N-dimensional affine Weyl-Heisenberg wavelets
Neue und alte Thesen über Wavelets.
New constructions of piecewise-constant wavelets.
New uncertainty principles for the continuous Gabor transform and the continuous wavelet transform.
Non-Lebesgue multiresolution analyses
Classical notions of wavelets and multiresolution analyses deal with the Hilbert space L²(ℝ) and the standard translation and dilation operators. Key in the study of these subjects is the low-pass filter, which is a periodic function h ∈ L²([0,1)) that satisfies the classical quadrature mirror filter equation |h(x)|²+|h(x+1/2)|² = 2. This equation is satisfied almost everywhere with respect to Lebesgue measure on the torus. Generalized multiresolution analyses and wavelets exist in abstract Hilbert...
Non-MSF Wavelets for the Hardy Space H²(ℝ)
All wavelets constructed so far for the Hardy space H²(ℝ) are MSF wavelets. We construct a family of H²-wavelets which are not MSF. An equivalence relation on H²-wavelets is introduced and it is shown that the corresponding equivalence classes are non-empty. Finally, we construct a family of H²-wavelets with Fourier transform not vanishing in any neighbourhood of the origin.
Non-separable bidimensional wavelet bases.
We build orthonormal and biorthogonal wavelet bases of L2(R2) with dilation matrices of determinant 2. As for the one dimensional case, our construction uses a scaling function which solves a two-scale difference equation associated to a FIR filter. Our wavelets are generated from a single compactly supported mother function. However, the regularity of these functions cannot be derived by the same approach as in the one dimensional case. We review existing techniques to evaluate the regularity of...
Numerical experiments in Fourier asymptotics of Cantor measures and wavelets.
On a property of continuous solutions of the dilation equation with positive coefficients
On a Theorem of Ingham.
On Discrete and Continous Norms in Paley-Wiener Spaces and Consequences for Exponential Frames.
On Discrete Frames Associated with Semidirect Products.
On extension of Gabor transform to Boehmians
On infinitely smooth almost-wavelets with compact support.
There are known wavelets with exponential decay on infinity [2,3,4] and wavelets with compact support [5]. But these functions have finite smoothness. It is known that there do not exist infinitely differentiable compactly supported wavelets.
On Multiresolution Analysis (MRA) Wavelets in ...
On multiscale denoising of spherical functions: basic theory and numerical aspects.
On Orthogonal Wavelets with the Oversampling Property.
On orthonormal product systems.
On the -problem in Weyl-Heisenberg frames
Let . We investigate the characterization problem which asks for a classification of all the triples such that the Weyl-Heisenberg system is a frame for . It turns out that the answer to the problem is quite complicated, see Gu and Han (2008) and Janssen (2003). Using a dilation technique, one can reduce the problem to the case where and only let and vary. In this paper, we extend the Zak transform technique and use the Fourier analysis technique to study the problem for the case of...
On the adaptive wavelet estimation of a multidimensional regression function under -mixing dependence: Beyond the standard assumptions on the noise
We investigate the estimation of a multidimensional regression function from observations of an -mixing process , where , represents the design and the noise. We concentrate on wavelet methods. In most papers considering this problem, either the proposed wavelet estimator is not adaptive (i.e., it depends on the knowledge of the smoothness of in its construction) or it is supposed that is bounded or/and has a known distribution. In this paper, we go far beyond this classical framework....