Modulus Of Continuity Research Articles

Insight into Spatially Colored Stochastic Heat Equation: Temporal Fractal Nature of the Solution

In this paper, the solution to a spatially colored stochastic heat equation (SHE) is studied. This solution is a random function of time and space. For a fixed point in space, the resulting random function of time has exact, dimension-dependent, global continuity moduli, and laws of the iterated logarithm (LILs). It is obtained that the set of fast points at which LILs fail in this process, and occur infinitely often, is a random fractal, the size of which is evaluated by its Hausdorff dimension. These points of this process are everywhere dense with the power of the continuum almost surely, and their hitting probabilities are determined by the packing dimension dimP(E) of the target set E.

A dynamical Thouless formula

In this paper we establish an abstract, dynamical Thouless-type formula for affine families of GL2(R) cocycles. This result extends the classical formula relating, via the Hilbert transform, the maximal Lyapunov exponent and the integrated density of states of a Schrödinger operator. Here, the role of the integrated density of states will be played by a more geometrical quantity, the fibered rotation number. As an application of this formula we present limitations on the modulus of continuity of random linear cocycles. Moreover, we derive Hölder-type continuity properties of the fibered rotation number for linear cocycles over various base dynamics.

Approximation properties by Bézier variant of the Baskakov–Schurer–Szász–Stancu operators

In this paper, we have construct a new version of the Bézier‐type Baskakov–Schurer–Szász–Stancu operators. For this new class of operators, uniform convergence is shown in any compact subset or positive real line. We prove Korovkin‐type theorem, Voronovskaya‐type theorem, and Grüss–Voronovskaya‐type theorem. Moreover, at the end, we express the behavior of the operators in the Lipschitz‐type space using the modulus of continuity and smoothness.

Approximation of classes of periodic functions of several variables with given majorant of mixed moduli of continuity

We obtain the exact-order estimates of approximation of the Nikol'skii-Besov-type classes $B^{\Omega}_{\infty,\theta}$ of periodic functions of several variables with a given function $\Omega(t)$ of a special form by using linear operators satisfying certain conditions. The approximation error is estimated in the metric of the space $L_{\infty}$. The obtained estimates of the considered approximation characteristic, in addition to independent interest, can be used to establish the lower bounds of the orthowidths of the corresponding functional classes.

Some Approximation Properties of Operators Including Degenerate Appell Polynomials

ABSTRACT This paper is interested in a new sequence of linear positive operators including degenerate Appell polynomials. We give a convergence theorem for these operators and obtain the quantitative estimation of the approximation by using modulus of continuity, Peetre’s 𝒦-functional, Lipschitz class functions and a Voronovskaja-type theorem. In addition, we give a Kantorovich modification of these operators and derive some approximation properties.

Distribution of eigenvalues of a perturbed differentiation operator on the interval

In this paper, we construct a characteristic determinant of the spectral problem of a first-order differential equation on an interval with an integral perturbation in the boundary value condition, which is an entire analytic function of the spectral parameter. Based on the formula for the characteristic determinant, conclusions are drawn about the asymptotic behavior of the spectrum of the perturbed spectral problem depending on the modulus of continuity of the subinteral function.

Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity

Asymptotic Estimates for the Growth of Deformed Hankel Transform by Modulus of Continuity

Neural network operators of generalized fractional integrals equipped with a vector-valued function

In this paper, motivated by the recent development of the approximation capability of neural network operators, we aim to construct neural network (NN) operators for multivariate fractional integrals of order α, which are equipped with a vector-valued function ψ and its Jacobian matrix. In this system, the connection strengths between output neurons are represented by fractional mean values, which depend on the vector-valued function ψ, of the approximated multivariate function. To estimate the rate of convergence (learning rate), the connection weights of NN are equipped with a decreasing sequence (λn). The activation function of the system is constructed using a linear collection ΩΨ, which consists of density functions generated by multivariate sigmoidal functions. Our goal is to construct a flexible and productive hybrid system by leveraging a diverse selection of the function ψ, the parameter α, the sequence (λn) and the activation function ΩΨ. The quantitative estimates of the operators are examined by means of the multivariate modulus of continuity. Moreover, we provide some illustrative examples with graphs to demonstrate the approximation performance of the operators through the selected activation functions. Finally, we present numerical results consisting of the maximum absolute errors of approximation for the proposed operators based on various selections of the vector-valued function ψ.

Approximation by exponential-type polynomials

In this paper, a family of exponential-type polynomials is introduced and studied. Both the uniform and the Lp convergence are established in suitable function spaces. In the Lp-case, some estimates are also achieved using an exponentially weighted version of the p-norm. Further, a Voronovskaja type formula is proved, finding the exact order of pointwise approximation in case of continuous functions having second derivative at some points. Finally, quantitative estimates for the order of approximation are established in both the continuous and the Lp-cases in terms of the modulus of continuity and K-functionals of the involved functions. In the latter estimate, a crucial role is played by the so-called Hardy-Littlewood maximal function.

Approximation Properties of Extended Beta-Type Szász–Mirakjan Operators

In this article, we introduce generalized beta extension of Szacute{a}sz-integral type operators and study their approximation properties. First, we calculate the some estimates for these operators. Further, we study the uniform convergence and order of approximation in terms of Korovkin-type theorem and modulus of continuity for the space of univariate continuous functions and bivariate continuous functions in their sections.. Moreover, numerical estimates and graphical representations for convergence of one- and two-dimensional sequences of operators are studied. In continuation, local and global approximation properties are studied in terms of the first- and second-order modulus of smoothness, Peetre’s K-functional and weight functions in various functional spaces.

A new type of Szász–Mirakjan operators based on q-integers

In this article, by using the notion of quantum calculus, we define a new type Szász–Mirakjan operators based on the q-integers. We derive a recurrence formula and calculate the moments Phi _{n,q}(t^{m};x) for m=0,1,2 and the central moments Phi _{n,q}((t-x)^{m};x) for m=1,2. We give estimation for the first and second-order central moments. We present a Korovkin type approximation theorem and give a local approximation theorem by using modulus of continuity. We obtain a local direct estimate for the new Szász–Mirakjan operators in terms of Lipschitz-type maximal function of order α. Finally, we prove a Korovkin type weighted approximation theorem.

Riemann–Liouville fractional integral type exponential sampling Kantorovich series

In the present paper, we introduce a new family of sampling Kantorovich type operators using fractional-type integrals. We study approximation properties of newly constructed operators and give a rate of convergence via a suitable modulus of continuity. Furthermore, we obtain an asymptotic formula considering locally regular functions. Secondly, we deal with logarithmic weighted spaces. By using a certain weighted logarithmic modulus of continuity, we obtain a rate of convergence and give a quantitative form of Voronovskaja-type theorem considering the remainder of Mellin–Taylor’s formula. Moreover, we give a relation between generalized exponential sampling operators and newly constructed operators. Finally, we present some examples of kernels satisfying the obtained results. The results are examined by illustrative numerical table and graphical representations.

Modulus of continuity of weak solutions to a class of singular elliptic equations

Modulus of continuity of weak solutions to a class of singular elliptic equations

Normal families and quasiregular mappings

AbstractBeardon and Minda gave a characterization of normal families of holomorphic and meromorphic functions in terms of a locally uniform Lipschitz condition. Here, we generalize this viewpoint to families of mappings in higher dimensions that are locally uniformly continuous with respect to a given modulus of continuity. Our main application is to the normality of families of quasiregular mappings through a locally uniform Hölder condition. This provides a unified framework in which to consider families of quasiregular mappings, both recovering known results of Miniowitz, Vuorinen and others and yielding new results. In particular, normal quasimeromorphic mappings, Yosida quasiregular mappings and Bloch quasiregular mappings can be viewed as classes of quasiregular mappings which arise through consideration of various metric spaces for the domain and range. We give several characterizations of these classes and obtain upper bounds on the rate of growth in each class.

An approach to preserve functions with exponential growth by using modified Lupaş-Kantrovich operators

As part of this study, we propose a modification of the so-known Lupaş-Kantrovich that preserve exponential function . To support this claim, we estimate the convergence rate of the operators in terms of both the usual and exponential modulus of continuity. Our analysis also includes a global estimate and quantitative Voronovskaya results. Demonstrating the effectiveness of modified operators, we provided a result and supporting graphs.

On a Family of Parameter-Based Bernstein Type Operators with Shape-Preserving Properties

This article aims to introduce a new linear positive operator with a parameter. Our focus lies in analyzing the distinct characteristics and inherent properties exhibited by this operator. Additionally, we provide a proof of the convergence rate and present a revised version of the Voronovskaja theorem specifically tailored for this newly defined operator. Furthermore, we provide an upper bound for the error according to the modulus of continuity. Finally, the preservation of monotonicity and convexity by the operator is being investigated.

Order of Approximation by a New Univariate Kantorovich Type Operator

In order to approximate Lebesgue integrable functions on [0, 1], a sequence of linear positive integral operators of Kantorovich type Lσ<sσ>f (x) with a parameter sσ is introduced. The estimates for rates of approximation for functions with a specific smoothness are proved using the appropriate modulus of continuity.

Neural network approximation of continuous functions in high dimensions with applications to inverse problems

The remarkable successes of neural networks in a huge variety of inverse problems has fueled their adoption in disciplines ranging from medical imaging to seismic analysis over the past decade. However, the high dimensionality of such inverse problems has simultaneously left current theory, which predicts that networks should scale exponentially in the dimension of the problem, unable to explain why the seemingly small networks used in these settings work as well as they do in practice. To reduce this gap between theory and practice, a general method for bounding the complexity required for a neural network to approximate a Hölder continuous function (as well as uniformly continuous functions that admit a specified modulus of continuity) defined on a high-dimensional set with a low-complexity structure is provided herein. The idea for the approach is based on the observation that the existence of a linear Johnson–Lindenstrauss embedding A∈Rd×D of a given high-dimensional set S⊂RD into a low dimensional cube [−M,M]d implies that for any Hölder (or uniformly) continuous function f:S→Rp, there exists a Hölder (or uniformly) continuous function g:[−M,M]d→Rp such that g(Ax)=f(x) for all x∈S. Hence, if one has a neural network which approximates g:[−M,M]d→Rp, then a layer can be added that implements the JL embedding A to obtain a neural network that approximates f:S→Rp. By pairing JL embedding results along with results on approximation of Hölder (or uniformly) continuous functions by neural networks, one then obtains results which bound the complexity required for a neural network to approximate Hölder (or uniformly) continuous functions on high dimensional sets. The end result is a general theoretical framework which can then be used to better explain the observed empirical successes of smaller networks in a wider variety of inverse problems than current theory allows.

Shape Preserving Properties of Parametric Szász Type Operators on Unbounded Intervals

The Be´zier-type operator has become a powerful tool in operator theory, neural networks, curve and surface design and representation because of its good shape-preserving properties. Motivated by the improvements of the operator in computational disciplines, we investigate some elementary properties of two kinds of modified Sza´sz type basis functions, depending on non-negative parameters. Using the derivative, the symmetry of variables, the modulus of continuity and the concave continuous modulus, we study some shape preserving properties of these operators concerning monotonicity, convexity, starshapedness, semi-additivity and the preservation of smoothness. Moreover, some illustrative examples are provided to demonstrate the approximation behavior of the proposed operators and the classical ones.

Charlier Polinomlarıyla İlişkili Kantorovich Tipi Operatörler Sınıfının Yaklaşım Özellikleri

In this paper, we introduce a kind of Charlier polynomial-based Szász-Kantorovich type operator. We begin by using Korovkin's theorem to demonstrate the uniform convergence of these operators. Second, using mathematical techniques like Peetre’s K-functional notion and the common modulus of the operators, we evaluate the order of convergence of the operators. Third, we use the Voronovskaya type approximation theorem to derive an asymptotic formula for the operator we gave. Finally, we give a numerical example using Maple 2022.