A Fractal Plancherel Theorem
- Autores
- Molter, Ursula Maria; Zuberman, Leandro
- Año de publicación
- 2009
- Idioma
- inglés
- Tipo de recurso
- artículo
- Estado
- versión publicada
- Descripción
- A measure µ on R n is called locally and uniformly h-dimensional if µ(Br(x)) ≤ h(r) for all x ∈ R n and for all 0 < r < 1, where h is a real valued function. If f ∈ L 2 (µ) and Fµf denotes its Fourier transform with respect to µ, it is not true (in general) that Fµf ∈ L 2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L 2 (µ) the L 2 -norm of its Fourier transform restricted to a ball of radius r has the same order of growth as r nh(r −1 ) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L 2 (µ)-norm of f (Theorem 3.2). By imposing certain restrictions on the measure µ, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which h(x) = x α .
Fil: Molter, Ursula Maria. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina
Fil: Zuberman, Leandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina - Materia
-
HAUSDORFF MEASURES
FOURIER TRANSFORM
DIMENSION
PLANCHEREL - Nivel de accesibilidad
- acceso abierto
- Condiciones de uso
- https://creativecommons.org/licenses/by-nc-sa/2.5/ar/
- Repositorio
.jpg)
- Institución
- Consejo Nacional de Investigaciones Científicas y Técnicas
- OAI Identificador
- oai:ri.conicet.gov.ar:11336/245148
Ver los metadatos del registro completo
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A Fractal Plancherel TheoremMolter, Ursula MariaZuberman, LeandroHAUSDORFF MEASURESFOURIER TRANSFORMDIMENSIONPLANCHERELhttps://purl.org/becyt/ford/1.1https://purl.org/becyt/ford/1A measure µ on R n is called locally and uniformly h-dimensional if µ(Br(x)) ≤ h(r) for all x ∈ R n and for all 0 < r < 1, where h is a real valued function. If f ∈ L 2 (µ) and Fµf denotes its Fourier transform with respect to µ, it is not true (in general) that Fµf ∈ L 2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L 2 (µ) the L 2 -norm of its Fourier transform restricted to a ball of radius r has the same order of growth as r nh(r −1 ) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L 2 (µ)-norm of f (Theorem 3.2). By imposing certain restrictions on the measure µ, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which h(x) = x α .Fil: Molter, Ursula Maria. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; ArgentinaFil: Zuberman, Leandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; ArgentinaMichigan State University Press2009-03info:eu-repo/semantics/articleinfo:eu-repo/semantics/publishedVersionhttp://purl.org/coar/resource_type/c_6501info:ar-repo/semantics/articuloapplication/pdfapplication/pdfapplication/pdfapplication/pdfhttp://hdl.handle.net/11336/245148Molter, Ursula Maria; Zuberman, Leandro; A Fractal Plancherel Theorem; Michigan State University Press; Real Analysis Exchange; 34; 1; 3-2009; 1-160147-19371930-1219CONICET DigitalCONICETenginfo:eu-repo/semantics/altIdentifier/url/https://projecteuclid.org/journals/real-analysis-exchange/volume-34/issue-1/A-Fractal-Plancherel-Theorem/rae/1242738921.fullinfo:eu-repo/semantics/openAccesshttps://creativecommons.org/licenses/by-nc-sa/2.5/ar/reponame:CONICET Digital (CONICET)instname:Consejo Nacional de Investigaciones Científicas y Técnicas2026-05-27T14:35:50Zoai:ri.conicet.gov.ar:11336/245148instacron:CONICETInstitucionalhttp://ri.conicet.gov.ar/Organismo científico-tecnológicoNo correspondehttp://ri.conicet.gov.ar/oai/requestdasensio@conicet.gov.ar; lcarlino@conicet.gov.arArgentinaNo correspondeNo correspondeNo correspondeopendoar:34982026-05-27 14:35:50.529CONICET Digital (CONICET) - Consejo Nacional de Investigaciones Científicas y Técnicasfalse |
| dc.title.none.fl_str_mv |
A Fractal Plancherel Theorem |
| title |
A Fractal Plancherel Theorem |
| spellingShingle |
A Fractal Plancherel Theorem Molter, Ursula Maria HAUSDORFF MEASURES FOURIER TRANSFORM DIMENSION PLANCHEREL |
| title_short |
A Fractal Plancherel Theorem |
| title_full |
A Fractal Plancherel Theorem |
| title_fullStr |
A Fractal Plancherel Theorem |
| title_full_unstemmed |
A Fractal Plancherel Theorem |
| title_sort |
A Fractal Plancherel Theorem |
| dc.creator.none.fl_str_mv |
Molter, Ursula Maria Zuberman, Leandro |
| author |
Molter, Ursula Maria |
| author_facet |
Molter, Ursula Maria Zuberman, Leandro |
| author_role |
author |
| author2 |
Zuberman, Leandro |
| author2_role |
author |
| dc.subject.none.fl_str_mv |
HAUSDORFF MEASURES FOURIER TRANSFORM DIMENSION PLANCHEREL |
| topic |
HAUSDORFF MEASURES FOURIER TRANSFORM DIMENSION PLANCHEREL |
| purl_subject.fl_str_mv |
https://purl.org/becyt/ford/1.1 https://purl.org/becyt/ford/1 |
| dc.description.none.fl_txt_mv |
A measure µ on R n is called locally and uniformly h-dimensional if µ(Br(x)) ≤ h(r) for all x ∈ R n and for all 0 < r < 1, where h is a real valued function. If f ∈ L 2 (µ) and Fµf denotes its Fourier transform with respect to µ, it is not true (in general) that Fµf ∈ L 2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L 2 (µ) the L 2 -norm of its Fourier transform restricted to a ball of radius r has the same order of growth as r nh(r −1 ) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L 2 (µ)-norm of f (Theorem 3.2). By imposing certain restrictions on the measure µ, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which h(x) = x α . Fil: Molter, Ursula Maria. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentina Fil: Zuberman, Leandro. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria; Argentina |
| description |
A measure µ on R n is called locally and uniformly h-dimensional if µ(Br(x)) ≤ h(r) for all x ∈ R n and for all 0 < r < 1, where h is a real valued function. If f ∈ L 2 (µ) and Fµf denotes its Fourier transform with respect to µ, it is not true (in general) that Fµf ∈ L 2 (e.g. [10]). However in this paper we prove that, under certain hypothesis on h, for any f ∈ L 2 (µ) the L 2 -norm of its Fourier transform restricted to a ball of radius r has the same order of growth as r nh(r −1 ) when r → ∞. Moreover we prove that the ratio between these quantities is bounded by the L 2 (µ)-norm of f (Theorem 3.2). By imposing certain restrictions on the measure µ, we can also obtain a lower bound for this ratio (Theorem 4.3). These results generalize the ones obtained by Strichartz in [10] where he considered the particular case in which h(x) = x α . |
| publishDate |
2009 |
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2009-03 |
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info:eu-repo/semantics/article info:eu-repo/semantics/publishedVersion http://purl.org/coar/resource_type/c_6501 info:ar-repo/semantics/articulo |
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article |
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publishedVersion |
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http://hdl.handle.net/11336/245148 Molter, Ursula Maria; Zuberman, Leandro; A Fractal Plancherel Theorem; Michigan State University Press; Real Analysis Exchange; 34; 1; 3-2009; 1-16 0147-1937 1930-1219 CONICET Digital CONICET |
| url |
http://hdl.handle.net/11336/245148 |
| identifier_str_mv |
Molter, Ursula Maria; Zuberman, Leandro; A Fractal Plancherel Theorem; Michigan State University Press; Real Analysis Exchange; 34; 1; 3-2009; 1-16 0147-1937 1930-1219 CONICET Digital CONICET |
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eng |
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eng |
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