International Journal of Wavelets, Multiresolution and Information Processing

In this paper we construct compactly supported biorthogonal wavelet bases of the interval, with dilation factor M. Next, the natural MRA on the cube arising from the tensor product of a multilevel decomposition of the unit interval is developed. New Jackson and Bernstein type inequalities are proved, providing a characterization for anisotropic Sobolev spaces.

We study the wavelet transform of functions invariant under a symmetry group, where the wavelet transform is associated to an irreducible unitary group representation. Among other results a new inversion formula and a new covariance principle are derived. As main examples we discuss the continuous wavelet transform and the short time Fourier transform of radially symmetric functions on ℝ^d.

This paper deals with the construction of generalized uncertainty relations. Starting with a two-operator parabola ansatz, we derive a new set of uncertainties by extending the parabola ansatz to quadratic forms. This setting can be applied to the computation of uncertainties where more than two operators are involved. For affine Weyl–Heisenberg groups, some examples are explored.

Wavelet and frames have become a widely used tool in mathematics, physics, and applied science during the last decade. In this paper we discuss the construction of frames for L²(ℝⁿ) using the action of closed reductive subgroups H ⊆ GL(n,ℝ) such that H has an open orbit in ℝⁿ under the action (h,ω) ↦ (h^-1)^T(ω). In particular, we show that if is an affine symmetric space, then there is a canonical way to construct frames.

In this paper we introduce quaternion-valued multi-resolution analysis. Applying the theory of matrix-valued wavelet analysis, we give the construction of scaling functions and wavelets by identifying the quaternion-valued function with the complex duplex matrix-valued function.

We give a sufficient condition for a function to belong to some sampling space. Also, we prove that a sampling space must be contained in another one which has a Riesz basis consisting of integer translations of a base function. Examples are given.

We ask what conditions can be placed on generators φ of principal shift invariant spaces to ensure the validity of analogues of the classical sampling theorem for bandlimited signals. Critical rate sampling schemes lead to expansion formulas in terms of samples, while oversampling schemes can lead to expansions in which function values depend only on nearby samples. The basic techniques for validating such schemes are built on the Zak transform and the Poisson summation formula. Validation conditions are phrased in terms of orthogonality, smoothness, and self-similarity, as well as bandlimitedness or compact support of the generator. Effective sampling rates which depend on the length of support of the generator or its Fourier transform are derived.

This paper presents a coarse quantization algorithm (TFΣΔ-II) for tight Gabor frame expansions of certain functions in L²(ℝ), an alternative to the TFΣΔ of Ref. 11. By using some a priori information about the function to be quantized and compromising the translation invariance of the TFΣΔ, TFΣΔ-II produces an approximation in L²(ℝ), as opposed to the weak type approximations of TFΣΔ. In particular, for a tight Gabor frame with frame bound A, we prove that the L²-approximation error corresponding to a kth-order TFΣΔ-II quantizer is of order O(A^-k). Furthermore, motivated by TFΣΔ-II, we construct an algorithm to coarsely quantize the Fourier coefficients of certain compactly supported functions.