Approximation Theorem Research Articles

Iterative Methods of Weak and Strong Convergence Theorems for the Split Common Solution of the Feasibility Problems, Generalized Equilibrium Problems, and Fixed Point Problems

The purpose of this paper is to introduce the extragradient methods for solving split feasibility problems, generalized equilibrium problems, and fixed point problems involved in nonexpansive mappings and pseudocontractive mappings. We establish the results of weak and strong convergence under appropriate conditions. As applications of our three main theorems, when the mappings and their domains take different types of cases, we can obtain nine iterative approximation theorems and corollas on fixed points, variational inequality solutions, and equilibrium points.

Robust Bayesian cluster enumeration based on the [formula omitted] distribution

A major challenge in cluster analysis is that the number of data clusters is mostly unknown and it must be estimated prior to clustering the observed data. In real-world applications, the observed data is often subject to heavy tailed noise and outliers which obscure the true underlying structure of the data. Consequently, estimating the number of clusters becomes challenging. To this end, we derive a robust cluster enumeration criterion by formulating the problem of estimating the number of clusters as maximization of the posterior probability of multivariate tν distributed candidate models. We utilize Bayes’ theorem and asymptotic approximations to come up with a robust criterion that possesses a closed-form expression. Further, we refine the derivation and provide a robust cluster enumeration criterion for data sets with finite sample size. The robust criteria require an estimate of cluster parameters for each candidate model as an input. Hence, we propose a two-step cluster enumeration algorithm that uses the expectation maximization algorithm to partition the data and estimate cluster parameters prior to the calculation of one of the robust criteria. The performance of the proposed algorithm is tested and compared to existing cluster enumeration methods using numerical and real data experiments.

On Durrmeyer Typeλ-Bernstein Operators via (p,q)-Calculus

In the present paper, Durrmeyer typeλ-Bernstein operators via (p,q)-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in someλcases than Durrmeyer type (p,q)-Bernstein operators.

Approximation by Parametric Extension of Szász-Mirakjan-Kantorovich Operators Involving the Appell Polynomials

The purpose of this article is to introduce a Kantorovich variant of Szász-Mirakjan operators by including the Dunkl analogue involving the Appell polynomials, namely, the Szász-Mirakjan-Jakimovski-Leviatan-type positive linear operators. We study the global approximation in terms of uniform modulus of smoothness and calculate the local direct theorems of the rate of convergence with the help of Lipschitz-type maximal functions in weighted space. Furthermore, the Voronovskaja-type approximation theorems of this new operator are also presented.

An interpolation method for the optimal control problem governed by the elliptic convection–diffusion equation

AbstractWe know that due to the Weierstrass approximation theorem any continuous function over a closed interval can be approximated by a polynomial of sufficiently high degree. Therefore, they are frequently used to approximate functions. We proposed a Lagrange polynomial‐based approach for solving the optimal control problem governed by the elliptic convection–diffusion partial differential equation. This paper solves it by the barycentric interpolation method as a class of strong‐form numerical methods. Thanks to Lagrangian multipliers, optimality system is derived and then the barycentric collocation method is employed to discretize the state variable and adjoint variable. Barycentric interpolation method is a Lagrange polynomial interpolation that is fast and deserves to be known as a method of polynomial interpolation. The convergence of the proposed method is also proved. At the end, numerical experiments are employed to illustrate the theoretical findings.

Inverse problem with final overdetermination for time-fractional differential equation in a Banach space

Abstract We consider in a Banach space E the inverse problem ( 𝐃 t α ⁢ u ) ⁢ ( t ) = A ⁢ u ⁢ ( t ) + ℱ ⁢ ( t ) ⁢ f , t ∈ [ 0 , T ] , u ⁢ ( 0 ) = u 0 , u ⁢ ( T ) = u T , 0 < α < 1 (\mathbf{D}_{t}^{\alpha}u)(t)=Au(t)+\mathcal{F}(t)f,\quad t\in[0,T],u(0)=u^{0}% ,u(T)=u^{T},\,0<\alpha<1 with operator A, which generates the analytic and compact α-times resolvent family { S α ⁢ ( t , A ) } t ≥ 0 {\{S_{\alpha}(t,A)\}_{t\geq 0}} , the function ℱ ⁢ ( ⋅ ) ∈ C 1 ⁢ [ 0 , T ] {\mathcal{F}(\,\cdot\,)\in C^{1}[0,T]} and u 0 , u T ∈ D ⁢ ( A ) {u^{0},u^{T}\in D(A)} are given and f ∈ E {f\in E} is an unknown element. Under natural conditions we have proved the Fredholm solvability of this problem. In the special case for a self-adjoint operator A, the existence and uniqueness theorems for the solution of the inverse problem are proved. The semidiscrete approximation theorem for this inverse problem is obtained.

Polynomial-Based Robust Adaptive Impedance Control of Electrically Driven Robots

SUMMARYThis paper presents a robust adaptive impedance controller for electrically driven robots using polynomials of degree N as a universal approximator. According to the universal approximation theorem, polynomials of degree N can approximate uncertainties including un-modeled dynamics and external disturbances. This fact is completely discussed and proved in this paper. The polynomial coefficients are estimated based on the adaptive law calculated in the stability analysis. A performance evaluation has been carried out to verify satisfactory performance of the controller. Simulation results on a two degree of freedom manipulator have been presented to guarantee its successful implementation.

Hierarchical fusion network for periocular and iris by neural network approximation and sparse autoencoder

The eye region is one of the most attractive sources for identification and verification due to the representative availability of such biometric modalities as periocular and iris. Many score-level fusion approaches have been proposed to combine these two modalities targeting to improve the robustness. The score-level approaches can be grouped into three categories: transformation-based, classification-based and density-based. Each category has its own benefits, if combined can lead to a robust fusion mechanism. In this paper, we propose a hierarchical fusion network to fuse multiple fusion approaches from transformation-based and classification-based categories into a unified framework for classification. The proposed hierarchical approach relies on the universal approximation theorem for neural networks to approximate each fusion approach as one child neural network and then ensemble them into a unified parent network. This mechanism takes advantage of both categories to improve the fusion performance, illustrated by an improved equal error rate of the multimodal biometric system. We subsequently force the parent network to learn the representation and interaction strategy between the child networks from the training data through a sparse autoencoder layer, leading to further improvements. Experiments on two public datasets (MBGC version 2 and CASIA-Iris-Thousand) and our own dataset validate the effectiveness of the proposed hierarchical fusion approach for periocular and iris modalities.

On Coincidence Points for a Zamfirescu Type Pair of Multi-Valued Mappings

A concept of a Zamfirescu type pair of multi-valued mappings between metric spaces is introduced. A coincidence existence theorem is proved for such pairs of mappings. It is shown that the result is a generalization of the main result of the recent joint work by K. Neammanee and A. Kaevkhao, in which the concept of a multi-valued Zamfirescu mapping was introduced and the fixed point existence and approximation theorems were proved. In addition, it is shown in this paper that all listed results are special cases of the principle of search for zeros of special $$(1,\lambda)$$ -search functionals proposed earlier by T. N. Fomenko.

A proof of validity for multiphase Whitham modulation theory.

It is proved that approximations which are obtained as solutions of the multiphase Whitham modulation equations stay close to solutions of the original equation on a natural time scale. The class of nonlinear wave equations chosen for the starting point is coupled nonlinear Schrödinger equations. These equations are not in general integrable, but they have an explicit family of multiphase wavetrains that generate multiphase Whitham equations, which may be elliptic, hyperbolic, or of mixed type. Due to the change of type, the function space set-up is based on Gevrey spaces with initial data analytic in a strip in the complex plane. In these spaces a Cauchy-Kowalevskaya-like existence and uniqueness theorem is proved. Building on this theorem and higher-order approximations to Whitham theory, a rigorous comparison of solutions, of the coupled nonlinear Schrödinger equations and the multiphase Whitham modulation equations, is obtained.

Functional approximations via Stein’s method of exchangeable pairs

We combine the method of exchangeable pairs with Stein's method for functional approximation. As a result, we give a general linearity condition under which an abstract Gaussian approximation theorem for stochastic processes holds. We apply this approach to estimate the distance of a sum of random variables, chosen from an array according to a random permutation, from a Gaussian mixture process. This result lets us prove a functional combinatorial central limit theorem. We also consider a graph-valued process and bound the speed of convergence of the distribution of its rescaled edge counts to a continuous Gaussian process.

The Minkowski chain and Diophantine approximation

The Hurwitz chain gives a sequence of pairs of Farey approximations to an irrational real number. Minkowski gave a criterion for a number to be algebraic by using a certain generalization of the Hurwitz chain. We apply Minkowski's generalization (the Minkowski chain) to give criteria for a real linear form to be either badly approximable or singular. We also give a variant of Dirichlet's approximation theorem for a real linear form that produces a whole basis of approximating integral vectors rather than a single one. This result holds if and only if the form is badly approximable. The proofs rely on properties of successive minima and reduced bases of lattices.

Product of deferred Cesàro and deferred weighted statistical probability convergence and its applications to Korovkin-type theorems

In the present work, we introduce and study the notion of statistical probability convergence for sequences of random variables as well as the idea of statistical convergence for sequences of real numbers, which are defined over a Banach space via the product of deferred Cesàro and deferred weighted summability means. We first establish a theorem presenting aconnection between them. Based upon our proposed methods, we then prove a Korovkin-type approximation theorem with algebraic test functions for a sequence of random variables on a Banach space, and demonstrate that our theorem effectively extends and improves most (if not all) of the previously existing results (in classical as well as in statistical versions). Furthermore, an illustrative example is presented here by means of the generalized Meyer–König and Zeller operators of a sequence of random variables in order to demonstrate that our established theorem is stronger than its traditional and statistical versions. Finally, we estimate the rate of the product of deferred Cesàro and deferred weighted statistical probability convergence, and accordingly establish a new result.

On Multivariate Bleimann, Butzer and Hahn Operators

In this paper, we state a Korovkin-type theorem for uniform approximation of functions, belonging to a class generated by multivariable function of modulus of continuity, by the sequence of multivariate positive linear operators. Then, using this theorem, we investigate the corresponding uniform approximation result for the multivariate Bleimann, Butzer and Hahn operators which are not in a tensor product design. Moreover, we give an elementary proof that these operators are non-increasing in n when the attached function is convex and non-increasing and we add a graphical example.

The uniform convergence of a double sequence of functions at a point and Korovkin-type approximation theorems

Abstract In this paper, we introduce an interesting kind of convergence for a double sequence called the uniform convergence at a point. We give an example and demonstrate a Korovkin-type approximation theorem for a double sequence of functions using the uniform convergence at a point. Then we show that our result is stronger than the Korovkin theorem given by Volkov and present several graphs. Finally, in the last section, we compute the rate of convergence.

An elementary proof of Mirsky's low rank approximation theorem

An elementary proof is given for Mirsky's result on best low rank approximations of a given matrix with respect to all unitarily invariant norms.

A Zero-One Law for Uniform Diophantine Approximation in Euclidean Norm

Abstract We study a norm-sensitive Diophantine approximation problem arising from the work of Davenport and Schmidt on the improvement of Dirichlet’s theorem. Its supremum norm case was recently considered by the 1st-named author and Wadleigh [ 17], and here we extend the set-up by replacing the supremum norm with an arbitrary norm. This gives rise to a class of shrinking target problems for one-parameter diagonal flows on the space of lattices, with the targets being neighborhoods of the critical locus of the suitably scaled norm ball. We use methods from geometry of numbers to generalize a result due to Andersen and Duke [ 1] on measure zero and uncountability of the set of numbers (in some cases, matrices) for which Minkowski approximation theorem can be improved. The choice of the Euclidean norm on $\mathbb{R}^2$ corresponds to studying geodesics on a hyperbolic surface, which visit a decreasing family of balls. An application of the dynamical Borel–Cantelli lemma of Maucourant [ 25] produces, given an approximation function $\psi $, a zero-one law for the set of $\alpha \in \mathbb{R}$ such that for all large enough $t$ the inequality $\left (\frac{\alpha q -p}{\psi (t)}\right )^2 + \left (\frac{q}{t}\right )^2 < \frac{2}{\sqrt{3}}$ has non-trivial integer solutions.

Notes on the Szegő minimum problem. I. Measures with deep zeroes

The classical Szegő polynomial approximation theorem states that the polynomials are dense in the space L2(ρ), where ρ is a measure on the unit circle, if and only if the logarithmic integral of the measure ρ diverges. In this note we give a quantitative version of Szegő’s theorem in the special case when the divergence of the logarithmic integral is caused by deep zeroes of the measure ρ on a sufficiently rare subset of the circle.

Speed of Convergence of Chernoff Approximations to Solutions of Evolution Equations

This communication is devoted to establishing the very first steps in study of the speed at which the error decreases while dealing with the based on the Chernoff theorem approximations to one-parameter semigroups that provide solutions to evolution equations.

Approximation of functions by a new construction ofBernstein‐Chlodowskyoperators: Theory and applications

AbstractThe main motivation of this paper is to provide a generalization of Bernstein‐Chlodowsky type operators which depend on functionτby means of two sequences of functions. The newly defined operators fix the test function set{1, τ, τ2}. Then we present the approximation properties of newly defined operators, such as weighted approximation, degree of approximation and Voronovskaya type theorems. Finally, we present a series of numerical examples demonstrating the effectiveness of this newly defined Bernstein‐Chlodowsky operators for computing function approximation.