This is a short introduction to Hilbert space frame theory and its applications for those outside the area who want an introduction to the subject. We will increase this over time. There are incomplete sections at this time. If anyone wants to add a section or fill in an incomplete section on ”their applications” contact Pete Casazza. 1. Basic Definitions For a more complete treatment of frame ...

Quaternionic Hilbert spaces play an important role in applied physical sciences especially quantum physics. In this paper, the operator valued frames on quaternionic are introduced and studied. terms of a class partial isometries spaces, parametrization Parseval is obtained. We extend to many properties vector process. Moreover, we show that all can be obtained from single frame. Finally, sever...

We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networks when their input is subject to an adversarial perturbation. The most impor...

Unit norm tight frames provide Parseval-like decompositions of vectors in terms of possibly nonorthogonal collections of unit norm vectors. One way to prove the existence of unit norm tight frames is to characterize them as the minimizers of a particular energy functional, dubbed the frame potential. We consider this minimization problem from a numerical perspective. In particular, we discuss h...

The shearlet representation has gained increasing recognition in recent years as a framework for the efficient representation of multidimensional data. This representation consists of a countable collection of functions defined at various locations, scales and orientations, where the orientations are obtained through the use of shearing matrices. While shearing matrices offer the advantage of p...

We introduce Parseval networks, a form of deep neural networks in which the Lipschitz constant of linear, convolutional and aggregation layers is constrained to be smaller than 1. Parseval networks are empirically and theoretically motivated by an analysis of the robustness of the predictions made by deep neural networks when their input is subject to an adversarial perturbation. The most impor...

In this dissertation, we study the structure of correlation minimizing frames. A correlation minimizing (N,d)-frame is any uniform Parseval frame of N vectors in dimension, d, such that the largest absolute value of the inner products of any pair of vectors is as small as possible. We call this value the correlation constant. These frames are important as they are optimal for the 2-erasures pro...

In this paper, we study the finite Gabor system for oversampling schemes. A characterization of dual tight frame using discrete time Zak transform is given. Also, a method to calculate coefficients expansion in case and necessary sufficient condition existence biorthogonal pair Riesz basis l2(ZL) Further, introduce notion block sliding Fourier (BSDFT) which reduces computational complexity give...

In this paper, we study the construction of irregular shearlet systems, i.e., systems of the form SH(ψ,Λ) = {a− 4 ψ(A−1 a S−1 s (x− t)) : (a, s, t) ∈ Λ}, where ψ ∈ L(R), Λ is an arbitrary sequence in R ×R×R2, Aa is a parabolic scaling matrix and Ss a shear matrix. These systems are obtained by appropriately sampling the Continuous Shearlet Transform. We derive sufficient conditions for such a d...