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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: closed (31 January 2024) | Viewed by 23779

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Prof. Dr. Roman Dmytryshyn

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Department of Mathematical and Functional Analysis, Vasyl Stefanyk Precarpathian National University, 57 Shevchenka Str., 76018 Ivano-Frankivsk, Ukraine
Interests: approximation theory; continued fractions and their generalizations; special functions; numerical analysis
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Prof. Dr. Tuncer Acar

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Faculty of Science, Department of Mathematics, Selcuk University, Selcuklu, 42003 Konya, Turkey
Interests: functional analysis; approximation theory
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Special Issue Information

Dear Colleagues,

The theory of approximations is one of the most intriguing sections of mathematics, as its field overlaps with both classical and modern analysis, as well as numerical analysis, and even various branches of applied mathematics. Nowadays, due to the development of computer technology and the requirements of natural and engineering sciences, interest in studying various approximation techniques has grown significantly. In the scientific community, this is a continuous stimulus to develop new and better-performing approximation techniques, able to grasp the particular features of the problem.

The primary purpose of the Special Issue is to highlight the advanced techniques of approximation theory, which have a practical application to a wide range of mathematics problems. This, in turn, will enrich mathematical science with profound and fruitful results.

Prof. Dr. Roman Dmytryshyn
Dr. Tuncer Acar
Guest Editors

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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.

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Keywords

interpolation in approximation theory
approximation by polynomials
spline approximation
approximation by trigonometric polynomials
inequalities in approximation
approximation by rational functions
Padé approximation
rate of convergence
inverse theorems in approximation theory
simultaneous approximation
approximation with constraints
approximation by special function classes
approximation by operators
saturation in approximation theory
best constants in approximation theory
approximation by arbitrary linear expressions
approximation by arbitrary nonlinear expressions
uniqueness of best approximation
best approximants
approximate quadratures
numerical approximation
series expansions
asymptotic approximations
asymptotic expansions
abstract approximation theory
remainders in approximation formulas
least-squares methods
continued fractions and their generalizations
convergence and divergence of infinite limiting processes
approximation of solutions of differential equations
approximation of solutions of functional-differential equations
approximation of solutions of integral equations

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Related Special Issue

Advanced Approximation Techniques and Their Applications, 2nd Edition in Axioms (9 articles)

Published Papers (15 papers)

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Research

12 pages, 287 KiB

Open AccessArticle

Szász–Durrmeyer Operators Involving Confluent Appell Polynomials

by Kadir Kanat and Selin Erdal

Axioms 2024, 13(3), 135; https://doi.org/10.3390/axioms13030135 - 20 Feb 2024

Cited by 1 | Viewed by 1089

Abstract

This article is concerned with the Durrmeyer-type generalization of Szász operators, including confluent Appell polynomials and their approximation properties. Also, the rate of convergence of the confluent Durrmeyer operators is found by using the modulus of continuity and Peetre’s

K

-functional. Then, we [...] Read more.

K

-functional. Then, we show that, under special choices of

A (t)

, the newly constructed operators reduce confluent Hermite polynomials and confluent Bernoulli polynomials, respectively. Finally, we present a comparison of newly constructed operators with the Durrmeyer-type Szász operators graphically. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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17 pages, 292 KiB

Open AccessArticle

Tractability of Approximation of Functions Defined over Weighted Hilbert Spaces

by Huichao Yan and Jia Chen

Axioms 2024, 13(2), 108; https://doi.org/10.3390/axioms13020108 - 5 Feb 2024

Viewed by 1210

Abstract

We investigate

L_{2}

-approximation problems in the worst case setting in the weighted Hilbert spaces

H (K_{R_{d, α, γ}})

with weights

R_{d, α, γ}

under parameters [...] Read more.

We investigate

L_{2}

-approximation problems in the worst case setting in the weighted Hilbert spaces

H (K_{R_{d, α, γ}})

with weights

R_{d, α, γ}

under parameters

1 \geq γ_{1} \geq γ_{2} \geq \dots \geq 0

and

1 < α_{1} \leq α_{2} \leq \dots

. Several interesting weighted Hilbert spaces

H (K_{R_{d, α, γ}})

appear in this paper. We consider the worst case error of algorithms that use finitely many arbitrary continuous linear functionals. We discuss tractability of

L_{2}

-approximation problems for the involved Hilbert spaces, which describes how the information complexity depends on d and

ε^{- 1}

. As a consequence we study the strongly polynomial tractability (SPT), polynomial tractability (PT), weak tractability (WT), and

(t_{1}, t_{2})

-weak tractability (

(t_{1}, t_{2})

-WT) for all

t_{1} > 1

and

t_{2} > 0

in terms of the introduced weights under the absolute error criterion or the normalized error criterion. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

30 pages, 1906 KiB

Open AccessArticle

Schröder-Based Inverse Function Approximation

by Roy M. Howard

Axioms 2023, 12(11), 1042; https://doi.org/10.3390/axioms12111042 - 8 Nov 2023

Cited by 2 | Viewed by 1223

Abstract

Schröder approximations of the first kind, modified for the inverse function approximation case, are utilized to establish general analytical approximation forms for an inverse function. Such general forms are used to establish arbitrarily accurate analytical approximations, with a set relative error bound, for an inverse function when an initial approximation, typically with low accuracy, is known. Approximations for arcsine, the inverse of x − sin(x), the inverse Langevin function and the Lambert W function are used to illustrate this approach. Several applications are detailed. For the root approximation of a function, Schröder approximations of the first kind, based on the inverse of a function, have an advantage over the corresponding generalization of the standard Newton–Raphson method, as explicit analytical expressions for all orders of approximation can be obtained. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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13 pages, 304 KiB

Open AccessArticle

Investigation of the F* Algorithm on Strong Pseudocontractive Mappings and Its Application

by Felix D. Ajibade, Francis Monday Nkwuda, Hussaini Joshua, Taiwo P. Fajusigbe and Kayode Oshinubi

Axioms 2023, 12(11), 1041; https://doi.org/10.3390/axioms12111041 - 8 Nov 2023

Viewed by 1148

Abstract

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the

F^{*}

iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the

F^{*}

iterative algorithm [...] Read more.

In the context of uniformly convex Banach space, this paper focuses on examining the strong convergence of the

F^{*}

iterative algorithm to the fixed point of a strongly pseudocontractive mapping. Furthermore, we demonstrate through numerical methods that the

F^{*}

iterative algorithm converges strongly and faster than other current iterative schemes in the literature and extends to the fixed point of a strong pseudocontractive mapping. Finally, under a nonlinear quadratic Volterra integral equation, the application of our findings is shown. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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14 pages, 310 KiB

Open AccessArticle

Approximation of Functions of the Classes

C_{β}^{ψ} H^{α}

by Linear Methods Summation of Their Fourier Series

by Yurii Kharkevych and Inna Kal’chuk

Axioms 2023, 12(11), 1010; https://doi.org/10.3390/axioms12111010 - 26 Oct 2023

Viewed by 1047

Abstract

In this paper, we considered arbitrary linear summation methods of Fourier series specified by a set of continuous functions dependent on the real parameter and established their approximation properties. We obtained asymptotic formulas for the exact upper bounds of the deviations of operators [...] Read more.

λ

-methods of Fourier series summation from the functions of the classes

C_{β}^{ψ} H^{α}

under certain restrictions on the functions

ψ

. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

17 pages, 3042 KiB

Open AccessArticle

Nonlinear 2D C¹ Quadratic Spline Quasi-Interpolants on Triangulations for the Approximation of Piecewise Smooth Functions

by Francesc Aràndiga and Sara Remogna

Axioms 2023, 12(10), 1002; https://doi.org/10.3390/axioms12101002 - 23 Oct 2023

Cited by 1 | Viewed by 1064

Abstract

The aim of this paper is to present and study nonlinear bivariate

C^{1}

The aim of this paper is to present and study nonlinear bivariate

C^{1}

quadratic spline quasi-interpolants on uniform criss-cross triangulations for the approximation of piecewise smooth functions. Indeed, by using classical spline quasi-interpolants, the Gibbs phenomenon appears when approximating near discontinuities. Here, we use weighted essentially non-oscillatory techniques to modify classical quasi-interpolants in order to avoid oscillations near discontinuities and maintain high-order accuracy in smooth regions. We study the convergence properties of the proposed quasi-interpolants and we provide some numerical and graphical tests confirming the theoretical results. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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17 pages, 1739 KiB

Open AccessArticle

A New Extension of Optimal Auxiliary Function Method to Fractional Non-Linear Coupled ITO System and Time Fractional Non-Linear KDV System

by Rashid Nawaz, Aaqib Iqbal, Hina Bakhtiar, Wissal Audah Alhilfi, Nicholas Fewster-Young, Ali Hasan Ali and Ana Danca Poțclean

Axioms 2023, 12(9), 881; https://doi.org/10.3390/axioms12090881 - 14 Sep 2023

Cited by 4 | Viewed by 1180

Abstract

In this article, we investigate the utilization of Riemann–Liouville’s fractional integral and the Caputo derivative in the application of the Optimal Auxiliary Function Method (OAFM). The extended OAFM is employed to analyze fractional non-linear coupled ITO systems and non-linear KDV systems, which feature equations of a fractional order in time. We compare the results obtained for the ITO system with those derived from the Homotopy Perturbation Method (HPM) and the New Iterative Method (NIM), and for the KDV system with the Laplace Adomian Decomposition Method (LADM). OAFM demonstrates remarkable convergence with a single iteration, rendering it highly effective. In contrast to other existing analytical approaches, OAFM emerges as a dependable and efficient methodology, delivering high-precision solutions for intricate problems while saving both computational resources and time. Our results indicate superior accuracy with OAFM in comparison to HPM, NIM, and LADM. Additionally, we enhance the accuracy of OAFM through the introduction of supplementary auxiliary functions. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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14 pages, 304 KiB

Open AccessArticle

Approximation Characteristics of Gel’fand Type in Multivariate Sobolev Spaces with Mixed Derivative Equipped with Gaussian Measure

by Yuqi Liu, Huan Li and Xuehua Li

Axioms 2023, 12(9), 804; https://doi.org/10.3390/axioms12090804 - 22 Aug 2023

Cited by 1 | Viewed by 852

Abstract

In this paper, we study the probabilistic Gel’fand

(N, δ)

-width of multivariate Sobolev spaces

M W_{2}^{r} (T^{d})

with mixed derivative that are equipped with Gaussian measure

μ

L_{q} (T^{d})

. The sharp asymptotic estimates are [...] Read more.

In this paper, we study the probabilistic Gel’fand

(N, δ)

-width of multivariate Sobolev spaces

M W_{2}^{r} (T^{d})

with mixed derivative that are equipped with Gaussian measure

μ

L_{q} (T^{d})

. The sharp asymptotic estimates are determined by employing the discretization method. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

22 pages, 352 KiB

Open AccessArticle

N-Widths of Multivariate Sobolev Spaces with Common Smoothness in Probabilistic and Average Settings in the S_q Norm

by Yuqi Liu, Xuehua Li and Huan Li

Axioms 2023, 12(7), 698; https://doi.org/10.3390/axioms12070698 - 17 Jul 2023

Cited by 1 | Viewed by 1099

Abstract

In this article, we give the sharp bounds of probabilistic Kolmogorov

(N, δ)

-widths and probabilistic linear

(N, δ)

-widths of the multivariate Sobolev space

W_{2}^{A}

with common smoothness on a

S_{q}

norm equipped with the Gaussian measure [...] Read more.

In this article, we give the sharp bounds of probabilistic Kolmogorov

(N, δ)

-widths and probabilistic linear

(N, δ)

-widths of the multivariate Sobolev space

W_{2}^{A}

with common smoothness on a

S_{q}

norm equipped with the Gaussian measure

μ

, where

A \subset R^{d}

is a finite set. And we obtain the sharp bounds of average width from the results of the probabilistic widths. These results develop the theory of approximation of functions and play important roles in the research of related approximation algorithms for Sobolev spaces. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

14 pages, 331 KiB

Open AccessArticle

Approximation Relations on the Posets of Pseudoultrametrics

by Svyatoslav Nykorovych, Oleh Nykyforchyn and Andriy Zagorodnyuk

Axioms 2023, 12(5), 438; https://doi.org/10.3390/axioms12050438 - 28 Apr 2023

Cited by 2 | Viewed by 1036

Abstract

In this paper we study pseudoultrametrics, which are a natural mixture of ultrametrics and pseudometrics. They satisfy a stronger form of the triangle inequality than usual pseudometrics and naturally arise in problems of classification and recognition. The text focuses on the natural partial order on the set of all pseudoultrametrics on a fixed (not necessarily finite) set. In addition to the “way below” relation induced by a partial order, we introduce its version which we call “weakly way below”. It is shown that a pseudoultrametric should satisfy natural conditions closely related to compactness, for the set of all pseudoultrametric weakly way below it to be non-trivial (to consist not only of the zero pseudoultrametric). For non-triviality of the set of all pseudoultrametrics way below a given one, the latter must be compact. On the other hand, each compact pseudoultrametric is the least upper bound of the directed set of all pseudoultrametrics way below it, which are compact as well. Thus it is proved that the set

C P s U (X)

of all compact pseudoultrametric on a set X is a continuous poset. This shows that compactness is a crucial requirement for efficiency of approximation in methods of classification by means of ultrapseudometrics. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

15 pages, 2559 KiB

Open AccessArticle

On Some Branched Continued Fraction Expansions for Horn’s Hypergeometric Function H₄(a,b;c,d;z₁,z₂) Ratios

by Tamara Antonova, Roman Dmytryshyn, Ilona-Anna Lutsiv and Serhii Sharyn

Axioms 2023, 12(3), 299; https://doi.org/10.3390/axioms12030299 - 15 Mar 2023

Cited by 12 | Viewed by 1425

Abstract

The paper deals with the problem of representation of Horn’s hypergeometric functions by branched continued fractions. The formal branched continued fraction expansions for three different Horn’s hypergeometric function

H_{4}

ratios are constructed. The method employed is a two-dimensional generalization of the classical [...] Read more.

H_{4}

ratios are constructed. The method employed is a two-dimensional generalization of the classical method of constructing of Gaussian continued fraction. It is proven that the branched continued fraction, which is an expansion of one of the ratios, uniformly converges to a holomorphic function of two variables on every compact subset of some domain

H,

H \subset C^{2},

and that this function is an analytic continuation of this ratio in the domain

H .

The application to the approximation of functions of two variables associated with Horn’s double hypergeometric series

H_{4}

is considered, and the expression of solutions of some systems of partial differential equations is indicated. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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14 pages, 347 KiB

Open AccessArticle

Composition and Decomposition of Positive Linear Operators (VIII)

by Ana Maria Acu, Ioan Raşa and Andra Seserman

Axioms 2023, 12(3), 228; https://doi.org/10.3390/axioms12030228 - 22 Feb 2023

Cited by 3 | Viewed by 1556

Abstract

In a series of papers, most of them authored or co-authored by H. Gonska, several authors investigated problems concerning the composition and decomposition of positive linear operators defined on spaces of functions. For example, given two operators with known properties, A and B [...] Read more.

A \circ B

, such as the eigenstructure, the inverse, the Voronovskaja formula, and the second-order central moments. One motivation for studying composed operators is the possibility to obtain better rates of approximation and better Voronovskaja formulas. Our paper will address such problems involving compositions of some classical positive linear operators. We present general results as well as numerical experiments. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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10 pages, 247 KiB

Open AccessEditor’s ChoiceArticle

Fixed Point Theorems for Generalized Classes of Operators

by Cristiana Ionescu

Axioms 2023, 12(1), 69; https://doi.org/10.3390/axioms12010069 - 9 Jan 2023

Cited by 2 | Viewed by 1941

Abstract

In this work, we consider weakly generalized operators, which extend the Geraghty mappings that are studied with regard to the existence and uniqueness of their fixed points, in the setting offered by strong b-metric spaces. Classic results are obtained as corollaries. An [...] Read more.

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

10 pages, 369 KiB

Open AccessEditor’s ChoiceArticle

New Classes of Degenerate Unified Polynomials

by Daniel Bedoya, Clemente Cesarano, Stiven Díaz and William Ramírez

Axioms 2023, 12(1), 21; https://doi.org/10.3390/axioms12010021 - 25 Dec 2022

Cited by 12 | Viewed by 1650

Abstract

In this paper, we introduce a class of new classes of degenerate unified polynomials and we show some algebraic and differential properties. This class includes the Appell-type classical polynomials and their most relevant generalizations. Most of the results are proved by using generating [...] Read more.

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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10 pages, 778 KiB

Open AccessEditor’s ChoiceArticle

Statistical Convergence via q-Calculus and a Korovkin’s Type Approximation Theorem

by Mohammad Ayman Mursaleen and Stefano Serra-Capizzano

Axioms 2022, 11(2), 70; https://doi.org/10.3390/axioms11020070 - 9 Feb 2022

Cited by 35 | Viewed by 3257

Abstract

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically

C_{1}^{q}

-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and [...] Read more.

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically

C_{1}^{q}

-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically

C_{1}^{q}

-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem. Full article

(This article belongs to the Special Issue Advanced Approximation Techniques and Their Applications)

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