 2016-06-29 Last updated: 2016-06-30Bibliographically approved

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Castro, Alejandro J.
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