Weighted frames of exponentials and stable recovery of multidimensional functions from nonuniform Fourier samples

In this paper, we consider the problem of recovering a compactly supported multivariate function from a collection of pointwise samples of its Fourier transform taken nonuniformly. We do this by using the concept of weighted Fourier frames. A seminal result of Beurling shows that sample points give rise to a classical Fourier frame provided they are relatively separated and of sufficient density. However, this result does not allow for arbitrary clustering of sample points, as is often the case in practice. Whilst keeping the density condition sharp and dimension independent, our first result removes the separation condition and shows that density alone suffices. However, this result does not lead to estimates for the frame bounds. A known result of Groechenig provides explicit estimates, but only subject to a density condition that deteriorates linearly with dimension. In our second result we improve these bounds by reducing the dimension dependence. In particular, we provide explicit frame bounds which are dimensionless for functions having compact support contained in a sphere. Next, we demonstrate how our two main results give new insight into a reconstruction algorithm---based on the existing generalized sampling framework---that allows for stable and quasi-optimal reconstruction in any particular basis from a finite collection of samples. Finally, we construct sufficiently dense sampling schemes that are often used in practice---jittered, radial and spiral sampling schemes---and provide several examples illustrating the effectiveness of our approach when tested on these schemes

On stable reconstructions from nonuniform Fourier measurements

We consider the problem of recovering a compactly-supported function from a finite collection of pointwise samples of its Fourier transform taking nonuniformly. First, we show that under suitable conditions on the sampling frequencies - specifically, their density and bandwidth - it is possible to recover any such function $f$ in a stable and accurate manner in any given finite-dimensional subspace; in particular, one which is well suited for approximating $f$. In practice, this is carried out using so-called nonuniform generalized sampling (NUGS). Second, we consider approximation spaces in one dimension consisting of compactly supported wavelets. We prove that a linear scaling of the dimension of the space with the sampling bandwidth is both necessary and sufficient for stable and accurate recovery. Thus wavelets are up to constant factors optimal spaces for reconstruction

Construction of scaling partitions of unity

Partitions of unity in ${\mathbf R}^d$ formed by (matrix) scales of a fixed function appear in many parts of harmonic analysis, e.g., wavelet analysis and the analysis of Triebel-Lizorkin spaces. We give a simple characterization of the functions and matrices yielding such a partition of unity. For invertible expanding matrices, the characterization leads to easy ways of constructing appropriate functions with attractive properties like high regularity and small support. We also discuss a class of integral transforms that map functions having the partition of unity property to functions with the same property. The one-dimensional version of the transform allows a direct definition of a class of nonuniform splines with properties that are parallel to those of the classical B-splines. The results are illustrated with the construction of dual pairs of wavelet frames

ON THE CHARACTERIZATION OF SCALING FUNCTIONS ON NON-ARCHEMEDEAN FIELDS

In real life application all signals are not obtained from uniform shifts; so there is a natural question regarding analysis and decompositions of these types of signals by a stable mathematical tool.  This gap was filled by Gabardo and Nashed [11]   by establishing a constructive algorithm based on the theory of spectral pairs for constructing non-uniform wavelet basis in $L^2(\mathbb R)$. In this setting, the associated translation set $\Lambda =\left\{ 0,r/N\right\}+2\,\mathbb Z$ is no longer a discrete subgroup of $\mathbb R$ but a spectrum associated with a certain one-dimensional spectral pair and the associated dilation is an even positive integer related to the given spectral pair. In this paper, we characterize the scaling function for non-uniform multiresolution analysis on local fields of positive characteristic (LFPC). Some properties of wavelet scaling function associated with non-uniform multiresolution analysis (NUMRA) on LFPC are also established

Nonuniform p - tight wavelet frames on positive half line

Wavelet frames have gained considerable popularity during the past decade, primarily due to their substantiated applications in diverse and widespread fields of engineering and science. Tight wavelet frames provide representations of signals and images where repetition of the representation is favored and the ideal reconstruction property of the associated filter bank algorithm, as in the case of orthonormal wavelets is kept. The main objective of this paper is to introduce a notion of nonuniform wavelet system in L² (R⁺) and provide a complete characterization of such systems to be tight nonuniform wavelet frames in L² (R⁺) by using Walsh-Fourier transform.Publisher's Versio