scholarly journals A MONOTONE C¹ FUNCTION AND A RIEMANN INTEGRABLE FUNCTION WHOSE COMPOSITION IS NOT RIEMANN INTEGRABLE

1996 ◽  
Vol 22 (1) ◽  
pp. 404
Author(s):  
de Bobadilla de Olazabal
Keyword(s):  
Integrable Function ◽  
Riemann Integrable Function

A Convergence Theorem for Certain Riemann Sums

1969 ◽  
Vol 12 (4) ◽  
pp. 523-525 ◽  
Author(s):  
Charles K. Chui
Keyword(s):  
Convergence Theorem ◽  
Integrable Function ◽  
Riemann Sums ◽  
Riemann Integrable Function ◽  
Image Position

For a Riemann integrable function f on the interval [0,1], letand consider the Riemann sums

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

2020 ◽  
Vol 15 (2) ◽  
pp. 99-112
Author(s):  
Fabrizio Durante ◽  
Juan Fernández-Sánchez ◽  
Claudio Ignazzi ◽  
Wolfgang Trutschnig
Keyword(s):  
Integrable Function ◽  
Average Distance ◽  
Extremal Problems ◽  
Riemann Integrable Function ◽  
Limit Points ◽  
Maximal Average ◽  
Uniformly Distributed Sequences

AbstractMotivated by the maximal average distance of uniformly distributed sequences we consider some extremal problems for functionals of type {\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.

Mean Value Theorem for the m-Integral of Dinculeanu

1972 ◽  
Vol 15 (2) ◽  
pp. 243-251 ◽  
Author(s):  
Pedro Morales
Keyword(s):  
Convex Hull ◽  
Integrable Function ◽  
Mean Value ◽  
Mean Value Theorem ◽  
Real Interval ◽  
Riemann Sums ◽  
Differentiable Functions ◽  
Riemann Integrable Function ◽  
Dominated Convergence ◽  
M Integral

The classical mean value theorem asserts that if f is a real, bounded, Riemann integrable function defined on a finite real interval a≤t≤b, then , where infa≤t≤bf(t)≤y0supa≤t≤bf(t). The extensions of Choquet [3], Price [15], and of this paper generalize the fact that y0 belongs to the closure of the convex hull of f([a, b]). The version of Choquet ([3, p. 38]) applies to a continuous function on a compact interval with values in a Banach space; that of Price ([15, p. 24]) applies to a bilinear integral of a special type containing the Birkhoff integral [2]. The m-integral of Dinculeanu [6] (specialization of Bartle's *- integral [1]) leaves intact the Lebesgue dominated convergence theorem and is strong enough to support an extended development. The paper is organized as follows: the object of §2 is to express the integral of a bounded m-integrable function as a limit of Riemann sums; §3 gives Price's generalization of "convex hull" [15]; the theorem of the paper is established in §4; §5 gives applications to vector differentiation which, for continuously differentiable functions, contain results of Dieudonné [5] and McLeod [13].

scholarly journals ALMOST RIEMANN INTEGRABLE FUNCTIONS

1995 ◽  
Vol 26 (3) ◽  
pp. 231-233
Author(s):  
MOUSTAFA DAMLAKHI
Keyword(s):  
Arbitrary Function ◽  
Integrable Function ◽  
Bounded Interval ◽  
Integrable Functions ◽  
Riemann Integrable Function ◽  
Riemann Integrable Functions

An arbitrary function $f$ on a bounded interval $[a,b]$ is  termed an almost $R$-integrable function if there exists a Riemann integrable function $g$ such that $f =g$ a.e. In this note a characterization of the class of almost $R$-integrable functions is obtained.

scholarly journals Fractal Frames of Functions on the Rectangle

2021 ◽  
Vol 5 (2) ◽  
pp. 42
Author(s):  
María A. Navascués ◽  
Ram Mohapatra ◽  
Md. Nasim Akhtar
Keyword(s):  
Tensor Product ◽  
Product Space ◽  
Integrable Function ◽  
Two Dimensional ◽  
Tensor Product Space ◽  
Fractal Functions

In this paper, we define fractal bases and fractal frames of L2(I×J), where I and J are real compact intervals, in order to approximate two-dimensional square-integrable maps whose domain is a rectangle, using the identification of L2(I×J) with the tensor product space L2(I)⨂L2(J). First, we recall the procedure of constructing a fractal perturbation of a continuous or integrable function. Then, we define fractal frames and bases of L2(I×J) composed of product of such fractal functions. We also obtain weaker families as Bessel, Riesz and Schauder sequences for the same space. Additionally, we study some properties of the tensor product of the fractal operators associated with the maps corresponding to each variable.

On the Integrability of the Ergodic Hilbert Transform for A Class of Functions with Equal Absolute Values

1998 ◽  
Vol 5 (2) ◽  
pp. 101-106
Author(s):  
L. Ephremidze
Keyword(s):  
Hilbert Transform ◽  
Measure Space ◽  
Integrable Function ◽  
Finite Measure ◽  
Ergodic Transformation ◽  
Finite Measure Space ◽  
Class Of Functions

Abstract It is proved that for an arbitrary non-atomic finite measure space with a measure-preserving ergodic transformation there exists an integrable function f such that the ergodic Hilbert transform of any function equal in absolute values to f is non-integrable.

scholarly journals General boundedness theorems to some second order nonlinear differential equation with integrable forcing term

1995 ◽  
Vol 18 (4) ◽  
pp. 823-824 ◽  
Author(s):  
Allan Kroopnick
Keyword(s):  
Differential Equation ◽  
Nonlinear Differential Equation ◽  
Real Line ◽  
Integrable Function ◽  
Second Order ◽  
Boundedness Theorem ◽  
Forcing Term ◽  
Order Nonlinear Differential Equation

In this note we present a boundedness theorem to the equationx″+c(t,x,x′)+a(t)b(x)=e(t)wheree(t)is a continuous absolutely integrable function over the nonnegative real line. We then extend the result to the equationx″+c(t,x,x′)+a(t,x)=e(t). The first theorem provides the motivation for the second theorem. Also, an example illustrating the theory is then given.

scholarly journals Fubini’s Theorem for Non-Negative or Non-Positive Functions

2018 ◽  
Vol 26 (1) ◽  
pp. 49-67
Author(s):  
Noboru Endou
Keyword(s):  
Integrable Function ◽  
Functional Space ◽  
The Other ◽  
Integration Theory ◽  
Fubini’S Theorem ◽  
Positive Functions ◽  
System 1 ◽  
Article 5 ◽  
Fubini's Theorem ◽  
Lebesgue Integration

Summary The goal of this article is to show Fubini’s theorem for non-negative or non-positive measurable functions [10], [2], [3], using the Mizar system [1], [9]. We formalized Fubini’s theorem in our previous article [5], but in that case we showed the Fubini’s theorem for measurable sets and it was not enough as the integral does not appear explicitly. On the other hand, the theorems obtained in this paper are more general and it can be easily extended to a general integrable function. Furthermore, it also can be easy to extend to functional space Lp [12]. It should be mentioned also that Hölzl and Heller [11] have introduced the Lebesgue integration theory in Isabelle/HOL and have proved Fubini’s theorem there.

scholarly journals Nonstandard Dirichlet problems with competing (p,q)-Laplacian, convection, and convolution

2021 ◽  
Vol 66 (1) ◽  
pp. 95-103
Author(s):  
Dumitru Motreanu ◽  
Viorica Venera Motreanu
Keyword(s):  
Fixed Point ◽  
Weak Solution ◽  
Dirichlet Problem ◽  
Elliptic Problem ◽  
Integrable Function ◽  
Generalized Solution ◽  
Dirichlet Problems ◽  
Reaction Term ◽  
Fixed Point Approach

"The paper focuses on a nonstandard Dirichlet problem driven by the operator $-\Delta_p +\mu\Delta_q$, which is a competing $(p,q)$-Laplacian with lack of ellipticity if $\mu>0$, and exhibiting a reaction term in the form of a convection (i.e., it depends on the solution and its gradient) composed with the convolution of the solution with an integrable function. We prove the existence of a generalized solution through a combination of fixed-point approach and approximation. In the case $\mu\leq 0$, we obtain the existence of a weak solution to the respective elliptic problem."

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