arXiv subjects

Diana T. Stoeva

Publications and source records attributed to Diana T. Stoeva.

At least 19 recordsLinked to original sources

arXiv2026

Oblique Pseudoinverses of Multipliers

Multipliers are operators on a Hilbert space that have been implicitly used in applications throughout the years. Multipliers are defined via three sequences and are mainly used in signal processing and psychoacoustics among many others. They modify a signal in three steps: analysis, term-wise scalar multiplication, and synthesis. While the study of multipliers stemmed from applications, its theory has earned an equally important place in the literature in the last few decades. One of the main questions in the theory of multipliers is the invertibility of multipliers. It is known that multipliers are not always invertible, and it has been shown that under mild conditions, the inverse of an invertible multiplier can be expressed as another multiplier of a very specific structure. In this work, we extend such result to the setting of possibly noninvertible multipliers. We solved the challenges of determining an appropriate generalized inverse to replace the inverse, and determining a proper duality relationship of the sequences of the resulting multiplier. We showed that these are nontrivial tasks that required the intricacies of the theory of frames for subspaces.

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arXiv2026

Celebrating Women in Mathematics

This paper was motivated by the worldwide May 12 initiative that aims to celebrate, encourage, and inspire women in mathematics. It presents in short how the May 12 initiative has arisen, what are some of the events in the first years, in particular the Generalized functions online workshop that started in 2021 in this context (and has continued as an annual event ever since), and a brief overview of some female mathematicians who have significant scientific contributions and who are the first women in some aspects: Maryam Mirzakhani (the first female mathematician who was awarded with the prestigious Fields medal; the May 12 initiative appeared in her honour), Hypatia (considered to be the earliest known female mathematician), Sofia Kovalevskaya (the first woman who has been awarded a doctorate in mathematics and considered to be the first woman who got a full professorship in mathematics in the modern academic sense), Emmy Noether (the first woman who gave a plenary lecture at the International Congress of Mathematicians), Karen Uhlenbeck (the first woman awarded with the prestigious Abel Prize), and Ingrid Daubechies (the first woman who became a full professor in mathematics at Princeton University, the first woman elected for President of the International Mathematical Union, the first woman who received the National Academy of Sciences Award in Mathematics). The paper was an invited contribution for the Akademie Intakt of the Austrian Academy of Sciences. It appeared in Akademie Intakt 2021, English Edition, Austrian Academy of Sciences, Vienna, 2021, 10-13, and it is submitted to arXiv with a permission from the publisher.

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arXiv2024

Constructions of dual frames compensating for erasures with implementation

Let $I\subseteq \Bbb N$ be a finite or infinite set and let ${(x_n)_{n\in I}}$ be a frame for a separable Hilbert space $\mathcal{H}$. Consider transmission of a signal $h\in\mathcal{H}$ where a finite subset $(\langle h,x_n\rangle)_{n\in E}$ of the frame coefficients $(\langle h,x_n\rangle)_{n\in I}$ is lost. There are several approaches in the literature aiming recovery of $h$. In this paper we focus on the approach based on construction of a dual frame of the reduced frame $(x_n)_{n\in I\setminus E}$ which is then used for perfect reconstruction from the preserved frame coefficients $(\langle h,x_n\rangle)_{n\in I\setminus E}$. There are several methods for such construction, starting from the canonical dual or any other dual frame of ${(x_n)_{n\in I}}$. We implemented the algorithms for these methods and performed tests to compare their computational efficiency.

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arXiv2024

Asymptotic analysis for generalized functions using frames

This paper is a short overview of the main Abelian- and Tauberian-type results from [4, 14, 26] regarding the asymptotic analysis of different classes of generalized functions in terms of appropriate frames. The Tauberian-type results provide a comprehensive characterization of the quasiasymptotic and S-asymptotic properties of distributions.

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arXiv2022

Dual frames compensating for erasures -- non-canonical case

In this paper we study the problem of recovering a signal from frame coefficients with erasures. Suppose that erased coefficients are indexed by a finite set $E$. Starting from a frame $(x_n)_{n=1}^\infty$ and its arbitrary dual frame, we give sufficient conditions for constructing a dual frame of $(x_n)_{n\in E^c}$ so that the perfect reconstruction can be obtained from the preserved frame coefficients. The work is motivated by methods using the canonical dual frame of $(x_n)_{n=1}^\infty$, which however do not extend automatically to the case when the canonical dual is replaced with another dual frame. The differences between the cases when the starting dual frame is the canonical dual and when it is not the canonical dual are investigated. We also give several ways of computing a dual of the reduced frame, among which we are the most interested in the iterative procedure for computing this dual frame. Computational tests show that in certain cases the iterative algorithm performs faster than the other considered procedures.

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arXiv2021

On compactly supported dual windows of Gabor frames

The main purpose of the paper is to give a characterization of all compactly supported dual windows of a Gabor frame. As an application, we consider an iterative procedure for approximation of the canonical dual window via compactly supported dual windows on every step. In particular, the procedure allows to have approximation of the canonical dual window via dual windows from certain modulation spaces or from the Schwartz space.

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arXiv2020

On a characterization of Riesz bases via biorthogonal sequences

It is well known that a sequence in a Hilbert space is a Riesz basis if and only if it is a complete Bessel sequence with biorthogonal sequence which is also a complete Bessel sequence. Here we prove that the completeness of one (any one) of the biorthogonal sequences can be omitted in the characterization.

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arXiv2019

Localization of Fréchet frames and expansion of generalized functions

Matrix type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fréchet spaces. Moreover, the localization of Fréchet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.

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arXiv2018

Operator representations of sequences and dynamical sampling

This paper is a contribution to the theory of dynamical sampling. Our purpose is twofold. We first consider representations of sequences in a Hilbert space in terms of iterated actions of a bounded linear operator. This generalizes recent results about operator representations of frames, and is motivated by the fact that only very special frames have such a representation. As our second contribution we give a new proof of a construction of a special class of frames that are proved by Aldroubi et al. to be representable via a bounded operator. Our proof is based on a single result by Shapiro \& Shields and standard frame theory, and our hope is that it eventually can help to provide more general classes of frames with such a representation.

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arXiv2018

A survey on the unconditional convergence and the invertibility of multipliers with implementation

The paper presents a survey over frame multipliers and related concepts. In particular, it includes a short motivation of why multipliers are of interest to consider, a review as well as extension of recent results, devoted to the unconditional convergence of multipliers, sufficient and/or necessary conditions for the invertibility of multipliers, and representation of the inverse via Neumann-like series and via multipliers with particular parameters. Multipliers for frames with specific structure, namely, Gabor and wavelet multipliers, are also considered. Some of the results for the representation of the inverse multiplier are implemented in Matlab codes and the algorithms are described.

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arXiv2017

Frame expansions of test functions, tempered distributions, and ultradistributions

The paper is devoted to frame expansions in Fréchet spaces. First we review some results which concern series expansions in general Fréchet spaces via Fréchet and General Fréchet frames. Then we present some new results on series expansions of tempered distributions and ultradistributions, and the corresponding test functions, via localized frames and coefficients in appropriate sequence spaces.

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arXiv2016

Characterization of atomic decompositions, Banach frames, Xd-frames, duals and synthesis-pseudo-duals, with application to Hilbert frame theory

In this paper we consider series expansions via a frame and a non-frame and the possibilities for interchange of the two sequences, both in the Hilbert and Banach space setting. First we give a characterization of frame-related concepts in Banach spaces (atomic decompositions, Banach frames, $X_d$-Riesz bases, $X_d$-frames, $X_d$-Bessel sequences, and sequences satisfying the lower $X_d$-frame condition). We also determine necessary and sufficient conditions for operators to preserve the type of the concepts listed above. Then we discuss differences and relationships between expansions in a Banach space and its dual space when interchanging the involved sequences. Finally, we apply some of the results to answer problems in Hilbert frame theory. We show that interchanging a frame and a non-Bessel sequence in series expansions is not always possible (leading to differentiation of analysis- and synthesis-pseudo-duals of a frame) and determine an appropriate subspace where interchange can be done. We characterize all the synthesis-pseudo-duals of a frame and determine a class of frames whose synthesis-pseudo-duals (resp. analysis-pseudo-duals) are necessarily frames. We also investigate connections between the lower frame condition and series expansions. Examples are given to illustrate statements in the paper and to show the optimality of some results.

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arXiv2016

On the dual frame induced by an invertible frame multiplier

Recently it has been established that given an invertible frame multiplier with semi-normalized symbol, a specific dual of any of the two involved frames can be determined for the inversion purpose. The inverse can be represented as a multiplier with the reciprocal symbol, this particular dual of one of the given frames, and any dual of the other frame. The specific dual is the only one having this property among all Bessel sequences. In this manuscript we extend the results showing that the specific dual with the above mentioned property is unique among all possible sequences. Furthermore, we allow the symbol to be not necessarily semi-normalized. Finally we characterize cases when the canonical dual frame and the new specific dual frame coincide.

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arXiv2015

On various R-duals and the duality principle

The duality principle states that a Gabor system is a frame if and only if the corresponding adjoint Gabor system is a Riesz sequence. In general Hilbert spaces and without the assumption of any particular structure, Casazza, Kutyniok and Lammers have introduced the so-called R-duals that also lead to a characterization of frames in terms of associated Riesz sequences; however, it is still an open question whether this abstract theory is a generalization of the duality principle. In this paper we prove that a modified version of the R-duals leads to a generalization of the duality principle that keeps all the attractive properties of the R-duals. In order to provide extra insight into the relations between a given sequence and its R-duals, we characterize all the types of R-duals that are available in the literature for the special case where the underlying sequence is a Riesz basis.

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arXiv2014

On R-duals and the duality principle in Gabor analysis

The concept of R-duals of a frame was introduced by Casazza, Kutyniok and Lammers in 2004, with the motivation to obtain a general version of the duality principle in Gabor analysis. For tight Gabor frames and Gabor Riesz bases the three authors were actually able to show that the duality principle is a special case of general results for R-duals. In this paper we introduce various alternative R-duals, with focus on what we call R-duals of type II and III. We show how they are related and provide characterizations of the R-duals of type II and III. In particular, we prove that for tight frames these classes coincide with the R-duals by Casazza et el., which is desirable in the sense that the motivating case of tight Gabor frames already is well covered by these R-duals. On the other hand, all the introduced types of R-duals generalize the duality principle for larger classes of Gabor frames than just the tight frames and the Riesz bases; in particular, the R-duals of type III cover the duality principle for all Gabor frames.

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arXiv2013

Representation of the inverse of a frame multiplier

Certain mathematical objects appear in a lot of scientific disciplines, like physics, signal processing and, naturally, mathematics. In a general setting they can be described as frame multipliers, consisting of analysis, multiplication by a fixed sequence (called the symbol), and synthesis. They are not only interesting mathematical objects, but also important for applications, for example for the realization of time-varying filters. In this paper we show a surprising result about the inverse of such operators, if existing, as well as new results about a core concept of frame theory, dual frames. We show that for semi-normalized symbols, the inverse of any invertible frame multiplier can always be represented as a frame multiplier with the reciprocal symbol and dual frames of the given ones. Furthermore, one of those dual frames is uniquely determined and the other one can be arbitrarily chosen. We investigate sufficient conditions for the special case, when both dual frames can be chosen to be the canonical duals. In connection to the above, we show that the set of dual frames determines a frame uniquely. Furthermore, for a given frame, the union of all coefficients of its dual frames is dense in $\ell^2$. We also investigate invertible Gabor multipliers. Finally, we give a numerical example for the invertibility of multipliers in the Gabor case.

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arXiv2012

Fréchet frames, general definition and expansions

We define an {\it $(X_1,Θ, X_2)$-frame} with Banach spaces $X_2\subseteq X_1$, $|\cdot|_1 \leq |\cdot|_2$, and a $BK$-space $(Θ, \snorm[\cdot])$. Then by the use of decreasing sequences of Banach spaces ${X_s}_{s=0}^\infty$ and of sequence spaces ${Θ_s}_{s=0}^\infty$, we define a general Fr\' echet frame on the Fr\' echet space $X_F=\bigcap_{s=0}^\infty X_s$. We give frame expansions of elements of $X_F$ and its dual $X_F^*$, as well of some of the generating spaces of $X_F$ with convergence in appropriate norms. Moreover, we give necessary and sufficient conditions for a general pre-Fr\' echet frame to be a general Fr\' echet frame, as well as for the complementedness of the range of the analysis operator $U:X_F\toΘ_F$.

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arXiv2011

Can any unconditionally convergent multiplier be transformed to have the symbol (1) and Bessel sequences by shifting weights?

Multipliers are operators that combine (frame-like) analysis, a multiplication with a fixed sequence, called the symbol, and synthesis. They are very interesting mathematical objects that also have a lot of applications for example in acoustical signal processing. It is known that bounded symbols and Bessel sequences guarantee unconditional convergence. In this paper we investigate necessary and equivalent conditions for the unconditional convergence of multipliers. In particular we show that, under mild conditions, unconditionally convergent multipliers can be transformed by shifting weights between symbol and sequence, into multipliers with symbol (1) and Bessel sequences.

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