Frames by orbits of two operators that commute

Reducing and invariant subspaces under two commuting shift operators

This paper is a contribution to frame theory. Frames in a Hilbert space are generalizations of orthonormal bases. In particular, Gabor frames of $$L^2(\mathbb {R})$$
 , which are made of translations and modulations of one or more windows, are often used in applications. More precisely, the paper deals with a question posed in the last years by Christensen and Hasannasab about the existence of overcomplete Gabor frames, with some ordering over $$\mathbb {Z}$$
 , which are orbits of bounded operators on $$L^2(\mathbb {R})$$
 . Two classes of overcomplete Gabor frames which cannot be ordered over $$\mathbb {Z}$$
 and represented by orbits of operators in $$GL(L^2(\mathbb {R}))$$
 are given. Some results about operator representation are stated in a general context for arbitrary frames, covering also certain wavelet frames.

/pdf/orbits-of-bounded-bijective-operators-and-gabor-frames-243252me57.pdf

Orbits of bounded bijective operators and Gabor frames

pdf/dynamical-sampling-for-the-recovery-of-spatially-constant-2fncsm3xpd.pdf

Dynamical sampling for the recovery of spatially constant source terms in dynamical systems

We analyze the problem of recovering a source term of the form $$h(t)=\sum _{j}h_j\phi (t-t_j)\chi _{[t_j, \infty )}(t)$$ 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 $$\phi $$ 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. 

/pdf/recovery-of-rapidly-decaying-source-terms-from-dynamical-18k0ghkf.pdf

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

Dynamical sampling for shift-preserving operators

In this note, we solve the dynamical sampling problem for a class of shift-preserving operators $L:V\to V$ acting on a finitely generated shift-invariant space $V$. We find conditions on $L$ and a finite set of functions of $V$ so that the iterations of the operator $L$ on the functions produce a frame generator set of $V$. This means that the integer translations of the generators form a frame of $V$.

Dynamical Sampling for Shift-preserving Operators

Handle everyday research tasks with reliable, citation-backed results

Your personal Research Agent to handle research tasks with citation-backed results

Popular Tasks used by Researchers

How can I help with your research?

Meet SciSpace

Get more enhanced response by uploading the PDFs you want me to reference.

No relevant PDFs in your library

SciSpace is the AI research assistant for academics. Run systematic literature reviews on 280M+ papers, and write papers with cited sources. Trusted by 1M+ students, PhDs & researchers.

SciSpace | AI for Research

Analyze PDFs

Code & Manuscripts

Funding & Grants

Literature & Patents

Medical & Clinical Data

Systematic Review

Visualize & Present

Web & Data

Build a Google Scholar-like website for your research.

Build a website

Create charts and images for your research

Create a Chart

Write a paper for submission to a journal

Draft a manuscript

Patent Search

Design eye-catching scientific posters in minutes.

Scientific Poster Generation

Systematic Literature Review

One task is running at the moment. Your messages will be shown right after.

Drag and drop or click here to browse

Loved by <highlight>1 million+</highlight> researchers

Extract a list of specific topics and their sources from unstructured text

Topics

Compare and analyze relevant papers that matches with your search

Papers

Get insights from PDFs and bookmarked papers from your library

My library

Recent searches

Try searching for:

Catch AI-generated content in scholarly and non-scholarly content

{ai} Detector

Ai Writer

Get PDF Summaries, highlighted text explanations 

Chat with PDF

Effortlessly create in-text citations and bibliographies in APA and 2,500 other formats

Citation generator

Get explanations, summaries, and answers on academic papers

Ease up your research workflow with {scispace}'s cohort of exciting AI tools

Elevate your academic writing skills and convey your ideas the way you want

Paraphraser

Explore our range of reading and writing tools

Your file is being prepared and should be ready in a few minutes. If it's a large file, it might take a bit longer. You can close this window, and we'll email you the file when it's done.

You have reached a maximum limit of <strong>{limit}</strong> columns in the table. Remove at least <strong>1</strong> column to add or create another one.

D. Carbajal

Author Tools

Chat about Author

Papers

Dynamical sampling for shift-preserving operators

Dynamical Sampling for Shift-preserving Operators