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Regularity analysis of functions and random processes using wavelets

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Wavelets and Their Applications

Part of the book series: NATO ASI Series ((ASIC,volume 442))

Abstract

In the first part of the paper, we recall how the pointwise smoothness of functions can be studied using orthonormal bases of wavelets, and we give three applications: pointwise regularity of elliptic operators, refined Sobolev imbeddings, and the construction of multifractal functions having a prescribed spectrum of singularities.

In the second part of the paper, we show how to construct wavelets that are orthonormal for the scalar product 〈Af, g〉 where A is a positive differential or pseudodifferential operator. We use these wavelets to decompose a large class of multidimensional Gaussian processes, including fractional Brownian motion, but also processes with nonstationary increments. This decomposition simplifies the simulation of the process (its wavelet coefficients are independent, identically distributed Gaussiane), and it allows us to calculate its exact local and global modulus of continuity.

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Author information

Authors and Affiliations

  1. CERMA (U.R.A. 1502 du C.N.R.S.), Ecole Nationale des Ponts et Chaussée la Courtine, 93167, Noisy le Grand, France

    Stéphane Jaffard

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  1. Stéphane Jaffard
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Editor information

Editors and Affiliations

  1. Prometheus Inc., Newport, Rhode Island, USA

    J. S. Byrnes

  2. University of Massachusetts, Boston, Massachusetts, USA

    J. S. Byrnes , Kathryn A. Hargreaves  & Karl Berry ,  & 

  3. Harvard University, Cambridge, Massachusetts, USA

    Jennifer L. Byrnes

  4. Interleaf, Cambridge, Massachusetts, USA

    Karl Berry

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© 1994 Springer Science+Business Media Dordrecht

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Jaffard, S. (1994). Regularity analysis of functions and random processes using wavelets. In: Byrnes, J.S., Byrnes, J.L., Hargreaves, K.A., Berry, K. (eds) Wavelets and Their Applications. NATO ASI Series, vol 442. Springer, Dordrecht. https://doi.org/10.1007/978-94-011-1028-0_4

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  • DOI: https://doi.org/10.1007/978-94-011-1028-0_4

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-010-4448-6

  • Online ISBN: 978-94-011-1028-0

  • eBook Packages: Springer Book Archive

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