Riesz–Jacobi Transforms as Principal Value Integrals
Abstract 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 ot...
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Gespeichert in:
| Autor*in: |
Castro, Alejandro J. [verfasserIn] |
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| Format: |
Artikel |
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| Sprache: |
Englisch |
| Erschienen: |
2015 |
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| Schlagwörter: |
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| Anmerkung: |
© Springer Science+Business Media New York 2015 |
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| Übergeordnetes Werk: |
Enthalten in: The journal of Fourier analysis and applications - Springer US, 1994, 22(2015), 3 vom: 30. Sept., Seite 493-541 |
|---|---|
| Übergeordnetes Werk: |
volume:22 ; year:2015 ; number:3 ; day:30 ; month:09 ; pages:493-541 |
| Links: |
|---|
| DOI / URN: |
10.1007/s00041-015-9430-1 |
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OLC2062946120 |
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| 520 | |a Abstract 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. | ||
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10.1007/s00041-015-9430-1 doi (DE-627)OLC2062946120 (DE-He213)s00041-015-9430-1-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn Castro, Alejandro J. verfasserin aut Riesz–Jacobi Transforms as Principal Value Integrals 2015 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Springer Science+Business Media New York 2015 Abstract 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. Jacobi expansion Jacobi operator Riesz transform Integral representation Principal value integral Nowak, Adam aut Szarek, Tomasz Z. aut Enthalten in The journal of Fourier analysis and applications Springer US, 1994 22(2015), 3 vom: 30. Sept., Seite 493-541 (DE-627)185271340 (DE-600)1233179-X (DE-576)079875580 1069-5869 nnns volume:22 year:2015 number:3 day:30 month:09 pages:493-541 https://doi.org/10.1007/s00041-015-9430-1 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_70 GBV_ILN_105 GBV_ILN_2088 AR 22 2015 3 30 09 493-541 |
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Abstract 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. © Springer Science+Business Media New York 2015 |
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Abstract 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. © Springer Science+Business Media New York 2015 |
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Abstract 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. © Springer Science+Business Media New York 2015 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">OLC2062946120</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230331231829.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">200819s2015 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s00041-015-9430-1</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2062946120</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s00041-015-9430-1-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Castro, Alejandro J.</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Riesz–Jacobi Transforms as Principal Value Integrals</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2015</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Springer Science+Business Media New York 2015</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract 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.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Jacobi expansion</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Jacobi operator</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Riesz transform</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Integral representation</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Principal value integral</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Nowak, Adam</subfield><subfield code="4">aut</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Szarek, Tomasz Z.</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">The journal of Fourier analysis and applications</subfield><subfield code="d">Springer US, 1994</subfield><subfield code="g">22(2015), 3 vom: 30. 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