Approximation Theory and Methods 2020

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

Deadline for manuscript submissions: closed (31 August 2021) | Viewed by 16992

Special Issue Editor

Section of Mathematics, International Telematic University, Corso Vittorio Emanuele II, 39, 00186 Roma, Italy
Interests: special functions; orthogonal polynomials; differential equations; operator theory; multivariate approximation theory; Lie algebra
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

The importance of approximation theory and related methods ranges from a need to represent functions in computer calculations to an interest in the mathematics of the subject; work in numerical analysis and in mathematical computation is one of the main links between these two extremes.

In this Special Issue, we will cover the field of spectral problems; exponential integrators for ODE systems; and some applications for the numerical solution of evolutionary PDE, also discretized, by using the concepts and the related formalism of special functions and orthogonal polynomials, which represent a powerful tool to simplify computation. Not only this, we will open computer users with efficient programs for general approximation calculations, in order that useful advances in the subject can be applied.

Prof. Dr. Clemente Cesarano
Guest Editor

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Keywords

  • Splines
  • Multivariate approximation
  • Approximation operators
  • Special functions and orthogonal polynomials
  • Theory of minimax approximation
  • Approximation to periodic functions

Published Papers (8 papers)

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Research

24 pages, 4883 KiB  
Article
Spline Approximation, Part 2: From Polynomials in the Monomial Basis to B-splines—A Derivation
by Nikolaj Ezhov, Frank Neitzel and Svetozar Petrovic
Mathematics 2021, 9(18), 2198; https://doi.org/10.3390/math9182198 - 08 Sep 2021
Cited by 5 | Viewed by 2432
Abstract
In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary [...] Read more.
In a series of three articles, spline approximation is presented from a geodetic point of view. In part 1, an introduction to spline approximation of 2D curves was given and the basic methodology of spline approximation was demonstrated using splines constructed from ordinary polynomials. In this article (part 2), the notion of B-spline is explained by means of the transition from a representation of a polynomial in the monomial basis (ordinary polynomial) to the Lagrangian form, and from it to the Bernstein form, which finally yields the B-spline representation. Moreover, the direct relation between the B-spline parameters and the parameters of a polynomial in the monomial basis is derived. The numerical stability of the spline approximation approaches discussed in part 1 and in this paper, as well as the potential of splines in deformation detection, will be investigated on numerical examples in the forthcoming part 3. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
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16 pages, 18541 KiB  
Article
Planar Typical Bézier Curves with a Single Curvature Extremum
by Chuan He, Gang Zhao, Aizeng Wang, Shaolin Li and Zhanchuan Cai
Mathematics 2021, 9(17), 2148; https://doi.org/10.3390/math9172148 - 03 Sep 2021
Cited by 2 | Viewed by 2016
Abstract
This paper focuses on planar typical Bézier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given [...] Read more.
This paper focuses on planar typical Bézier curves with a single curvature extremum, which is a supplement of typical curves with monotonic curvature by Y. Mineur et al. We have proven that the typical curve has at most one curvature extremum and given a fast calculation formula of the parameter at the curvature extremum. This will allow designers to execute a subdivision at the curvature extremum to obtain two pieces of typical curves with monotonic curvature. In addition, we put forward a sufficient condition for typical curve solutions under arbitrary degrees for the G1 interpolation problem. Some numerical experiments are provided to demonstrate the effectiveness and efficiency of our approach. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
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12 pages, 278 KiB  
Article
On the Approximation by Balázs–Szabados Operators
by Adrian Holhoş
Mathematics 2021, 9(14), 1588; https://doi.org/10.3390/math9141588 - 06 Jul 2021
Cited by 2 | Viewed by 1340
Abstract
We present three new approximation properties of the Balázs–Szabados operators. Firstly, we prove that, in certain cases, these operators approximate some super-exponential functions on compact intervals. Next, we provide a new estimate of the error of approximation using a suitable modulus of continuity. [...] Read more.
We present three new approximation properties of the Balázs–Szabados operators. Firstly, we prove that, in certain cases, these operators approximate some super-exponential functions on compact intervals. Next, we provide a new estimate of the error of approximation using a suitable modulus of continuity. Finally, we characterize the functions which can be uniformly approximated in the weighted norm of polynomial weight spaces. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
9 pages, 995 KiB  
Article
Solving the Coriolis Vibratory Gyroscope Motion Equations by Means of the Angular Rate B-Spline Approximation
by Mikhail Basarab and Boris Lunin
Mathematics 2021, 9(3), 292; https://doi.org/10.3390/math9030292 - 02 Feb 2021
Cited by 2 | Viewed by 1969
Abstract
The exact solution of the movement equation of the Coriolis vibratory gyroscope (CVG) with a linear law of variation of the angular rate of rotation of the base is given. The solution is expressed in terms of the Weber functions (the parabolic cylinder [...] Read more.
The exact solution of the movement equation of the Coriolis vibratory gyroscope (CVG) with a linear law of variation of the angular rate of rotation of the base is given. The solution is expressed in terms of the Weber functions (the parabolic cylinder functions) and their asymptotic representations. On the basis of the obtained solution, an analytical solution to the equation of the ring dynamics in the case of piecewise linear approximation of an arbitrary angular velocity profile on a time grid is derived. The piecewise linear solution is compared with the more rough piecewise constant solution and the dependence of the error of such approximations on the sampling step in time is estimated numerically. The results obtained make it possible to significantly reduce the number of operations when it is necessary to study long-range dynamics of oscillations of the system, as well as quantitatively and qualitatively control the convergence of finite-difference schemes for solving the movement equations of the Coriolis vibratory gyroscope. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
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16 pages, 297 KiB  
Article
Using Parameter Elimination to Solve Discrete Linear Chebyshev Approximation Problems
by Nikolai Krivulin
Mathematics 2020, 8(12), 2210; https://doi.org/10.3390/math8122210 - 13 Dec 2020
Cited by 2 | Viewed by 1490
Abstract
We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many [...] Read more.
We consider discrete linear Chebyshev approximation problems in which the unknown parameters of linear function are fitted by minimizing the least maximum absolute deviation of errors. Such problems find application in the solution of overdetermined systems of linear equations that appear in many practical contexts. The least maximum absolute deviation estimator is used in regression analysis in statistics when the distribution of errors has bounded support. To derive a direct solution of the problem, we propose an algebraic approach based on a parameter elimination technique. As a key component of the approach, an elimination lemma is proved to handle the problem by reducing it to a problem with one parameter eliminated, together with a box constraint imposed on this parameter. We demonstrate the application of the lemma to the direct solution of linear regression problems with one and two parameters. We develop a procedure to solve multidimensional approximation (multiple linear regression) problems in a finite number of steps. The procedure follows a method that comprises two phases: backward elimination and forward substitution of parameters. We describe the main components of the procedure and estimate its computational complexity. We implement symbolic computations in MATLAB to obtain exact solutions for two numerical examples. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
10 pages, 1843 KiB  
Article
To Google or Not: Differences on How Online Searches Predict Names and Faces
by Carmen Moret-Tatay, Abigail G. Wester and Daniel Gamermann
Mathematics 2020, 8(11), 1964; https://doi.org/10.3390/math8111964 - 05 Nov 2020
Cited by 5 | Viewed by 1587
Abstract
Word and face recognition are processes of interest for a large number of fields, including both clinical psychology and computer calculations. The research examined here aims to evaluate the role of an online frequency’s ability to predict both face and word recognition by [...] Read more.
Word and face recognition are processes of interest for a large number of fields, including both clinical psychology and computer calculations. The research examined here aims to evaluate the role of an online frequency’s ability to predict both face and word recognition by examining the stability of these processes in a given amount of time. The study will further examine the differences between traditional theories and current contextual frequency approaches. Reaction times were recorded through both a logarithmic transformation and through a Bayesian approach. The Bayes factor notation was employed as an additional test to support the evidence provided by the data. Although differences between face and name recognition were found, the results suggest that latencies for both face and name recognition are stable for a period of six months and online news frequencies better predict reaction time for both classical frequentist analyses. These findings support the use of the contextual diversity approach. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
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20 pages, 1872 KiB  
Article
The Optimal Shape Parameter for the Least Squares Approximation Based on the Radial Basis Function
by Sanpeng Zheng, Renzhong Feng and Aitong Huang
Mathematics 2020, 8(11), 1923; https://doi.org/10.3390/math8111923 - 02 Nov 2020
Cited by 7 | Viewed by 2375
Abstract
The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. The RBF is especially suitable for scattered data approximation and high dimensional function approximation. The smoothness and approximation accuracy of the RBF are affected by [...] Read more.
The radial basis function (RBF) is a class of approximation functions commonly used in interpolation and least squares. The RBF is especially suitable for scattered data approximation and high dimensional function approximation. The smoothness and approximation accuracy of the RBF are affected by its shape parameter. There has been some research on the shape parameter, but the research on the optimal shape parameter of the least squares based on the RBF is scarce. This paper proposes a way for the measurement of the optimal shape parameter of the least squares approximation based on the RBF and an algorithm to solve the corresponding optimal parameter. The method consists of considering the shape parameter as an optimization variable of the least squares problem, such that the linear least squares problem becomes nonlinear. A dimensionality reduction is applied to the nonlinear least squares problem in order to simplify the objective function. To solve the optimization problem efficiently after the dimensional reduction, the derivative-free optimization is adopted. The numerical experiments indicate that the proposed method is efficient and reliable. Multiple kinds of RBFs are tested for their effects and compared. It is found through the experiments that the RBF least squares with the optimal shape parameter is much better than the polynomial least squares. The method is successfully applied to the fitting of real data. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
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13 pages, 274 KiB  
Article
Stancu Type Baskakov—Durrmeyer Operators and Approximation Properties
by Adem Kilicman, Mohammad Ayman Mursaleen and Ahmed Ahmed Hussin Ali Al-Abied
Mathematics 2020, 8(7), 1164; https://doi.org/10.3390/math8071164 - 15 Jul 2020
Cited by 27 | Viewed by 2157
Abstract
In this article, we introduce Stancu type generalization of Baskakov–Durrmeyer operators by using inverse Pólya–Eggenberger distribution. We discuss some basic results and approximation properties. Moreover, we study the statistical convergence for these operators. Full article
(This article belongs to the Special Issue Approximation Theory and Methods 2020)
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