Riemann Integrable Functions Research Articles

Statistical convergence of integral form of modified Szász–Mirakyan operators: an algorithm and an approach for possible applications

In this study, we take into account the of modified Szász–Mirakyan–Kantorovich operators to obtain their rate of convergence using the modulus of continuity and for the functions in Lipschitz space. Then, we obtain the statistical convergence of this form. In addition, we determine the weighted statistical convergence and compare it with the statistical one for the same operator. Medical applications and traditional mathematics; one way to get a close approximation of the Riemann integrable functions is through the use of the Kantorovich modification of positive linear operators. The use of Kantorovich operators is tremendously helpful from a medical point of view. Their application is shown as an approximation of the rate of convergence in respect of modulus of continuity.

Banach spaces with the Lebesgue property of Riemann integrability

A Banach space is said to have the Lebesgue property if every Riemann-integrable function f:[0,1]→X is Lebesgue almost everywhere continuous. We give a characterization of the Lebesgue property in terms of a new sequential asymptotic structure that is strictly between the notions of spreading and asymptotic models. We also reproduce an apparently lost theorem of Pelczynski and da Rocha Filho that a subspace X⊂L1[0,1] has the Lebesgue property if every spreading model of X is equivalent to the unit vector basis of ℓ1.

Discover an accurate approximation of dynamical system without prior information and customized design

Modeling dynamical systems is fraught with challenges when data can be collected but thorough analysis of the mechanism is absent. We design a method to discover unknown dynamical systems from data. The method discovers an accurate approximation of the model without the prior information and the customized design for each problem. The identification steps are straightforward as bringing in the data and then obtaining the model. The method begins with the simple idea that the equations of motion of many practical problems are Riemann integrable functions. For this reason, the Fourier series can decompose the equations of motion. In order to improve the accuracy, we design an extension that helps us to approximate unknown functions by the Fourier series with a high rate of convergence. The idea converts the difficulty of modeling the dynamical system into finding its Fourier series approximation. Convenient procedures enable the modeling of different problems. Numerical examples show that the new method discovers linear and nonlinear dynamical systems in the same steps and without the prior information.

Riemann Integral on Fractal Structures

In this work we start developing a Riemann-type integration theory on spaces which are equipped with a fractal structure. These topological structures have a recursive nature, which allows us to guarantee a good approximation to the true value of a certain integral with respect to some measure defined on the Borel σ-algebra of the space. We give the notion of Darboux sums and lower and upper Riemann integrals of a bounded function when given a measure and a fractal structure. Furthermore, we give the notion of a Riemann-integrable function in this context and prove that each μ-measurable function is Riemann-integrable with respect to μ. Moreover, if μ is the Lebesgue measure, then the Lebesgue integral on a bounded set of Rn meets the Riemann integral with respect to the Lebesgue measure in the context of measures and fractal structures. Finally, we give some examples showing that we can calculate improper integrals and integrals on fractal sets.

Discontinuous Riemann integrable functions emerging from cellular automata

Discontinuous Riemann integrable functions emerging from cellular automata

Approximation and interpolation with neural network

In this paper we show that multilayer feedforward networks with one single hidden layer.and certain types of activation functions can approximate univariant continuous functions defined on a compact set. arbitrarily well. In particular, our results contain some usual activation functions such as sigmoidal functions, RELU functions and threshold functions. Besides, since interpolation problems are highly related to approximation problem, we demonstrate that a wide range of functions have the ability to interpolate and generalize our results to functions which are not polynomial on R. Compared to existing results by numerous work, our methods are more intuitive and less technical. Lastly, the paper discusses the possibility of combining interpolation property and approximating property together, and demonstrates that given any Riemann integrable functions on a compact set in R, with several points on its graph, the finite combination of monotone sigmoidal functions can pass through these points and approximate the given function arbitrarily well with respect to L^1 (dx) (in the sense of Riemann integral) when the number of points getting large.

A concise proof of Benford’s law

This article presents a concise proof of the famous Benford’s law when the distribution has a Riemann integrable probability density function and provides a criterion to judge whether a distribution obeys the law. The proof is intuitive and elegant, accessible to anyone with basic knowledge of calculus, revealing that the law originates from the basic property of human number system. The criterion can bring great convenience to the field of fraud detection.

Characterization of the Extension of Lattice Linear Space of Continuous Bounded Functions Generated by $$\boldsymbol{\mu}$$-Riemann-integrable Functions by Means of Order Boundaries

Characterization of the Extension of Lattice Linear Space of Continuous Bounded Functions Generated by $$\boldsymbol{\mu}$$-Riemann-integrable Functions by Means of Order Boundaries

Derivation of New Degrees for Best COCUNP Weighted Approximation: II

Approximation Theory is a branch of analysis and applied mathematics requiring that the approximation process preserves certain <img src=image/13417154_01.gif>-shaped properties defined at a finite interval <img src=image/13417154_02.gif>, such as convexity in all or parts of the interval. The (Co)convex and Unconstrained Polynomial (COCUNP) approximation is one of the key estimations of the approximation theory that Kopotun has recently raised for ten years. Numerous studies have been conducted on modern methods of weighted approximation to construct the best degree of approximation. In developing COCUNP a novel technique, the Lebesgue Stieltjes integral-i technique is used to resolve certain disadvantages, such as Riemann's integrable functions, which do not have the degree of the best approximation in norm space. In order to achieve the main goal, Derivation of New Degree (DOND) of the best COCUNP approximation was constructions. The theoretical results revealed that, in general, the new degrees of best approximation were able to smaller errors compared to the existing literature in the same estimating. In conclusion, this study has successfully developed DOND for the best (Co)convex Polynomial (COCP) weighted approximation.

Strong derivative and the essentially Riemann integral

We characterize the primitives of Riemann integrable functions using the concept of strong differentiability. We also introduce the notion of essentially Riemann integrable function and characterize it descriptively using its primitive. We also give a characterization of this function that parallels Lebesgue's criterion for Riemann integrability.

A Study on the Approximations of Bounded Functions in the Locally Global Spaces (L δ ,P ), (0 < P ≤ 1)

The purpose of the present paper is to evaluate the error of the approximation of the function F defined in [0,1] by Bernstein polynomial in L P space (0 < P ≤ 1).We study the direct relation between the degrees of approximation of Riemann integrable functions with respect to algebraic polynomials with average modulus of smoothness the locally global spaces (Lδ,P ), (0 < P ≤ 1) wich is importance for the approximations theory.

Irradiance transport equation applied to propagation of wavefront obtained by the Bi-Ronchi test using point cloud

In this work, we show a Bi-Ronchi test (BRT) proposal using point cloud for sensing the whole surface and wavefront of spherical mirrors as a fast computational test with efficiency comparable with the Ronchi test. We also show an application of the Irradiance Transport equation based on the differential form of the circular Zernike polynomials, to obtain the Phase Transport equation, with capabilities to study the wavefront propagation. To this aim, we experimentally analyze the propagation of W with the BRT, to validate the experimental propagation performed with the Irradiance Transport Equation, giving place to a Phase Transport equation. For this purpose, we use square rulings to observe bi-Ronchigrams and their corresponding Transversal Aberrations as well as their W. We validated our results by comparing the BRT with the Ronchi Test for several measurements of a given wavefront W with Ronchi rulings in the same position. Three different bRR were placed in several locations before and beyond the mirror curvature Radius in order to characterize the experimental propagation of W and test the reflection relation of W before and beyond the mirror Curvature Radius. Finally, we use the Phase Transport equation to analyze the propagation of an efficient W in terms of the differential circular Zernike Polynomials in order to obtain a Riemann-integrable function to numerically find a solution for validating the experimental propagation, the symmetry of the wavefronts, the existence of a symmetry matrix, and the BRT.

A new limit result in change point analysis

Change points are abrupt changes in time ordered data. The cumulative sum (CUSUM) test statistic is an important statistic for testing for a change point. In this article, we prove a functional central limit theorem for heterogeneous α-mixing triangular array in assuming a Riemann integrable variance function. We investigate the asymptotic behavior of the CUSUM statistic under the null and alternative hypotheses through a simulation.

Atypical Series Representations of Riemann-Integrable Functions

Consider the following series and the usual questions such a thing brings up: does it converge and, if so, then what to? (*) 122+232−242+452−462+472−482+−⋯. For clarity, we note that the next eight...

On Extremal Problems for Pairs of Uniformly Distributed Sequences and Integrals with Respect to Copula Measures

Abstract Motivated by the maximal average distance of uniformly distributed sequences we consider some extremal problems for functionals of type μ C ↦ ∫ 0 1 ∫ 0 1 F d μ C , {\mu _C} \mapsto \int_0^1 {{{\int_0^1 {Fd} }_\mu }_C,} where µC is a copula measure and F is a Riemann integrable function on [0, 1]2 of a specific type. Such problems have been considered in [4] and are of interest in the study of limit points of two uniformly distributed sequences.

Distinct Continuous Maps with All Riemann Sums Equal

Do examples as in the title exist? It depends on how the term Riemann sum is understood. For the standard, left, or right Riemann sums such examples do not exist. However, as we will see, they do exist for the lower and upper Riemann sums. Nevertheless, there are only a few examples of such pairs and they have a very simple structure. In this article, we describe all such pairs among Riemann integrable functions from an interval into . We also show that such pairs have an especially nice format when we restrict our attention to continuous maps. All the arguments presented are elementary. In particular, the part concerning continuous functions is self-contained and presented in a format accessible to good undergraduate students.

A Sobolev Space Inroad to Riemann Integrability

A conditioned equivalence is proved for a certain weighted Sobolev space to the space of Riemann integrable functions. An equivalence representing a new result that not only asserts the sufficiency (but non-necessity) nature of bounded variation of functions for their Riemann integrability, but also reveals a potential for some novel computational findings.

The N-Integral

In this paper, we introduced a Henstock-type integral named N-integral of a real valued function f on a closed and bounded interval [a,b]. The set N-integrable functions lie entirely between Riemann integrable functions and Henstock-Kurzweil integrable functions. Furthermore, this new integral integrates all improper Riemann integrable functions even if they are not Lebesgue integrable. It was shown that for a Henstock-Kurzweil integrable function f on [a,b], the following are equivalent:The function f is N-integrable;There exists a null set S for which given epsilon &gt;0 there exists a gauge delta such that for any delta-fine partial division D={(xi,[u,v])} of [a,b] we have [(phi_S(D) Gamma_epsilon) sum |f(v)-f(u)||v-u|&lt;epsilon] where phi_S(D)={(xi,[u,v])in D:xi not in S} and [Gamma_epsilon={(xi,[u,v]): |f(v)-f(u)|&lt;= epsilon}] andThe function f is continuous almost everywhere. A characterization of continuous almost everywhere functions was also given.

A Generalization of the Riemann-Lebesgue Theorem for Riemann Integrability

A classical theorem of Riemann and Lebesgue says that a bounded function defined on a compact interval is Riemann integrable if and only if it is continuous almost everywhere. In this note, we generalize their result and show that the difference between the upper and lower Riemann integrals of a not necessarily Riemann integrable function equals the upper Riemann integral of its oscillation function.

The Bounded Convergence Theorem

We revisit the nineteenth-century version of the bounded convergence theorem formulated by C. Arzelà in 1885 for Riemann integrable functions and, independently, by W. F. Osgood in 1897 for continuous functions.