Pointwise Convergence Theorem Research Articles

Exponential Sampling Type Kantorovich Max-Product Neural Network Operators

This article deals with the study of approximation behavior of the exponential sampling type Kantorovich max-product neural network operators based on sigmoidal activation function. We study point-wise and uniform convergence theorems in the space of continuous functions for these operators and determine the rate of convergence by means of logarithmic modulus of continuity. We also investigate the approximation of p -th ( 1 ≤ p < ∞ ) Lebesgue integrable functions by the aforementioned operators and study the rate of convergence exploiting Peetre’s K − functional.

The approximation capabilities of Durrmeyer-type neural network operators

AbstractIn this paper, a new family of neural network (NN) operators is introduced. The idea is to consider a Durrmeyer-type version of the widely studied discrete NN operators by Costarelli and Spigler (Neural Netw 44:101–106, 2013). Such operators are constructed using special density functions generated from suitable sigmoidal functions, while the reconstruction coefficients are based on a convolution between a general kernel function $$\chi $$ χ and the function being reconstructed, f. Here, we investigate their approximation capabilities, establishing both pointwise and uniform convergence theorems for continuous functions. We also provide quantitative estimates for the approximation order thanks to the use of the modulus of continuity of f; this turns out to be strongly influenced by the asymptotic behaviour of the sigmoidal function $$\sigma $$ σ . Our study also shows that the estimates we provide are, under suitable assumptions, the best possible. Finally, $$L^p$$ L p -approximation is also established. At the end of the paper, examples of activation functions are discussed.

Polynomial sequences in discrete nilpotent groups of step 2

Abstract We discuss some of our work on averages along polynomial sequences in nilpotent groups of step 2. Our main results include boundedness of associated maximal functions and singular integrals operators, an almost everywhere pointwise convergence theorem for ergodic averages along polynomial sequences, and a nilpotent Waring theorem. Our proofs are based on analytical tools, such as a nilpotent Weyl inequality, and on complex almost-orthogonality arguments that are designed to replace Fourier transform tools, which are not available in the noncommutative nilpotent setting. In particular, we present what we call a nilpotent circle method that allows us to adapt some of the ideas of the classical circle method to the setting of nilpotent groups.

On the convergence properties of sampling Durrmeyer‐type operators in Orlicz spaces

AbstractHere, we provide a unifying treatment of the convergence of a general form of sampling‐type operators, given by the so‐called sampling Durrmeyer‐type series. The main result consists of the study of a modular convergence theorem in the general setting of Orlicz spaces . From the latter theorem, the convergence in , in , and in the exponential spaces can be obtained as particular cases. For the completeness of the theory, we provide a pointwise and uniform convergence theorem on , in case of bounded continuous and bounded uniformly continuous functions; in this context, we also furnish a quantitative estimate for the order of approximation, using the modulus of continuity of the function to be approximated. Finally, applications and examples with graphical representations are given for several sampling series with special kernels.

Convergence of generalized sampling series in weighted spaces

Abstract The present paper deals with an extension of approximation properties of generalized sampling series to weighted spaces of functions. A pointwise and uniform convergence theorem for the series is proved for functions belonging to weighted spaces. A rate of convergence by means of weighted moduli of continuity is presented and a quantitative Voronovskaja-type theorem is obtained.

A Class of Integral Operators that Fix Exponential Functions

In this paper we introduce a general class of integral operators that fix exponential functions, containing several recent modified operators of Gauss–Weierstrass, or Picard or moment type operators. Pointwise convergence theorems are studied, using a Korovkin-type theorem and a Voronovskaja-type formula is obtained.

Noncommutative pointwise convergence of orthogonal expansions of several variables

AbstractIn this paper, the boundedness of the maximal function for an operator‐valued weighted L1 space on the unit sphere of , in which the weight functions are invariant under finite reflection groups, is established. We use it to prove the boundedness of orthogonal expansions in h‐harmonics and this result applies to various methods of summability. Furthermore, we obtain the corresponding pointwise convergence theorems. At last, we give some results in the special reflection group .

On nonlinear bivariate singular integral operators

In this paper, we give some pointwise convergence and Fatou type convergence theorems for a family of nonlinear bivariate singular integral operators in the following form: urn:x-wiley:mma:media:mma5425:mma5425-math-0006 where m1,m2 ≥ 1 are fixed natural numbers, and ω ∈ Ω, Ω denotes a nonempty set of indices endowed with a topology. Here, denotes a family of kernel functions and f belongs to the space of Lebesgue integrable functions . Some numerical examples and graphical illustrations supporting the results are also given.

Weak boundedness of operator-valued Bochner–Riesz means for the Dunkl transform

We consider operator-valued Bochner–Riesz means with weight function hκ2 under a finite reflection group for the Dunkl transform. We establish the maximal inequality of the weighted Hardy–Littlewood maximal function, and we apply it to the maximal inequality of operator-valued Bochner–Riesz means BRδ(hκ2;f)(x) for δ>λκ:=d−12+∑j=1dκj. Furthermore, we also obtain the corresponding pointwise convergence theorem.

Approximation by generalized bivariate Kantorovich sampling type series

The purpose of this paper is to construct a bivariate generalization of new family of Kantorovich type sampling operators $$(K_w^{\varphi }f)_{w>0}.$$ First, we give the pointwise convergence theorem and a Voronovskaja type theorem for these Kantorovich generalized sampling series. Further, we obtain the degree of approximation by means of modulus of continuity and quantitative version of Voronovskaja type theorem for the family $$(K_w^{\varphi }f)_{w>0}.$$ Finally, we give some examples of kernels such as box spline kernels and Bochner–Riesz kernel to which the theory can be applied.

A generalization of the exponential sampling series and its approximation properties

Abstract Here we introduce a generalization of the exponential sampling series of optical physics and establish pointwise and uniform convergence theorem, also in a quantitative form. Moreover we compare the error of approximation for Mellin band-limited functions using both classical and generalized exponential sampling series.

Harmonic Analysis on Quantum Tori

This paper is devoted to the study of harmonic analysis on quantum tori. We consider several summation methods on these tori, including the square Fejér means, square and circular Poisson means, and Bochner-Riesz means. We first establish the maximal inequalities for these means, then obtain the corresponding pointwise convergence theorems. In particular, we prove the noncommutative analogue of the classical Stein theorem on Bochner-Riesz means. The second part of the paper deals with Fourier multipliers on quantum tori. We prove that the completely bounded L p Fourier multipliers on a quantum torus are exactly those on the classical torus of the same dimension. Finally, we present the Littlewood-Paley theory associated with the circular Poisson semigroup on quantum tori. We show that the Hardy spaces in this setting possess the usual properties of Hardy spaces, as one can expect. These include the quantum torus analogue of Fefferman’s H1-BMO duality theorem and interpolation theorems. Our analysis is based on the recent developments of noncommutative martingale/ergodic inequalities and Littlewood-Paley-Stein theory.

On Convergence Properties for a Class of Kantorovich Discrete Operators

Here we give some pointwise convergence theorems and asymptotic formulae of Voronovskaja type for a general class of Kantorovich discrete operators. Applications to the Kantorovich version of some discrete operators are given.

Pointwise convergence for expansions in spherical monogenics

We offer a new approach to deal with the pointwise convergence of Fourier-Laplace series on the unit sphere of even-dimensional Euclidean spaces. By using spherical monogenics defined through the generalized Cauchy-Riemann operator, we obtain the spherical monogenic expansions of square integrable functions on the unit sphere. Based on the generalization of Fueter's theorem inducing monogenic functions from holomorphic functions in the complex plane and the classical Carleson's theorem, a pointwise convergence theorem on the new expansion is proved. The result is a generalization of Carleson's theorem to the higher dimensional Euclidean spaces. The approach is simpler than those by using special functions, which may have the advantage to induce the singular integral approach for pointwise convergence problems on the spheres.

Pointwise convergence and Ascoli theorems for nearness spaces

We first study subspaces and product spaces in the context of nearness spaces and prove that U-N spaces, C-N spaces, PN spaces and totally bounded nearness spaces are nearness hereditary; T-N spaces and compact nearness spaces are N-closed hereditary. We prove that N2 plus compact implies N-closed subsets. We prove that totally bounded, compact and N2 are productive. We generalize the concepts of neighborhood systems into the nearness spaces and prove that the nearness neighborhood systems are consistent with existing concepts of neighborhood systems in topological spaces, uniform spaces and proximity spaces respectively when considered in the respective sub-categories. We prove that a net of functions is convergent under the pointwise convergent nearness structure if and only if its cross-section at each point is convergent. We have also proved two Ascoli-Arzela type of theorems.

On the Approximation Properties of a Class of Convolution Type Nonlinear Singular Integral Operators

Abstract In the present paper we obtain both the pointwise convergence and the rate of pointwise convergence theorems of a class of operators defined by as (𝑥,λ) → (𝑥0,λ 0) in 𝐿1 〈𝑎,𝑏〉, where 〈𝑎,𝑏〉 is an arbitrary interval in 𝑅. Here λ ∈ Λ and Λ is a nonempty set of indices.

Clifford algebra approach to pointwise convergence of Fourier series on spheres

We offer an approach by means of Clifford algebra to convergence of Fourier series on unit spheres of even-dimensional Euclidean spaces. It is based on generalizations of Fueter's Theorem inducing quaternionic regular functions from holomorphic functions in the complex plane. We, especially, do not rely on the heavy use of special functions. Analogous Riemann-Lebesgue theorem, localization principle and a Dini's type pointwise convergence theorem are proved.

Approximation properties in abstract modular spaces for a class of general sampling-type operators

In this article we study approximation properties for the class of general integral operators of the form where G is a locally compact Hausdorff topological space, (Hw )w>0 is a net of closed subsets of G with suitable properties and, for every w>0, μ Hw is a regular measure on Hw We give pointwise, uniform and modular convergence theorems in abstract modular spaces and we apply the results to some kind of discrete operators including the sampling-type series.

A non-conventional ergodic theorem for a nilsystem

We prove a non conventional pointwise convergence theorem for a nilsystem, and give an explicit formula for the limit.

A POINTWISE CONVERGENCE THEOREM FOR SOME ERGODIC MEANS

A pointwise convergence theorem is proved for automorphisms with quasi-discrete spectrum.