Discrete Weierstrass Transform: Generalisations
The classical Weierstrass transform is an isometric operator mapping elements of the weighted L 2 - space L 2 ( R , exp ( - x 2 / 2 ) ) to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis in one dimension, where functions are defined on a discr...
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| Veröffentlicht in: | Complex analysis and operator theory 2024-02, Vol.18 (2), Article 20 |
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| creator | Massé, A. De Ridder, H. |
| description | The classical Weierstrass transform is an isometric operator mapping elements of the weighted
L
2
-
space
L
2
(
R
,
exp
(
-
x
2
/
2
)
)
to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis in one dimension, where functions are defined on a discrete line rather than the continuous space. This transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. The aim of this paper is to extend the definition to higher dimensions, where we must take into account the anticommutativity of the basic Clifford elements and use the generalised discrete Hermite polynomials. Furthermore, we investigate, back in dimension 1, the asymptotic behaviour if the mesh width approaches 0. |
| doi_str_mv | 10.1007/s11785-023-01464-3 |
| format | Article |
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L
2
-
space
L
2
(
R
,
exp
(
-
x
2
/
2
)
)
to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis in one dimension, where functions are defined on a discrete line rather than the continuous space. This transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. The aim of this paper is to extend the definition to higher dimensions, where we must take into account the anticommutativity of the basic Clifford elements and use the generalised discrete Hermite polynomials. Furthermore, we investigate, back in dimension 1, the asymptotic behaviour if the mesh width approaches 0.</description><identifier>ISSN: 1661-8254</identifier><identifier>EISSN: 1661-8262</identifier><identifier>DOI: 10.1007/s11785-023-01464-3</identifier><language>eng</language><publisher>Cham: Springer International Publishing</publisher><subject>Analysis ; Asymptotic properties ; Hermite polynomials ; Higher Dimensional Geometric Function Theory and Hypercomplex Analysis ; Mathematical analysis ; Mathematics ; Mathematics and Statistics ; Operator Theory ; Operators (mathematics)</subject><ispartof>Complex analysis and operator theory, 2024-02, Vol.18 (2), Article 20</ispartof><rights>The Author(s), under exclusive licence to Springer Nature Switzerland AG 2024. Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-6f84101dc961fa0b4ef1ef5d75731ec26c329ddd2d3cdad0e8f1b57354db3fcb3</cites></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s11785-023-01464-3$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s11785-023-01464-3$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>Massé, A.</creatorcontrib><creatorcontrib>De Ridder, H.</creatorcontrib><title>Discrete Weierstrass Transform: Generalisations</title><title>Complex analysis and operator theory</title><addtitle>Complex Anal. Oper. Theory</addtitle><description>The classical Weierstrass transform is an isometric operator mapping elements of the weighted
L
2
-
space
L
2
(
R
,
exp
(
-
x
2
/
2
)
)
to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis in one dimension, where functions are defined on a discrete line rather than the continuous space. This transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. The aim of this paper is to extend the definition to higher dimensions, where we must take into account the anticommutativity of the basic Clifford elements and use the generalised discrete Hermite polynomials. Furthermore, we investigate, back in dimension 1, the asymptotic behaviour if the mesh width approaches 0.</description><subject>Analysis</subject><subject>Asymptotic properties</subject><subject>Hermite polynomials</subject><subject>Higher Dimensional Geometric Function Theory and Hypercomplex Analysis</subject><subject>Mathematical analysis</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Operator Theory</subject><subject>Operators (mathematics)</subject><issn>1661-8254</issn><issn>1661-8262</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2024</creationdate><recordtype>article</recordtype><recordid>eNp9kE1Lw0AQhhdRsFb_gKeC57U7-5XEm1StQsFLxeOy2Z2VlDapO-nBf280ojdPMzDP-w48jF2CuAYhijkBFKXhQiouQFvN1RGbgLXAS2nl8e9u9Ck7I9oIYUVRVRM2v2soZOxx9ooNZuqzJ5qts28pdXl3M1tii9lvG_J907V0zk6S3xJe_Mwpe3m4Xy8e-ep5-bS4XfEgC9Fzm0oNAmKoLCQvao0JMJlYmEIBBmmDklWMUUYVoo8CywT1cDM61iqFWk3Z1di7z937Aal3m-6Q2-GlkxUAKGVAD5QcqZA7oozJ7XOz8_nDgXBfYtwoxg1i3LcYp4aQGkM0wO0b5r_qf1KfpXNmLQ</recordid><startdate>20240201</startdate><enddate>20240201</enddate><creator>Massé, A.</creator><creator>De Ridder, H.</creator><general>Springer International Publishing</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope></search><sort><creationdate>20240201</creationdate><title>Discrete Weierstrass Transform: Generalisations</title><author>Massé, A. ; De Ridder, H.</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-6f84101dc961fa0b4ef1ef5d75731ec26c329ddd2d3cdad0e8f1b57354db3fcb3</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2024</creationdate><topic>Analysis</topic><topic>Asymptotic properties</topic><topic>Hermite polynomials</topic><topic>Higher Dimensional Geometric Function Theory and Hypercomplex Analysis</topic><topic>Mathematical analysis</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Operator Theory</topic><topic>Operators (mathematics)</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>Massé, A.</creatorcontrib><creatorcontrib>De Ridder, H.</creatorcontrib><collection>CrossRef</collection><jtitle>Complex analysis and operator theory</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Massé, A.</au><au>De Ridder, H.</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Discrete Weierstrass Transform: Generalisations</atitle><jtitle>Complex analysis and operator theory</jtitle><stitle>Complex Anal. Oper. Theory</stitle><date>2024-02-01</date><risdate>2024</risdate><volume>18</volume><issue>2</issue><artnum>20</artnum><issn>1661-8254</issn><eissn>1661-8262</eissn><abstract>The classical Weierstrass transform is an isometric operator mapping elements of the weighted
L
2
-
space
L
2
(
R
,
exp
(
-
x
2
/
2
)
)
to the Fock space. We defined an analogue version of this transform in discrete Hermitian Clifford analysis in one dimension, where functions are defined on a discrete line rather than the continuous space. This transform is based on the classical definition, in combination with a discrete version of the Gaussian function and discrete counterparts of the classical Hermite polynomials. The aim of this paper is to extend the definition to higher dimensions, where we must take into account the anticommutativity of the basic Clifford elements and use the generalised discrete Hermite polynomials. Furthermore, we investigate, back in dimension 1, the asymptotic behaviour if the mesh width approaches 0.</abstract><cop>Cham</cop><pub>Springer International Publishing</pub><doi>10.1007/s11785-023-01464-3</doi></addata></record> |
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| ispartof | Complex analysis and operator theory, 2024-02, Vol.18 (2), Article 20 |
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| language | eng |
| recordid | cdi_proquest_journals_2911133514 |
| source | SpringerLink Journals - AutoHoldings |
| subjects | Analysis Asymptotic properties Hermite polynomials Higher Dimensional Geometric Function Theory and Hypercomplex Analysis Mathematical analysis Mathematics Mathematics and Statistics Operator Theory Operators (mathematics) |
| title | Discrete Weierstrass Transform: Generalisations |
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