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

Deadline for manuscript submissions: 28 November 2025 | Viewed by 7921

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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
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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
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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
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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 (5 papers)

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Research

24 pages, 1140 KiB

Open AccessArticle

Flexible and Efficient Iterative Solutions for General Variational Inequalities in Real Hilbert Spaces

by Emirhan Hacıoğlu, Müzeyyen Ertürk, Faik Gürsoy and Gradimir V. Milovanović

Axioms 2025, 14(4), 288; https://doi.org/10.3390/axioms14040288 - 11 Apr 2025

Viewed by 268

Abstract

This paper introduces a novel Picard-type iterative algorithm for solving general variational inequalities in real Hilbert spaces. The proposed algorithm enhances both the theoretical framework and practical applicability of iterative algorithms by relaxing restrictive conditions on parametric sequences, thereby expanding their scope of use. We establish convergence results, including a convergence equivalence with a previous algorithm, highlighting the theoretical relationship while demonstrating the increased flexibility and efficiency of the new approach. The paper also addresses gaps in the existing literature by offering new theoretical insights into the transformations associated with variational inequalities and the continuity of their solutions, thus paving the way for future research. The theoretical advancements are complemented by practical applications, such as the adaptation of the algorithm to convex optimization problems and its use in real-world contexts like machine learning. Numerical experiments confirm the proposed algorithm’s versatility and efficiency, showing superior performance and faster convergence compared to an existing method. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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15 pages, 660 KiB

Open AccessArticle

Approximation Results: Szász–Kantorovich Operators Enhanced by Frobenius–Euler–Type Polynomials

by Nadeem Rao, Mohammad Farid and Mohd Raiz

Axioms 2025, 14(4), 252; https://doi.org/10.3390/axioms14040252 - 27 Mar 2025

Viewed by 177

Abstract

This research focuses on the approximation properties of Kantorovich-type operators using Frobenius–Euler–Simsek polynomials. The test functions and central moments are calculated as part of this study. Additionally, uniform convergence and the rate of approximation are analyzed using the classical Korovkin theorem and the modulus of continuity for Lebesgue measurable and continuous functions. A Voronovskaja-type theorem is also established to approximate functions with first- and second-order continuous derivatives. Numerical and graphical analyses are presented to support these findings. Furthermore, a bivariate sequence of these operators is introduced to approximate a bivariate class of Lebesgue measurable and continuous functions in two variables. Finally, numerical and graphical representations of the error are provided to check the rapidity of convergence. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

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14 pages, 290 KiB

Open AccessArticle

Approximation and the Multidimensional Moment Problem

by Octav Olteanu

Axioms 2025, 14(1), 59; https://doi.org/10.3390/axioms14010059 - 14 Jan 2025

Viewed by 502

Abstract

The aim of this paper is to apply polynomial approximation by sums of squares in several real variables to the multidimensional moment problem. The general idea is to approximate any element of the positive cone of the involved function space with sums whose terms are squares of polynomials. First, approximations on a Cartesian product of intervals by polynomials taking nonnegative values on the entire

R^{2}

, or on

R_{+}^{2}

, are considered. Such results are discussed in

L_{μ}^{1} (R^{2})

and in

C (S_{1} \times S_{2})

-type spaces, for a large class of measures,

μ,

for compact subsets

S_{i}, i = 1, 2

of the interval

[0, + \infty)

. Thus, on such subsets, any nonnegative function is a limit of sums of squares. Secondly, applications to the bidimensional moment problem are derived in terms of quadratic expressions. As is well known, in multidimensional cases, such results are difficult to prove. Directions for future work are also outlined. Full article

(This article belongs to the Special Issue Numerical Methods and Approximation Theory)

9 pages, 234 KiB

Open AccessArticle

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 4613

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

Open AccessArticle

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 1579

Abstract

The Z-function is the real function given by

Z (t) = e^{i θ (t)} ζ (\frac{1}{2} + i t)

, 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) = e^{i θ (t)} ζ (\frac{1}{2} + i t)

, 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

Z_{N} (t) : = \sum_{k = 1}^{N} \frac{cos (θ (t) - ln (k) t)}{\sqrt{k}}

for any

N \in N

. Classically, these sections approximate the Z-function via the Hardy–Littlewood approximate functional equation (AFE)

Z (t) \approx 2 Z_{\tilde{N} (t)} (t)

for

\tilde{N} (t) = [\sqrt{\frac{t}{2 π}}]

. 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) \approx Z_{N (t)} (t)

for

N (t) = [\frac{t}{2}]

, 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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