arXiv subjects

Ilya Krishtal

Publications and source records attributed to Ilya Krishtal.

15 recordsLinked to original sources

arXiv2026

Frames from Functional Calculus

We introduce frames generated by applying families of functions to a diagonal operator through functional calculus, with diagonal entries forming a Carleson interpolating sequence. First, for separated spectra contained in a Stolz domain and an angular sector, we prove that systems of fractional operator powers remain frames whenever the step size satisfies the natural angular restriction. This extends earlier results for positive real spectra and permits noninteger powers. We also show that the angular restriction is sharp in general. Second, we give an $\ell^2$-perturbation criterion for frames generated by more general continuous functions, assuming a density condition on a curve containing the spectrum. The results broaden the class of structured frames available in dynamical sampling and related operator-orbit problems.

math.FA↗

arXiv2026

Block Diagonal Carleson Frames

We introduce \textit{block diagonal Carleson frames}, i.e. singly generated dynamical frames in which the generating operator is block diagonal. We classify all such frames when the generating operator is a direct sum of Jordan blocks, the sizes of which are uniformly bounded. Furthermore, block diagonal Carleson frames are shown to enjoy the high redundancy properties observed for Carleson frames. When the generating operator has nonnegative spectrum, we provide a complete description of the redundancy of such frames under the assumption of the existence of natural density.

math.FA↗

arXiv2025

Dynamical Sampling: A Survey

Dynamical sampling refers to a class of problems in which space-time samples are taken from a signal evolving under an underlying dynamical system. The goal is to use these samples to recover relevant information about the system, such as the initial state, the evolution operator, or the sources and sinks driving the dynamics. These problems are tightly connected to frame theory, operator theory, functional analysis, and other foundational areas of mathematics; they also give rise to new theoretical questions and have applications across engineering and the sciences. This survey provides an overview of the theoretical underpinnings of dynamical sampling, summarizes recent results, and outlines directions for future work, including open problems and conjectures.

math.FA↗

arXiv2025

Demystifying Carleson Frames

We study spanning properties of Carleson systems and prove a recent conjecture on frame subsequences of Carleson frames. In particular, we show that if $\{T^k\varphi\}_{k=0}^\infty$ is a Carleson frame, then every subsequence of the form $\{T^{Nk+j_k}\varphi\}_{k=0}^\infty$ where $N\in\mathbb{N}$ and $0 \leq j_k < N$ is also a frame.

math.FA↗

arXiv2024

Reconstructing Graph Signals from Noisy Dynamical Samples

We investigate the dynamical sampling space-time trade-off problem within a graph setting. Specifically, we derive necessary and sufficient conditions for space-time sampling that enable the reconstruction of an initial band-limited signal on a graph. Additionally, we develop and test numerical algorithms for approximating the optimal placement of sensors on the graph to minimize the mean squared error when recovering signals from time-space measurements corrupted by i.i.d.~additive noise. Our numerical experiments demonstrate that our approach outperforms previously proposed algorithms for related problems.

cs.IT↗

arXiv2024

Kadec-type theorems for sampled group orbits

We extend the classical Kadec 1/4 theorem for systems of exponential functions on an interval to frames and atomic decompositions formed by sampling an orbit of a vector under an isometric group representation.

math.FA↗

arXiv2024

Reconstruction algorithms for source term recovery from dynamical samples in catalyst models

This paper investigates the problem of recovering source terms in abstract initial value problems (IVP) commonly used to model various scientific phenomena in physics, chemistry, economics, and other fields. We consider source terms of the form $F=h+\eta$, where $\eta$ is a Lipschitz continuous background source. The primary objective is to estimate the unknown parameters of non-instantaneous sources $h(t)=\sum\limits_{j=0}^M h_je^{-\rho_j(t-t_j)}\chi_{[t_j,\infty)}(t)$, such as the decay rates, initial intensities and activation times. We present two novel recovery algorithms that employ distinct sampling methods of the solution of the IVP. Algorithm 1 combines discrete and weighted average measurements, whereas Algorithm 2 uses a different variant of weighted average measurements. We analyze the performance of these algorithms, providing upper bounds on the recovery errors of the model parameters. Our focus is on the structure of the dynamical samples used by the algorithms and on the error guarantees they yield.

math.DS↗

arXiv2023

On Low-Rank Convex-Convex Quadratic Fractional Programming

We present an efficient algorithm for solving fractional programming problems whose objective functions are the ratio of a low-rank quadratic to a positive definite quadratic with convex constraints. The proposed algorithm for these convex-convex problems is based on the Shen-Yu Quadratic Transform which finds stationary points of concave-convex sum-of-ratios problems. We further use elements of the algorithm proposed in [arXiv:1802.10192] and the classic Dinkelbach approach to ensure convergence. We show that our algorithm performs better than previous algorithms for low-rank problems.

math.OC↗

arXiv2022

Recovery of rapidly decaying source terms from dynamical samples in evolution equations

We analyze the problem of recovering a source term of the form $h(t)=\sum_{j}h_jϕ(t-t_j)χ_{[t_j, \infty)}(t)$ from space-time samples of the solution $u$ of an initial value problem in a Hilbert space of functions. In the expression of $h$, the terms $h_j$ belong to the Hilbert space, while $ϕ$ is a generic real-valued function with exponential decay at $\infty$. The design of the sampling strategy takes into account noise in measurements and the existence of a background source.

math.DS↗

arXiv2021

Predictive algorithms in dynamical sampling for burst-like forcing terms

In this paper, we consider the problem of recovery of a burst-like forcing term in an initial value problem (IVP) in the framework of dynamical sampling. We introduce an idea of using two particular classes of samplers that allow one to predict the solution of the IVP over a time interval without a burst. This leads to two different algorithms that stably and accurately approximate the burst-like forcing term even in the presence of a measurement acquisition error and a large background source.

cs.IT↗

arXiv2021

Sampling the flow of a bandlimited function

We analyze the problem of reconstruction of a bandlimited function $f$ from the space-time samples of its states $f_t=ϕ_t\ast f$ resulting from the convolution with a kernel $ϕ_t$. It is well-known that, in natural phenomena, uniform space-time samples of $f$ are not sufficient to reconstruct $f$ in a stable way. To enable stable reconstruction, a space-time sampling with periodic nonuniformly spaced samples must be used as was shown by Lu and Vetterli. We show that the stability of reconstruction, as measured by a condition number, controls the maximal gap between the spacial samples. We provide a quantitative statement of this result. In addition, instead of irregular space-time samples, we show that uniform dynamical samples at sub-Nyquist spatial rate allow one to stably reconstruct the function $\widehat f$ away from certain, explicitly described blind spots. We also consider several classes of finite dimensional subsets of bandlimited functions in which the stable reconstruction is possible, even inside the blind spots. We obtain quantitative estimates for it using Remez-Turán type inequalities. En route, we obtain a Remez-Turán inequality for prolate spheroidal wave functions. To illustrate our results, we present some numerics and explicit estimates for the heat flow problem.

math.CA↗

arXiv2018

Dynamical sampling with additive random noise

Dynamical sampling deals with signals that evolve in time under the action of a linear operator. The purpose of the present paper is to analyze the performance of the basic dynamical sampling algorithms in the finite dimensional case and study the impact of additive noise. The algorithms are implemented and tested on synthetic and real data sets, and denoising techniques are integrated to mitigate the effect of the noise. We also develop theoretical and numerical results that validate the algorithm for recovering the driving operators, which are defined via a real symmetric convolution.

math.NA↗

arXiv2014

Krylov Subspace Methods in Dynamical Sampling

Let $B$ be an unknown linear evolution process on $\mathbb C^d\simeq l^2(\mathbb Z_d)$ driving an unknown initial state $x$ and producing the states $\{B^\ell x, \ell = 0,1,\ldots\}$ at different time levels. The problem under consideration in this paper is to find as much information as possible about $B$ and $x$ from the measurements $Y=\{x(i)$, $Bx(i)$, $\dots$, $B^{\ell_i}x(i): i \in Ω\subset \mathbb Z^d\}$. If $B$ is a "low-pass" convolution operator, we show that we can recover both $B$ and $x$, almost surely, as long as we double the amount of temporal samples needed in \cite{ADK13} to recover the signal propagated by a known operator $B$. For a general operator $B$, we can recover parts or even all of its spectrum from $Y$. As a special case of our method, we derive the centuries old Prony's method \cite{BDVMC08, P795, PP13} which recovers a vector with an $s$-sparse Fourier transform from $2s$ of its consecutive components.

cs.IT↗

arXiv2013

Exact Reconstruction of Spatially Undersampled Signals in Evolutionary Systems

We consider the problem of spatiotemporal sampling in which an initial state $f$ of an evolution process $f_t=A_tf$ is to be recovered from a combined set of coarse samples from varying time levels $\{t_1,\dots,t_N\}$. This new way of sampling, which we call dynamical sampling, differs from standard sampling since at any fixed time $t_i$ there are not enough samples to recover the function $f$ or the state $f_{t_i}$. Although dynamical sampling is an inverse problem, it differs from the typical inverse problems in which $f$ is to be recovered from $A_Tf$ for a single time $T$. In this paper, we consider signals that are modeled by $\ell^2(\mathbb Z)$ or a shift invariant space $V\subset L^2(\mathbb R)$.

cs.OH↗

arXiv2013

Localization of Matrix Factorizations

Matrices with off-diagonal decay appear in a variety of fields in mathematics and in numerous applications, such as signal processing, statistics, communications engineering, condensed matter physics, and quantum chemistry. Numerical algorithms dealing with such matrices often take advantage (implicitly or explicitly) of the empirical observation that this off-diagonal decay property seems to be preserved when computing various useful matrix factorizations, such as the Cholesky factorization or the QR-factorization. There is a fairly extensive theory describing when the inverse of a matrix inherits the localization properties of the original matrix. Yet, except for the special case of band matrices, surprisingly very little theory exists that would establish similar results for matrix factorizations. We will derive a comprehensive framework to rigorously answer the question when and under which conditions the matrix factors inherit the localization of the original matrix for such fundamental matrix factorizations as the LU-, QR-, Cholesky, and Polar factorization.

math.NA↗