Eric Weber

Frames for Undergraduates

Introduction Linear algebra review Finite-dimensional operator theory Introduction to finite frames Frames in

Geometric aspects of frame representations of abelian groups

\mathbb{R}^2

On the Translation Invariance of Wavelet Subspaces

The dilation property of frames Dual and orthogonal frames Frame operator decompositions Harmonic and group frames Sampling theory Student presentations Anecdotes on frame theory projects by undergraduates Bibliography List of symbols Index.

Continuous and discrete Fourier frames for fractal measures

hhWe consider frames arising from the action of a unitary representation of a discrete countable abelian group. We show that the range of the analysis operator can be determined by computing which characters appear in the representation. This allows one to compare the ranges of two such frames, which is useful for determining similarity and also for multiplexing schemes. Our results then partially extend to Bessel sequences arising from the action of the group. We apply the results to sampling on bandlimited functions and to wavelet and Weyl-Heisenberg frames. This yields a sufficient condition for two sampling transforms to have orthogonal ranges, and two analysis operators for wavelet and Weyl-Heisenberg frames to have orthogonal ranges. The sufficient condition is easy to compute in terms of the periodization of the Fourier transform of the frame generators.

Encryption Schemes using Finite Frames and Hadamard Arrays

An examination of the translation invariance of V0 under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.

Riesz wavelets and multiresolution structures

Motivated by the existence problem of Fourier frames on fractal measures, we introduce Bessel and frame measures for a given finite measure on R d , as extensions of the notions of Bessel and frame spectra that correspond to bases of exponential functions. Not every finite compactly supported Borel measure admits frame measures. We present a general way of constructing Bessel/frame measures for a given measure. The idea is that if a convolution of two measures admits a Bessel measure then one can use the Fourier transform of one of the measures in the convolution as a weight for the Bessel measure to obtain a Bessel measure for the other measure in the convolution. The same is true for frame measures, but with certain restrictions. We investigate some general properties of frame measures and their Beurling dimensions. In particular we show that the Beurling dimension is invariant under convolution (with a probability measure) and under a certain type of discretization. Moreover, if a measure admits a frame measure then it admits an atomic one, and hence a weighted Fourier frame. We also construct some examples of frame measures for self-similar measures.

Finite Dimensional Dynamical Sampling: An Overview

We propose a cipher similar to the one-time pad and McEliece cipher based on a subband coding scheme. The encoding process is an approximation to the one-time pad encryption scheme. We present results of numerical experiments that suggest that a brute force attack on the proposed scheme does not result in all possible plaintexts, as the one-time pad does, but the brute force attack does not compromise the system. However, we demonstrate that the cipher is vulnerable to a chosenplaintext attack.

Fourier Series for Singular Measures

Multiresolution structures are important in applications, but they are also useful for analyzing properties of associated wavelets. Given a nonorthogonal (multi-) wavelet in a Hilbert space, we construct a core subspace. Subsequently, the dilates of the core subspace defines a ladder of nested subspaces. Of fundamental importance are two questions: 1) when is the core subspace shift invariant; and if yes, then 2) when is the core subspace generated by shifts of a single vector, i.e. there exists a scaling vector. If the wavelet generates a Riesz basis then the answer to question 1) is yes if and only if the wavelet is a biorthogonal wavelet. Additionally, if the wavelet generates a tight frame of arbitrary frame constant, then the core subspace is shift invariant. Question 1) is still open in case the wavelet generates a non-tight frame. We also present some known results to question 2) and provide some preliminary improvements. Our analysis here arises from investigating the dimension function and the multiplicity function of a wavelet. These two functions agree if the wavelet is orthogonal. Finally, we discuss how these questions are important for considering linear perturbation of wavelets. Utilizing the idea of the local commutant of a unitary system developed by Dai and Larson, we show that nearly all linear perturbations of two orthonormal wavelets form a Riesz wavelet. If in fact these wavelets correspond to a von Neumann algebra in the local commutant of a base wavelet, then the interpolated wavelet is biorthogonal. Moreover, we demonstrate that in this case the interpolated wavelets have a scaling vector if the base wavelet has a scaling vector.

Fourier Frames for the Cantor-4 Set

Dynamical sampling is an emerging paradigm for studying signals that evolve in time. In this chapter we present many of the available results pertaining to dynamical sampling in the finite dimensional setting. We also provide a brief survey of the latest results in the infinite dimensional setting.

The Kadison-Singer problem and the uncertainty principle

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure