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Ψec: A local spectral exterior calculus.

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Title: Ψec: A local spectral exterior calculus.
Authors: Lessig, Christian1 (AUTHOR) christian.lessig@ovgu.de
Source: Applied & Computational Harmonic Analysis. Mar2021, Vol. 51, p56-103. 48p.
Subjects: Differential forms, Differential calculus, Calculus, Fourier transforms, Stokes equations, Wavelets (Mathematics), Integrals
Abstract: We introduce Ψ ec (R n) , a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential r -form wavelets with flexible directional localization that provide tight frames for the spaces Ω r (R n) of forms in R 2 and R 3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the k -dimensional integral of an r -form is an (r − k) -form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of Ψ ec (R n) builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators. [ABSTRACT FROM AUTHOR]
Copyright of Applied & Computational Harmonic Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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  Data: <searchLink fieldCode="DE" term="%22Differential+forms%22">Differential forms</searchLink><br /><searchLink fieldCode="DE" term="%22Differential+calculus%22">Differential calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Calculus%22">Calculus</searchLink><br /><searchLink fieldCode="DE" term="%22Fourier+transforms%22">Fourier transforms</searchLink><br /><searchLink fieldCode="DE" term="%22Stokes+equations%22">Stokes equations</searchLink><br /><searchLink fieldCode="DE" term="%22Wavelets+%28Mathematics%29%22">Wavelets (Mathematics)</searchLink><br /><searchLink fieldCode="DE" term="%22Integrals%22">Integrals</searchLink>
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  Data: We introduce Ψ ec (R n) , a discretization of Cartan's exterior calculus of differential forms using wavelets. Our construction consists of differential r -form wavelets with flexible directional localization that provide tight frames for the spaces Ω r (R n) of forms in R 2 and R 3. By construction, the wavelets satisfy the de Rahm co-chain complex, the Hodge decomposition, and that the k -dimensional integral of an r -form is an (r − k) -form. They also verify Stokes' theorem for differential forms, with the most efficient finite dimensional approximation attained using directionally localized, curvelet- or ridgelet-like forms. The construction of Ψ ec (R n) builds on the geometric simplicity of the exterior calculus in the Fourier domain. We establish this structure by extending existing results on the Fourier transform of differential forms to a frequency description of the exterior calculus, including, for example, a Plancherel theorem for forms and a description of the symbols of all important operators. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Applied & Computational Harmonic Analysis is the property of Academic Press Inc. and its content may not be copied or emailed to multiple sites without the copyright holder's express written permission. Additionally, content may not be used with any artificial intelligence tools or machine learning technologies. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract.</i> (Copyright applies to all Abstracts.)
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      – Type: doi
        Value: 10.1016/j.acha.2020.10.003
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      – Code: eng
        Text: English
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        PageCount: 48
        StartPage: 56
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      – SubjectFull: Differential forms
        Type: general
      – SubjectFull: Differential calculus
        Type: general
      – SubjectFull: Calculus
        Type: general
      – SubjectFull: Fourier transforms
        Type: general
      – SubjectFull: Stokes equations
        Type: general
      – SubjectFull: Wavelets (Mathematics)
        Type: general
      – SubjectFull: Integrals
        Type: general
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      – TitleFull: Ψec: A local spectral exterior calculus.
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              Text: Mar2021
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              Y: 2021
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              Value: 51
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