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Mathematics
Special Issues
New Perspectives in Harmonic Analysis
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New Perspectives in Harmonic Analysis

A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "E: Applied Mathematics".

Deadline for manuscript submissions: 28 February 2026 | Viewed by 1365

Special Issue Editor

Prof. Dr. Stephen Casey

E-Mail Website
Guest Editor
Department of Mathematics and Statistics, American University, Washington, DC, USA
Interests: complex analysis; harmonic analysis; differential geometry; number theory with applications to signal and image processing

Special Issue Information

Dear Colleagues,

This Special Issue of Mathematics, entitled New Perspectives in Harmonic Analysis, aims to present recent advances in harmonic analysis and to explore new trends and directions in related areas. We therefore welcome papers that address a range of topics including, but not limited to,  signal and image processing, compressed sensing, coding theory, control theory computational neuroscience, deep learning, information theory, and theoretical, applied, and computational harmonic analysis.

Prof. Dr. Stephen Casey
Guest Editor

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Mathematics is an international peer-reviewed open access semimonthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

  • shannon sampling
  • irregular sampling
  • interpolation theory
  • Gabor systems
  • wavelets
  • frame theory
  • information theory
  • deep learning

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Published Papers (2 papers)

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Research

22 pages, 1027 KB  
Article
Probing the Topology of the Space of Tokens with Structured Prompts
by Michael Robinson, Sourya Dey and Taisa Kushner
Mathematics 2025, 13(20), 3320; https://doi.org/10.3390/math13203320 - 17 Oct 2025
Viewed by 229
Abstract
Some large language models (LLMs) are open source and are therefore fully open for scientific study. However, many LLMs are proprietary, and their internals are hidden, which hinders the ability of the research community to study their behavior under controlled conditions. For instance, [...] Read more.
Some large language models (LLMs) are open source and are therefore fully open for scientific study. However, many LLMs are proprietary, and their internals are hidden, which hinders the ability of the research community to study their behavior under controlled conditions. For instance, the token input embedding specifies an internal vector representation of each token used by the model. If the token input embedding is hidden, latent semantic information about the set of tokens is unavailable to researchers. This article presents a general and flexible method for prompting an LLM to reveal its token input embedding, even if this information is not published with the model. Moreover, this article provides strong theoretical justification—a mathematical proof for generic LLMs—for why this method should be expected to work. If the LLM can be prompted systematically and certain benign conditions about the quantity of data collected from the responses are met, the topology of the token embedding is recovered. With this method in hand, we demonstrate its effectiveness by recovering the token subspace of the Llemma-7BLLM. We demonstrate the flexibility of this method by performing the recovery at three different times, each using the same algorithm applied to different information collected from the responses. While the prompting can be a performance bottleneck depending on the size and complexity of the LLM, the recovery runs within a few hours on a typical workstation. The results of this paper apply not only to LLMs but also to general nonlinear autoregressive processes. Full article
(This article belongs to the Special Issue New Perspectives in Harmonic Analysis)
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22 pages, 266 KB  
Article
Spectral Theory and Hardy Spaces for Bessel Operators in Non-Standard Geometries
by Saeed Hashemi Sababe
Mathematics 2025, 13(4), 565; https://doi.org/10.3390/math13040565 - 8 Feb 2025
Viewed by 756
Abstract
This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov [...] Read more.
This paper develops novel results in the harmonic analysis of Bessel operators, extending their theory to higher-dimensional and non-Euclidean spaces. We present a refined framework for Hardy spaces associated with Bessel operators, emphasizing atomic decompositions, dual spaces, and connections to Sobolev and Besov spaces. The spectral theory of families of boundary-interpolating operators is also expanded, offering precise eigenvalue estimates and functional calculus applications. Furthermore, we explore Bessel operators under non-standard measures, such as fractal and weighted geometries, uncovering new analytical phenomena. Key implications include advanced insights into singular integrals, heat kernel behavior, and the boundedness of Riesz transforms, with potential applications in fractal geometry, constrained wave propagation, and mathematical physics. Full article
(This article belongs to the Special Issue New Perspectives in Harmonic Analysis)
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