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A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis.

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Title: A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis.
Authors: Aimar, Hugo1, Boasso, Juliana1 jboasso@santafe-conicet.gov.ar
Source: Mathematical Modelling & Analysis. 2026, Vol. 31 Issue 3, p431-451. 21p.
Subjects: Lipschitz continuity, Haar function, Measure theory, Wavelet transforms, Image processing, Wavelets (Mathematics)
Abstract: The a-Lipschitz character of a time series or an image summarizes, in the single parameter a, some persistence properties of the original function modeling the given signal. Such is the case of the Hurst exponent widely used in hydrology and other sciences. It is known that the continuous wavelet transform is useful to characterize the a-Lipschitz condition of a function. Recent results show that, in one dimension, the behavior of the Haar coefficients characterize the a-Lipschitz regularity of functions with respect to the induced dyadic metric. In this note, we extend the characterization of dyadic Lipschitz regularity of functions to non-atomic probability spaces, using generalized Haar systems. We also provide some two dimensional examples that can be designed and used to reflect specific textures in images. [ABSTRACT FROM AUTHOR]
Copyright of Mathematical Modelling & Analysis is the property of Vilnius Gediminas Technical University 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: A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis.
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  Data: <searchLink fieldCode="AR" term="%22Aimar%2C+Hugo%22">Aimar, Hugo</searchLink><relatesTo>1</relatesTo><br /><searchLink fieldCode="AR" term="%22Boasso%2C+Juliana%22">Boasso, Juliana</searchLink><relatesTo>1</relatesTo><i> jboasso@santafe-conicet.gov.ar</i>
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  Data: <searchLink fieldCode="DE" term="%22Lipschitz+continuity%22">Lipschitz continuity</searchLink><br /><searchLink fieldCode="DE" term="%22Haar+function%22">Haar function</searchLink><br /><searchLink fieldCode="DE" term="%22Measure+theory%22">Measure theory</searchLink><br /><searchLink fieldCode="DE" term="%22Wavelet+transforms%22">Wavelet transforms</searchLink><br /><searchLink fieldCode="DE" term="%22Image+processing%22">Image processing</searchLink><br /><searchLink fieldCode="DE" term="%22Wavelets+%28Mathematics%29%22">Wavelets (Mathematics)</searchLink>
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  Label: Abstract
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  Data: The a-Lipschitz character of a time series or an image summarizes, in the single parameter a, some persistence properties of the original function modeling the given signal. Such is the case of the Hurst exponent widely used in hydrology and other sciences. It is known that the continuous wavelet transform is useful to characterize the a-Lipschitz condition of a function. Recent results show that, in one dimension, the behavior of the Haar coefficients characterize the a-Lipschitz regularity of functions with respect to the induced dyadic metric. In this note, we extend the characterization of dyadic Lipschitz regularity of functions to non-atomic probability spaces, using generalized Haar systems. We also provide some two dimensional examples that can be designed and used to reflect specific textures in images. [ABSTRACT FROM AUTHOR]
– Name: AbstractSuppliedCopyright
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  Group: Ab
  Data: <i>Copyright of Mathematical Modelling & Analysis is the property of Vilnius Gediminas Technical University 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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        Value: 10.3846/mma.2026.24576
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      – Code: eng
        Text: English
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        PageCount: 21
        StartPage: 431
    Subjects:
      – SubjectFull: Lipschitz continuity
        Type: general
      – SubjectFull: Haar function
        Type: general
      – SubjectFull: Measure theory
        Type: general
      – SubjectFull: Wavelet transforms
        Type: general
      – SubjectFull: Image processing
        Type: general
      – SubjectFull: Wavelets (Mathematics)
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      – TitleFull: A measure theoretic approach to Lipschitz regularity and its Haar type wavelet analysis.
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              Text: 2026
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              Y: 2026
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