Approximation Theorem Research Articles

A Korovkin Type Approximation Theorem For Balázs Type Bleimann, Butzer and Hahn Operators via Power Series Statistical Convergence

AbstractIn this paper, we obtain a Korovkin type approximation theorem for power series statistical convergence of functions belonging to the class produced by multivariable modulus of continuity function. As an application of this theorem, we construct a non-tensor product Balázs type BBH operator which does not converge in ordinary sense. Moreover, we study promised approximation properties of this operator and compute the rate of convergence. Finally, we prove that our new approximation result works but its classical case fails.

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

In this paper, we define and study q-statistical limit point, q-statistical cluster point, q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. We establish relationships of q-statistical convergence with q-statistically Cauchy, q-strongly Cesàro and statistically C1q-summable sequences. Further, we apply q-statistical convergence to prove a Korovkin type approximation theorem.

Convergence and gradient algorithm of a class of neural networks based on the polygonal fuzzy numbers representation

As a special case of general fuzzy numbers, the polygonal fuzzy number can describe a fuzzy object by means of an ordered representation of finite real numbers. Different from general fuzzy numbers, the polygonal fuzzy numbers overcome the shortcoming of complex operations based on Zadeh’s traditional expansion principle, and can maintain the closeness of arithmetic operation. Hence, it is feasible to use a polygonal fuzzy number to approximate a general fuzzy number. First, an extension theorem of continuous functions on a real compact set is given according to open set construction theorem. Then using Weierstrass approximation theorem and ordered representation of the polygonal fuzzy numbers, the convergence of a single hidden layer feedforward polygonal fuzzy neural network is proved. Secondly, the gradient vector of the approximation error function and the optimization parameter vector of the network are given by using the ordered representation of polygonal fuzzy numbers, and then the gradient descent algorithm is used to train the optimal parameters of the polygonal fuzzy neural network iteratively. Finally, two simulation examples are given to verify the approximation ability of the network. Simulation result shows that the proposed network and the gradient descent algorithm are effective, and the single hidden layer feedforward network have good abilities in learning and generalization.

On some approximation properties of Gauss-Weierstrass singular operators

Approximation theorems were formulated for function continuous in the neighborhood of some point x, -∞< x < ∞. Namely, the upper bounds were obtained for the function approximations by their Gauss-Weierstrass singular operators in terms of a majorant function for the first- and second-order continuity moduli of the relevant functions.

Aggregation Operators with Undefined Inputs or Outputs

We consider a generalization of aggregation and fusion functions when some of their inputs or outputs are undefined. For simplicity, we represent undefined values by a single NaN (not-a-number) value. We define four main methods of treating undefined values, observe some of their properties, and compare them with several mainstream implementations and norms for handling NaN values. Finally, as a case study demonstrating the apparatus's usefulness, the formalism is applied to the discrete fuzzy transform technique. The formalism allows us to prove a generalized approximation theorem for fuzzy transforms, which removes previously required restrictions on fuzzy partitions for sparse input data.

Bernstein and Bernstein-Like Inequalities for Modulation Spaces

It is shown that the modulation spaces Mwp can be characterized by the approximation behavior of their elements using Local Fourier bases. In analogy to the Local Fourier bases, we show that the modulation spaces can also be characterized by the approximation behavior of their elements using Gabor frames. We derive direct and inverse approximation theorems that describe the best approximation by linear combinations of N terms of a given function using its modulates and translates.

Approximate Diagonal Integral Representations and Eigenmeasures for Lipschitz Operators on Banach Spaces

A new stochastic approach for the approximation of (nonlinear) Lipschitz operators in normed spaces by their eigenvectors is shown. Different ways of providing integral representations for these approximations are proposed, depending on the properties of the operators themselves whether they are locally constant, (almost) linear, or convex. We use the recently introduced notion of eigenmeasure and focus attention on procedures for extending a function for which the eigenvectors are known, to the whole space. We provide information on natural error bounds, thus giving some tools to measure to what extent the map can be considered diagonal with few errors. In particular, we show an approximate spectral theorem for Lipschitz operators that verify certain convexity properties.

Interval universal approximation for neural networks

To verify safety and robustness of neural networks, researchers have successfully applied abstract interpretation , primarily using the interval abstract domain. In this paper, we study the theoretical power and limits of the interval domain for neural-network verification. First, we introduce the interval universal approximation (IUA) theorem. IUA shows that neural networks not only can approximate any continuous function f (universal approximation) as we have known for decades, but we can find a neural network, using any well-behaved activation function, whose interval bounds are an arbitrarily close approximation of the set semantics of f (the result of applying f to a set of inputs). We call this notion of approximation interval approximation . Our theorem generalizes the recent result of Baader et al. from ReLUs to a rich class of activation functions that we call squashable functions . Additionally, the IUA theorem implies that we can always construct provably robust neural networks under ℓ ∞ -norm using almost any practical activation function. Second, we study the computational complexity of constructing neural networks that are amenable to precise interval analysis. This is a crucial question, as our constructive proof of IUA is exponential in the size of the approximation domain. We boil this question down to the problem of approximating the range of a neural network with squashable activation functions. We show that the range approximation problem (RA) is a Δ 2 -intermediate problem, which is strictly harder than NP -complete problems, assuming coNP ⊄ NP . As a result, IUA is an inherently hard problem : No matter what abstract domain or computational tools we consider to achieve interval approximation, there is no efficient construction of such a universal approximator. This implies that it is hard to construct a provably robust network, even if we have a robust network to start with.

On the large O−rates of convergence in limit theorems for compound random sums of arrays of row-wise independent random variables

Compound random sums are extensions of random sums when the random number of summands is a partial sum of independent and identically distributed positive integer-valued random variables, which assumed independent of summands. In the paper, upper bounds for the large O?rates of convergence in weak limit theorems for compound random sums of arrays of row-wise independent random variables, in term of Trotter distance are studied. The main results are approximation theorems which give the Trotter distance between normalized compound random sums of the given independent random variables and the compound ??decomposable random variables. By these results the converging rates in central limit theorem, weak law of large numbers and stable limit theorem for compound random sums are then established. The obtained results in this paper are closely related to the classical ones.

Applications of relative statistical convergence and associated approximation theorem

&lt;abstract&gt;&lt;p&gt;In this work, we investigate a new type of convergence known as relative statistical convergence through the use of the deferred Nörlund and deferred Riesz means. We demonstrate that the idea of deferred Nörlund and deferred Riesz statistically relative uniform convergence is significantly stronger than deferred Nörlund and deferred Riesz statistically uniform convergence. We provide some interesting examples which explain the validity of the theoretical results and effectiveness of constructed sequence spaces. Furthermore, as an application point of view we prove the Korovkin-type approximation theorem in the context of relative equi-statistical convergence for real valued functions and demonstrate that our theorem effectively extends and most of the earlier existing results. Finally, we present an example involving the Meyer-König-Zeller operator of real sequences proving that our theorem is a stronger approach than its classical and statistical version.&lt;/p&gt;&lt;/abstract&gt;

Multivalued Impulsive SDEs Driven by G‐Brownian Noise: Periodic Averaging Result

This paper aims to study two approximation theorems in view of the periodic averaging results for non‐Lipschitz multivalued stochastic differential equations with impulses and G‐Brownian motion (MISDEGs). By adopting G‐Itô’s formula and non‐Lipschitz condition, the solutions to the simplified MSDEGs without impulses may replace those of the initial MISDEGs in view of approximation in L2‐sense and capacity. Finally, we bring a couple of two examples to enhance our theoretical results.

Statistical convergence of Szász-Mirakjan-Kantorovich-type operators and their bivariate extension

Our main aim is to investigate the approximation properties of the summation integral type operators in a statistical sense. In this regard, we prove the statistical convergence theorem using well known Korovkin theorem and the degree of approximation is determined. Also using the weight function, the weighted statistical convergence theorem with the help of the Korovkin theorem is obtained. The statistical rate of convergence in the terms of modulus of continuity and function belonging to the Lipschitz class is determined. To support the convergence results of the proposed operators to the function, graphical representations take place and a comparison is shown with Sz?sz-Mirakjan-Kantorovich operators through examples. The last section deals with, a bivariate extension of the proposed operators to determine the approximation of the function of two variables, additionally, the rate of convergence is estimated as well as the convergence of the bivariate operators is shown by graphical representations.

Holomorphic Legendrian curves in projectivised cotangent bundles

We study holomorphic Legendrian curves in the standard complex contact structure on the projectivised cotangent bundle $X=\mathbb P(T^*Z)$ of a complex manifold $Z$ of dimension at least $2$. We provide a detailed analysis of Legendrian curves degenerating to vertical curves and obtain several approximation and general position theorems. In particular, we prove that any vertical holomorphic curve $M\to X$ from a compact bordered Riemann surface $M$ can be deformed to a horizontal Legendrian curve by an arbitrarily small deformation. A similar result is proved in the parametric setting, provided that all vertical curves under consideration are nondegenerate. Stronger results are obtained when the base $Z$ is an Oka manifold or a Stein manifold with the density property. Finally, we establish basic and 1-parametric h-principles for holomorphic Legendrian curves in $X$.

Artificial Intelligence-Assisted Design and Virtual Diagnostic for the Initial Condition of a Storage-Ring-Based Quantum Information System

Developments in Artificial Intelligence (AI) are helping to solve complex physical problems that otherwise may be too computationally demanding to solve using traditional approaches. Universal Approximation Theorems tell us that we can model any physical system if we can approximate the system with some continuous function (i.e., compact convergence topology and algorithmically generated sets of functions, such as the convolutional neural network), whether for an arbitrary depth or arbitrary width neural network. We consider the problem of solving a set of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> coupled algebraic equations as <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> becomes very large and apply machine learning (ML) to solve this problem for any value of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$N$ </tex-math></inline-formula> . The physical problem we are focusing on is to model the equilibrium positions of ions in an ion trap. A storage ring quantum computer could contain well over tens of thousands of ions. Quickly determining the equilibrium positions will be important to minimize the time to target and observe each ion. As each ion serves as a single qubit, this is important for setting and measuring the individual qubit states. The phonon modes from a collection of ions acts as another qubit, useful for gate operations. Measuring the phonon modes, where ions are oscillating around their respective equilibrium positions also means understanding the equilibrium positions very well. Turning all of this into a virtual diagnostic allows real time prediction and comparison to ensure unique definition of each ion.

A-statistical convergence with a rate and applications to approximation

A = (ank) be a regular summability matrix. In the present paper we deal with subspaces of the space of A?statistically convergent sequences obtained by the rate at which the A?statistical limit tends to zero. We prove that a sequence is the A?strongly convergent if and only if it is the A?statistically convergent and the A?uniformly integrable with the rate of o (an) where a = (an) is a positive nonincreasing sequence. We also make a link between the A?strong convergence and the A?distributional convergence with the rate of o (an). Finally, as an application we present an approximation theorem of Korovkin type.

On some Statistical Approximation Properties of Generalized Lupas-Stancu Operators

The purpose of this paper is to introduce Stancu variant of generalized Lupaş operators whose construction depends on a continuously differentiable, increasing and unbounded function ρ. Depending on the selection of γ and δ, these operators are more flexible than the generalized Lupaş operators while retaining their approximation properties. For these operators we give weighted approximation, Voronovskaya type theorem, quantitative estimates for the local approximation. Finally, we investigate the statistical approximation property of the new operators with the aid of a Korovkin type statistical approximation theorem.

WHITNEY APPROXIMATION FOR SMOOTH CW COMPLEX

We show a Whitney Approximation Theorem for a continuous map from a manifold to a smooth CW complex. This enables us to show that a topological CW complex is homotopy equivalent to a smooth CW complex in a category of topological spaces. It is also shown that, for any open covering of a smooth CW complex, there exists a partition of unity subordinate to the open covering.

On class number relations and intersections over function fields

The aim of this paper is to study class number relations over function fields and the intersections of Hirzebruch-Zagier type divisors on the Drinfeld-Stuhler modular surfaces. The main bridge is a particular "harmonic" theta series with nebentypus. Using the strong approximation theorem, the Fourier coefficients of this series are expressed in two ways; one comes from modified Hurwitz class numbers and another gives the intersection numbers in question.

Liouville-Green approximation for linearly coupled systems: Asymptotic analysis with applications to reaction-diffusion systems

&lt;p style='text-indent:20px;'&gt;Asymptotic analysis has become a common approach in investigations of reaction-diffusion equations and pattern formation, especially when considering generalizations of the original model, such as spatial heterogeneity, where finding an analytic solution even to the linearized equations is generally not possible. The Liouville-Green approximation (also known as WKBJ method), one of the more robust asymptotic approaches for investigating dissipative phenomena captured by linear equations, has recently been applied to the Turing model in a heterogeneous environment. It demonstrated the anticipated modifications to the results obtained in a homogeneous setting, such as localized patterns and local Turing conditions. In this context, we attempt a generalization of the scalar Liouville-Green approximation to multicomponent systems. Our broader mathematical approach results in general approximation theorems for systems of ODEs without turning points. We discuss the cases of exponential and oscillatory behaviour first before treating the general case. Subsequently, we demonstrate the spectral properties utilized in the approximation theorems for a typical Turing system, hence showing that Liouville-Green approximation is plausible for an arbitrary number of coupled species outside of turning points and generally valid for fast growing modes as long as the diffusivities are distinct. Note that our line of approach is via showing that the solution is close (using suitable weight functions for measuring the error) to a linear combination of Airy-like functions.&lt;/p&gt;

Lipschitz graphs and currents in Heisenberg groups

AbstractThe main result of the present article is a Rademacher-type theorem for intrinsic Lipschitz graphs of codimension$k\leq n$in sub-Riemannian Heisenberg groups${\mathbb H}^{n}$. For the purpose of proving such a result, we settle several related questions pertaining both to the theory of intrinsic Lipschitz graphs and to the one of currents. First, we prove an extension result for intrinsic Lipschitz graphs as well as a uniform approximation theorem by means of smooth graphs: both of these results stem from a new definition (equivalent to the one introduced by B. Franchi, R. Serapioni and F. Serra Cassano) of intrinsic Lipschitz graphs and are valid for a more general class of intrinsic Lipschitz graphs in Carnot groups. Second, our proof of Rademacher’s theorem heavily uses the language of currents in Heisenberg groups: one key result is, for us, a version of the celebrated constancy theorem. Inasmuch as Heisenberg currents are defined in terms of Rumin’s complex of differential forms, we also provide a convenient basis of Rumin’s spaces. Eventually, we provide some applications of Rademacher’s theorem including a Lusin-type result for intrinsic Lipschitz graphs, the equivalence between${\mathbb H}$-rectifiability and ‘Lipschitz’${\mathbb H}$-rectifiability and an area formula for intrinsic Lipschitz graphs in Heisenberg groups.