Decomposition of Finite Schmidt Rank Bounded Operators on the Tensor Product of Separable Hilbert Spaces

Inverse formulas for the tensor product are used to develop an algorithm to compute Schmidt decompositions of Finite Schmidt Rank (FSR) bounded operators on the tensor product of separable Hilbert spaces. The algorithm is then applied to solve inverse problems related to the tensor product of bounded operators. In particular, we show how properties of a FSR bounded operator are reflected by the operators involved in its Schmidt decomposition. These properties include compactness of FSR bounded operators and convergence of sequences whose terms are FSR bounded operators

On the Linear Independence of Finite Gabor and Wavelet Systems

Gabor and Wavelet Systems are some of the most important families of integrable functions with great potential in applications. Those applications include numerical analysis, signal processing (sound, images), and many other areas of physics and engineering. In this talk, we will present some partial results on a conjecture that states each finite Gabor system is linearly independent. We will also present cases of linearly independent and cases of linearly dependent finite wavelet systems

On the Linear Independence of Finite Wavelet Systems Generated by Schwartz Functions or Functions with Certain Behavior at Infinity

One of the motivations to state HRT conjecture on the linear independence of finite Gabor systems was the fact that there are linearly dependent Finite Wavelet Systems (FWS). Meanwhile, there are also many examples of linearly independent FWS, some of which are presented in this paper. We prove the linear independence of every three point FWS generated by a nonzero Schwartz function and with any number of points if the FWS is generated by a nonzero Schwartz function, for which the absolute value of the Fourier transform is decreasing at infinity. We also prove the linear independence of any FWS generated by a nonzero square integrable function, for which the Fourier transform has certain behavior at infinity. Such a function can be any square integrable function that is a linear complex combination of real valued rational and exponential functions

Beurling Weighted Spaces, Product-Convolution Operators, and the Tensor Product of Frames

G. Gaudry solved the multiplier problem for Beurling algebras, i.e., he identified the space of all multipliers of a Beurling algebra with a weighted space of bounded measures. In the first part of this thesis, we solve multiplier problems for some Beurling weighted spaces. We identify the space of all multipliers of some Beurling weighted spaces with the dual of spaces of Figa-Talamanca type. A paper by R.C Busby and H.A.Smith gives necessary and sufficient conditions for the compactness of product-convolution operators. In the second part of this thesis, we present some applications of the result of R.C Busby and H.A. Smith; and we prove that the eigenfunctions of certain product-convolution operators can be obtained as solutions of some differential equations. Incidentally, we obtain classical special functions as eigenfunctions of these product-convolution operators. In the third part of this thesis, we prove that the tensor product of two sequences is a frame(Riesz basis) if and only if each part of this tensor product is a frame (Riesz basis). We use this result to extend the Lyubarskii and Seip Wallsten theorem, characterizing Gabor frames generated by the Gaussian function, to higher dimensions

The Duals of *-Operator Frames for End*A(H)

Frames play significant role in signal and image processing, which leads to many applications in differents fields. In this paper we define the dual of ∗-operator frames and we show their propreties obtained in Hilbert A-modules and we establish some results

The HRT Conjecture

Stated more than 20 years ago by C. Heil, J. Ramanathan, and P. Topiwala, the HRT conjecture is about the linear independence of a collection of finitely many time-frequency shifts of an arbitrary nonzero square integrable function. Despite the simplicity of its statement, the conjecture is still open for the general case. In this talk the author will present results based on a paper by Dr. J. Benedetto and the author. The paper proves HRT conjecture for a class of functions with certain behavior at infinity. This class includes some square integrable functions built by combining polynomial, exponential, and logarithmic functions. For example, we prove HRT for any finite collection of time-frequency shifts of e{-|x|}