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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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
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