Search for Articles:
Title / Keyword
Author / Affiliation / Email
Journal
Article Type
 
 
Advanced Search
 
Section
Special Issue
Volume
Issue
Number
Page
 
3.5
2.4
Journals
Mathematics
Special Issues
Approximation Theory and Applications
share announcement

Approximation Theory and Applications

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

Deadline for manuscript submissions: 30 June 2024 | Viewed by 5690

Special Issue Editors

Dr. Paola Lamberti

E-Mail Website
Guest Editor
Department of Mathematics, University of Turin, 10123 Turin, Italy
Interests: univariate and multivariate spline approximation; numerical methods for cagd; numerical integration with applications; numerical algorithms and scientific software
Prof. Dr. Incoronata Notarangelo

E-Mail Website
Guest Editor
Department of Mathematics, University of Turin, 10123 Turin, Italy
Interests: approximation theory; numerical analysis

Special Issue Information

Dear Colleagues,

Approximation theory is an important bridge between pure and applied mathematics since it consists of a theoretical study of methods that use numerical approximation to solve problems of mathematical analysis by means of computational algorithms and computer simulation. After more than a century from the seminal works of Bernstein and Chebyshev (among others), approximation theory is now a very extensive branch of mathematics, interlacing with many various other scientific fields. In fact, it plays a central role in the analysis of numerical methods for mathematical, physical, medical, engineering, and social sciences and provides directions for future research. For example, polynomial approximation is the basis for the study of Gaussian rules; spline and radial basis function interpolation is an important tool in the geometric design of automotive and aerospatial vessels, while wavelets and their generalizations are used for the compression of large digital images and videos. Recently, approximation methods have been applied to the construction of numerical methods for integral and partial differential equations, fractional calculus, signal theory, and deep learning.

The aim of this Special Issue of Mathematics is to collect the state-of-the-art improvements of several aspects of approximation theory and of its both theoretical and applied branches and connections with other applied and computational sciences. Thus, research papers and review articles, considering the developments of approximation theory, as well as problems in which approximation theory plays a significant role, are welcome.

Dr. Paola Lamberti
Prof. Dr. Incoronata Notarangelo
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. 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

  • approximation by polynomials, splines, radial basis functions
  • approximation on bounded and unbounded domains
  • approximation by linear and nonlinear operators
  • interpolation and quasi-interpolation
  • rate of convergence of best approximation
  • orthogonal systems and fourier series
  • CAGD
  • numerical integration
  • numerical methods for integral and differential equations
  • applications to integral and differential problems, fractional calculus, and machine learning

Published Papers (6 papers)

Download All Papers
Order results
Result details
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:

Research

27 pages, 844 KiB  
Article
A Type of Interpolation between Those of Lagrange and Hermite That Uses Nodal Systems Satisfying Some Separation Properties
by Elías Berriochoa, Alicia Cachafeiro, Héctor García Rábade and José Manuel García-Amor
Mathematics 2024, 12(6), 869; https://doi.org/10.3390/math12060869 - 15 Mar 2024
Viewed by 483
Abstract
In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation [...] Read more.
In this paper, we study a method of polynomial interpolation that lies in-between Lagrange and Hermite methods. The novelty is that we use very general nodal systems on the unit circle as well as on the bounded interval only characterized by a separation property. The way in which we interpolate consists in considering all the nodes for the prescribed values and only half for the derivatives. Firstly, we develop the theory on the unit circle, obtaining the main properties of the nodal polynomials and studying the convergence of the interpolation polynomials corresponding to continuous functions with some kind of modulus of continuity and with general conditions on the prescribed values for half of the derivatives. We complete this first part of the paper with the study of the convergence for smooth functions obtaining the rate of convergence, which is slightly slower than that when equidistributed nodal points are considered. The second part of the paper is devoted to solving a similar problem on the bounded interval by using nodal systems having good properties of separation, generalizing the Chebyshev–Lobatto system, and well related to the nodal systems on the unit circle studied before. We obtain an expression of the interpolation polynomials as well as results about their convergence in the case of continuous functions with a convenient modulus of continuity and, particularly, for differentiable functions. Finally, we present some numerical experiments related to the application of the method with the nodal systems dealt with. Full article
(This article belongs to the Special Issue Approximation Theory and Applications)
►▼ Show Figures

Figure 1

17 pages, 932 KiB  
Article
A Nyström Method for 2D Linear Fredholm Integral Equations on Curvilinear Domains
by Anna Lucia Laguardia and Maria Grazia Russo
Mathematics 2023, 11(23), 4859; https://doi.org/10.3390/math11234859 - 3 Dec 2023
Viewed by 950
Abstract
This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good [...] Read more.
This paper is devoted to the numerical treatment of two-dimensional Fredholm integral equations, defined on general curvilinear domains of the plane. A Nyström method, based on a suitable Gauss-like cubature formula, recently proposed in the literature is proposed. The convergence, stability and good conditioning of the method are proved in suitable subspaces of continuous functions of Sobolev type. The cubature formula, on which the Nyström method is constructed, has an error that behaves like the best polynomial approximation of the integrand function. Consequently, it is also shown how the Nyström method inherits this property and, hence, the proposed numerical strategy is fast when the involved known functions are smooth. Some numerical examples illustrate the efficiency of the method, also in comparison with other methods known in the literature. Full article
(This article belongs to the Special Issue Approximation Theory and Applications)
►▼ Show Figures

Figure 1

18 pages, 330 KiB  
Article
On the Analytic Continuation of Lauricella–Saran Hypergeometric Function FK(a1,a2,b1,b2;a1,b2,c3;z)
by Tamara Antonova, Roman Dmytryshyn and Vitaliy Goran
Mathematics 2023, 11(21), 4487; https://doi.org/10.3390/math11214487 - 30 Oct 2023
Cited by 5 | Viewed by 877
Abstract
The paper establishes an analytical extension of two ratios of Lauricella–Saran hypergeometric functions FK with some parameter values to the corresponding branched continued fractions in their domain of convergence. The PC method used here is based on the correspondence between a formal [...] Read more.
The paper establishes an analytical extension of two ratios of Lauricella–Saran hypergeometric functions FK with some parameter values to the corresponding branched continued fractions in their domain of convergence. The PC method used here is based on the correspondence between a formal triple power series and a branched continued fraction. As additional results, analytical extensions of the Lauricella–Saran hypergeometric functions FK(a1,a2,1,b2;a1,b2,c3;z) and FK(a1,1,b1,b2;a1,b2,c3;z) to the corresponding branched continued fractions were obtained. To illustrate this, we provide some numerical experiments at the end. Full article
(This article belongs to the Special Issue Approximation Theory and Applications)
27 pages, 360 KiB  
Article
Approximation of Brownian Motion on Simple Graphs
by George A. Anastassiou and Dimitra Kouloumpou
Mathematics 2023, 11(20), 4329; https://doi.org/10.3390/math11204329 - 18 Oct 2023
Cited by 1 | Viewed by 1153
Abstract
This article is based on chapters 9 and 19 of the new neural network approximation monograph written by the first author. We use the approximation properties coming from the parametrized and deformed neural networks based on the parametrized error and q-deformed and [...] Read more.
This article is based on chapters 9 and 19 of the new neural network approximation monograph written by the first author. We use the approximation properties coming from the parametrized and deformed neural networks based on the parametrized error and q-deformed and β-parametrized half-hyperbolic tangent activation functions. So, we implement a univariate theory on a compact interval that is ordinary and fractional. The result is the quantitative approximation of Brownian motion on simple graphs: in particular over a system S of semiaxes emanating from a common origin radially arranged and a particle moving randomly on S. We produce a large variety of Jackson-type inequalities, calculating the degree of approximation of the engaged neural network operators to a general expectation function of this kind of Brownian motion. We finish with a detailed list of approximation applications related to the expectation of important functions of this Brownian motion. The differentiability of our functions is taken into account, producing higher speeds of approximation. Full article
(This article belongs to the Special Issue Approximation Theory and Applications)
17 pages, 349 KiB  
Article
Algebraic Solution of Tropical Best Approximation Problems
by Nikolai Krivulin
Mathematics 2023, 11(18), 3949; https://doi.org/10.3390/math11183949 - 17 Sep 2023
Viewed by 947
Abstract
We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on [...] Read more.
We introduce new discrete best approximation problems, formulated and solved in the framework of tropical algebra, which deals with semirings and semifields with idempotent addition. Given a set of samples, each consisting of the input and output of an unknown function defined on an idempotent semifield, the problem is to find a best approximation of the function, by tropical Puiseux polynomial and rational functions. A new solution approach is proposed, which involves the reduction of the problem of polynomial approximation to the best approximate solution of a tropical linear vector equation with an unknown vector on one side (a one-sided equation). We derive a best approximate solution to the one-sided equation, and we evaluate the inherent approximation error in a direct analytical form. Furthermore, we reduce the rational approximation problem to the best approximate solution of an equation with unknown vectors on both sides (a two-sided equation). A best approximate solution to the two-sided equation is obtained in numerical form, by using an iterative alternating algorithm. To illustrate the new technique developed, we solve example approximation problems in terms of a real semifield, where addition is defined as maximum and multiplication as arithmetic addition (max-plus algebra), which corresponds to the best Chebyshev approximation by piecewise linear functions. Full article
(This article belongs to the Special Issue Approximation Theory and Applications)
►▼ Show Figures

Figure 1

10 pages, 308 KiB  
Article
Superconvergent Nyström Method Based on Spline Quasi-Interpolants for Nonlinear Urysohn Integral Equations
by Sara Remogna, Driss Sbibih and Mohamed Tahrichi
Mathematics 2023, 11(14), 3236; https://doi.org/10.3390/math11143236 - 23 Jul 2023
Cited by 1 | Viewed by 660
Abstract
Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the [...] Read more.
Integral equations play an important role for their applications in practical engineering and applied science, and nonlinear Urysohn integral equations can be applied when solving many problems in physics, potential theory and electrostatics, engineering, and economics. The aim of this paper is the use of spline quasi-interpolating operators in the space of splines of degree d and of class Cd−1 on uniform partitions of a bounded interval for the numerical solution of Urysohn integral equations, by using a superconvergent Nyström method. Firstly, we generate the approximate solution and we obtain outcomes pertaining to the convergence orders. Additionally, we examine the iterative version of the method. In particular, we prove that the convergence order is (2d+2) if d is odd and (2d+3) if d is even. In case of even degrees, we show that the convergence order of the iterated solution increases to (2d+4). Finally, we conduct numerical tests that validate the theoretical findings. Full article
(This article belongs to the Special Issue Approximation Theory and Applications)
Show export options expand_more Show export options expand_less
Select all
Export citation of selected articles as:
 
Displaying articles 1-6
Back to TopTop