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Axioms
Special Issues
Numerical Methods and Approximation Theory
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Numerical Methods and Approximation Theory

A special issue of Axioms (ISSN 2075-1680). This special issue belongs to the section "Mathematical Analysis".

Deadline for manuscript submissions: 30 September 2024 | Viewed by 2136

Special Issue Editors

Prof. Dr. Marius Birou

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Guest Editor
Department of Mathematics, Technical University of Cluj Napoca, 400114 Cluj-Napoca, Romania
Interests: approximation by positive linear operators; numerical analysis
Special Issues, Collections and Topics in MDPI journals
Prof. Dr. Ana-Maria Acu

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Guest Editor
Department of Mathematics and Informatics, Lucian Blaga University of Sibiu, 550012 Sibiu, Romania
Interests: approximation theory; numerical analysis; probability and statistics
Special Issues, Collections and Topics in MDPI journals
Dr. Carmen Violeta Muraru

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Guest Editor
Associate Professor, Department of Mathematics, Informatics, “Vasile Alecsandri” University of Bacău, 600115 Bacău, Romania
Interests: approximation theory using linear and positive operators; probability and statistics; numerical methods
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The purpose of this Special Issue is to collect new articles on topics from the fields of numerical methods and approximation theory. These fields have developed tremendously in recent years, obtaining many applications. For example, the literature in the field of approximation of functions with positive and linear operators is quite rich. Numerical methods have been developed well for approximating the solutions of different types of equations (differential equations, equations with partial derivatives, and integral equations).

The connection between pure and applied mathematics is important (practical applications must be supported by rigorously proven theoretical results). In many studies, practical examples are presented to justify the applicability of theoretical results. Articles related to the two fields are welcome in this Special Issue (for example, inequalities or special functions that appear in the field of approximation of functions).

Prof. Dr. Marius Birou
Prof. Dr. Ana-Maria Acu
Dr. Carmen Violeta Muraru
Guest Editors

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. Axioms is an international peer-reviewed open access monthly 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 2400 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

  • approximation by positive linear operators
  • degree of approximation
  • iterates of positive linear operators
  • interpolation operators
  • quadrature formulas
  • numerical algorithms
  • numerical methods for partial differential equations
  • fixed point theory
  • special functions in approximations
  • inequalities in approximations

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Published Papers (2 papers)

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Research

9 pages, 234 KiB  
Article
The Invariant Subspace Problem for Separable Hilbert Spaces
by Roshdi Khalil, Abdelrahman Yousef, Waseem Ghazi Alshanti and Ma’mon Abu Hammad
Axioms 2024, 13(9), 598; https://doi.org/10.3390/axioms13090598 - 2 Sep 2024
Viewed by 1095
Abstract
In this paper, we prove that every bounded linear operator on a separable Hilbert space has a non-trivial invariant subspace. This answers the well-known invariant subspace problem. Full article
(This article belongs to the Special Issue Numerical Methods and Approximation Theory)
8 pages, 728 KiB  
Article
On the Approximation of the Hardy Z-Function via High-Order Sections
by Yochay Jerby
Axioms 2024, 13(9), 577; https://doi.org/10.3390/axioms13090577 - 25 Aug 2024
Viewed by 713
Abstract
The Z-function is the real function given by Z(t)=eiθ(t)ζ12+it, where ζ(s) is the Riemann zeta function, and θ(t) is [...] Read more.
The Z-function is the real function given by Z(t)=eiθ(t)ζ12+it, where ζ(s) is the Riemann zeta function, and θ(t) is the Riemann–Siegel theta function. The function, central to the study of the Riemann hypothesis (RH), has traditionally posed significant computational challenges. This research addresses these challenges by exploring new methods for approximating Z(t) and its zeros. The sections of Z(t) are given by ZN(t):=∑k=1Ncos(θ(t)−ln(k)t)k for any N∈N. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE) Z(t)≈2ZN˜(t)(t) for N˜(t)=t2π. While historically important, the Hardy–Littlewood AFE does not sufficiently discern the RH and requires further evaluation of the Riemann–Siegel formula. An alternative, less common, is Z(t)≈ZN(t)(t) for N(t)=t2, which is Spira’s approximation using higher-order sections. Spira conjectured, based on experimental observations, that this approximation satisfies the RH in the sense that all of its zeros are real. We present a proof of Spira’s conjecture using a new approximate equation with exponentially decaying error, recently developed by us via new techniques of acceleration of series. This establishes that higher-order approximations do not need further Riemann–Siegel type corrections, as in the classical case, enabling new theoretical methods for studying the zeros of zeta beyond numerics. Full article
(This article belongs to the Special Issue Numerical Methods and Approximation Theory)
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