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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: 31 August 2026 | Viewed by 3739

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Editor

Dr. Paolo Leonetti

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Guest Editor

Department of Economics, Università degli Studi dell'Insubria, 21100 Varese, Italy
Interests: analysis and applications

Special Issue Information

Dear Colleagues,

This Special Issue’s focus is “Functional Analysis” and all its applications. Indeed, today, functional analysis techniques play an important role in many research areas related to pure and applied mathematics, where the linear structure strongly interplays with the topological and order and algebraic properties of the underlying vector space. Thus, this Special Issue will focus on pure mathematical tools and all their applications. In this Special Issue, original research articles and reviews are welcome. Research areas may include (but are not limited to) the following: the geometry of Banach spaces, functional equations, summability, statistical and ideal convergence, partial orders, lineability, and all related applications.

Dr. Paolo Leonetti
Guest Editor

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Keywords

sequence spaces
functional equations
geometry of banach spaces
ideal and filter convergence
partial orders
linear operators
summability
convexity
topological vector spaces
lineability

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Published Papers (6 papers)

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Research

15 pages, 230 KB

Open AccessArticle

The Dynamic String-Averaging Method for Inverse Strongly-Monotone Operators with Summable Errors

by Alexander J. Zaslavski

Axioms 2026, 15(8), 594; https://doi.org/10.3390/axioms15080594 - 7 Aug 2026

Viewed by 107

Abstract

In the work by W. Takahashi and M. Toyoda (2003) it was introduced and studied an iterative process for solving a variational inequality problem which is induced by a inverse strongly-monotone mapping. They showed the weak convergence of the iteration process. Recently we established that most of exact iterates of the same iterative process are approximate solutions of the variational inequality. In the present work we use the dynamic string-averaging algorithm for finding a common solution of a finite family of variational inequality problems, generated by inverse strongly-monotone mappings, and a finite family of fixed point problems. We study this algorithm in the presence of summable computational errors. It is shown that the cardinality of the set of iterates which are not approximate solutions is finite and does not exceed a certain constant which is calculated. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

40 pages, 2447 KB

Open AccessArticle

On Runge–Kutta Methods Based on the Operators

J^{\pm}

by Gerd Baumann

Axioms 2026, 15(8), 573; https://doi.org/10.3390/axioms15080573 - 1 Aug 2026

Viewed by 227

Abstract

Using an integral method based on indefinite integrals

J^{\pm}

Using an integral method based on indefinite integrals

J^{\pm}

, we will give new numerical methods for tackling a variety of initial value problems. Within the scope of this study, a number of example cases of stiff and non-stiff initial value problems are examined in order to validate the newly developed computational approaches. The approximation sequence and the number of discretization steps may be more easily controlled with the help of this new technique. The methods provide an a priori evaluation of the approximation error, facilitating access to an optimal solution. The approach entails modeling indefinite integrals by discretization using conformal maps or the roots of orthogonal polynomials. We will analyze the techniques employed for several basis functions, including Sinc, Sinc-Gaussian, Lagrange, and generalized Lagrange polynomials. Utilizing these collocation approaches, matrices

A^{+}

for implicit Runge–Kutta procedures are automatically generated. In addition to being novel in the realm of Sinc techniques, this approach is also novel in the methodology of polynomial approximation. It eliminates the combinatorial problems that are linked with Radau techniques while also generalizing them. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

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18 pages, 286 KB

Open AccessArticle

Umbral Methods, Function Factorisation and Mittag–Leffler Fourier-Type Integral Transform

by Giuseppe Dattoli, Roberto Ricci and Tommaso Severati

Axioms 2026, 15(7), 520; https://doi.org/10.3390/axioms15070520 - 10 Jul 2026

Viewed by 397

Abstract

We propose a systematic way to construct trigonometric-like functions beyond the classical sine–cosine pair by factorising rational expressions in the umbral operator and then evaluating their action on the vacuum. The guiding idea is simple: the usual trigonometric functions may be viewed as cyclic components arising from a finite factorisation, and the same principle can be extended to an n-fold decomposition of rational umbral expressions. For each integer

n \geq 2

, the construction produces n functions which play the role of higher-order trigonometric-like components: their sum reconstructs the corresponding umbral function, while the individual components isolate the different cyclic sectors of its expansion. The construction is developed first in the formal umbral setting. The quadratic case

n = 2

gives the Gaussian-trigonometric functions, in which the cosine-like component is a Gaussian and the sine-like component is its natural umbral companion. The cubic case

n = 3

yields a three-component cyclic system and shows how the same idea extends beyond the usual even–odd decomposition. These examples suggest that trigonometric factorisation is not restricted to ordinary rotations, but belongs to a broader cyclic principle in umbral calculus. We then reinterpret the same formal identities through the recently developed analytic umbral framework. In this second step, the cyclic components are realised by Mellin–Barnes pairings, and the root-of-unity decomposition is related to the splitting of the corresponding spectral kernel. This analytic formulation provides contour representations and local expansions for the functions obtained formally, while also fixing the branch and residue conventions needed for their analytic continuation. Finally, we indicate how the same cyclic kernels act as deformations of the Fourier transform. The resulting framework presents higher-order umbral trigonometric functions as natural cyclic components of factorised rational or exponential umbral operators. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

13 pages, 301 KB

Open AccessArticle

On Functional Independence of Beurling Zeta-Functions

by Antanas Laurinčikas and Darius Šiaučiūnas

Axioms 2026, 15(5), 345; https://doi.org/10.3390/axioms15050345 - 7 May 2026

Viewed by 376

Abstract

Let

P

be a system of generalized prime numbers, and

N_{P}

the corresponding system of generalized integers. Assuming that

\underset{m \in N_{P}}{\sum_{m ⩽ x}} 1 - a x ≪ x^{β}

with

a > 0

and [...] Read more.

Let

P

be a system of generalized prime numbers, and

N_{P}

the corresponding system of generalized integers. Assuming that

\underset{m \in N_{P}}{\sum_{m ⩽ x}} 1 - a x ≪ x^{β}

with

a > 0

and

0 ⩽ β < 1

, we consider the Beurling zeta-function

ζ_{P} (s)

s = σ + i t

. Beurling zeta-functions constitute a wide class of non-standard zeta-functions which pose interesting mathematical problems. Numerous authors are searching for restrictions on the systems

P

and

N_{P}

that the corresponding Beurling zeta-functions have some properties similar to those of classical zeta-functions. One of such properties is the functional independence which was initiated by O. Hölder and D. Hilbert, and, in the most general form, by S.M. Voronin. This is a motivation to obtain the functional independence in the Voronin sense for a certain class of Beurling zeta-functions. Under a certain additional condition involving the generalized von Mangoldt function, we obtain the functional independence of the function

ζ_{P} (s)

. We prove that the function

ζ_{P} (s)

does not satisfy the equation

\sum_{k = 0}^{r} s^{k} F_{k} (ζ_{P} (s), ζ_{P}^{'} (s), \dots, ζ_{P}^{(n - 1)} (s)) = 0

with continuous functions

F_{k}

k = 0, \dots, r

. The proof is based on the universality property of

ζ_{P} (s)

on approximation of analytic functions by shifts

ζ_{P} (s + i τ)

τ \in R

. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

29 pages, 426 KB

Open AccessArticle

Umbral Theory and the Algebra of Formal Power Series

by Roberto Ricci

Axioms 2026, 15(3), 237; https://doi.org/10.3390/axioms15030237 - 21 Mar 2026

Cited by 3 | Viewed by 823

Abstract

Umbral theory, formulated in its modern version by S. Roman and G. C. Rota, has been reconsidered in more recent times by G. Dattoli and collaborators with the aim of devising a working computational tool in the framework of special function theory. Concepts like the umbral image and umbral vacuum have been introduced as pivotal elements of the discussion which, albeit effective, lack generality. This article is directed towards endowing the formalism with a rigorous formulation within the context of formal power series with complex coefficients

(C ⟦ t ⟧, \partial)

. The new formulation is founded on the definition of the umbral operator

u

as a functional in the “umbral ground state” subalgebra of analytically convergent formal series

φ \in C {t}

. We consider in detail some specific classes of umbral ground states

φ

and analyse the conditions for analytic convergence of the corresponding umbral identities, defined as formal series resulting from the action on

φ

of operators of the form

f (ζ u^{μ})

with

f \in C {t}

and

μ, ζ \in C

. For these umbral states, we exploit the Gevrey classification of formal power series to establish a connection with the theory of Borel–Laplace resummation, allowing us to make rigorous sense of a large class of—even divergent—umbral identities. As an application of the proposed theoretical framework, we introduce and investigate the properties of new umbral images for the Gaussian trigonometric functions, which emphasise the trigonometric-like nature of these functions and enable defining the concept of a “Gaussian Fourier transform”, a potentially powerful tool for applications. Full article

(This article belongs to the Special Issue Applications in Functional Analysis)

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18 pages, 292 KB

Open AccessArticle

Exponential Tail Estimates for Lacunary Trigonometric Series

by Maria Rosaria Formica, Eugeny Ostrovsky and Leonid Sirota

Axioms 2026, 15(1), 5; https://doi.org/10.3390/axioms15010005 - 22 Dec 2025

Viewed by 871

Abstract

We establish precise exponential tail estimates for lacunary trigonometric sums of the form

f_{N} (x) = \sum_{k = 1}^{N} c_{k} \cos (2 π n_{k} x)

, under the Hadamard gap condition. Using cumulant expansions [...] Read more.

We establish precise exponential tail estimates for lacunary trigonometric sums of the form

f_{N} (x) = \sum_{k = 1}^{N} c_{k} \cos (2 π n_{k} x)

, under the Hadamard gap condition. Using cumulant expansions and moment-generating function techniques, we obtain non-asymptotic upper bounds for the tail probabilities, including third-order corrections that refine the classical central limit theorem estimates. Furthermore, several examples illustrate these bounds for various choices of coefficients, highlighting the transition from subgaussian to stretched-exponential tail behavior. Full article

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