Matthias Holschneider

Wavelet analysis of potential fields

It is shown how a continuous wavelet technique may be used to locate and characterize homogeneous point sources from the field they generate measured in a distant hyperplane. For this a class of wavelets is introduced on which the Poisson semi-group essentially acts as a dilation.

On the wavelet transformation of fractal objects

The wavelet transformation is briefly presented. It is shown how the analysis of the local scaling behavior of fractals can be transformed into the investigation of the scaling behavior of analytic functions over the half-plane near the boundary of its domain of analyticity. As an example, a “Weierstrass-like” fractal function is considered, for which the wavelet transform is related to a Jacobi theta function. Some of the scalings of this theta function are analyzed, and give some information about the scaling behavior of this fractal.

Inverse Radon transforms through inverse wavelet transforms

The author shows, by considering the Radon inversion problem as an example, how to use the inverse wavelet transform technique to invert data obtained from nonorthogonal projections having some underlying symmetry group.

Continuous wavelet transforms on the sphere

In this very short paper we shall construct a continuous wavelet analysis based on dilations translations and rotations on the sphere. It is the analog of the construction proposed by Murenzi [in his thesis, 1990], on R2. At small scale we shall recover the Euclidian structure of the sphere. At large scale we obtain that the wavelet transform decays rapidly because the sphere is compact.

Wavelet analysis on the circle

The construction of a wavelet analysis over the circle is presented. The spaces of infinitely times differentiable functions, tempered distributions, and square integrable functions over the circle are analyzed by means of the wavelet transform.

Fractal Wavelet Dimensions and Localization

AbstractIn this paper we want to give a new definition of fractal dimensions as small scale behavior of theq-energy of wavelet transforms. This is a generalization of previous multi-fractal approaches. With this particular definition we will show that the 2-dimension (=correlation dimension) of the spectral measure determines the long time behavior of the time evolution generated by a bounded self-adjoint operator acting in some Hilbert space ℋ. It will be proved that for φ, ψ∈ℋ we have

Electromagnetic scattering from multi-scale rough surfaces

\mathop {\lim \inf }\limits_{T \to \infty } \frac{{\log \int_0^T {d\omega \left| {\left\langle {\psi \left| {e^{ - iA\omega } } \right.\phi } \right\rangle } \right|^2 } }}{{\log T}} = - \kappa ^ + (2)

Localization properties of wavelet transforms

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