Lebesgue Integral Research Articles

Positive Solutions for Second‐Order Singular Semipositone Differential Equations Involving Stieltjes Integral Conditions

By means of the fixed point theory in cones, we investigate the existence of positive solutions for the following second‐order singular differential equations with a negatively perturbed term: −u′′(t) = λ[f(t, u(t)) − q(t)], 0 < t < 1, , where λ > 0 is a parameter; f : (0, 1) × (0, ∞) → [0, ∞) is continuous; f(t, x) may be singular at t = 0, t = 1, and x = 0, and the perturbed term q : (0, 1) → [0, +∞) is Lebesgue integrable and may have finitely many singularities in (0, 1), which implies that the nonlinear term may change sign.

The behavior at infinity of an integrable function

Given a density d defined on the Borel subsets of [0,∞), the limit at infinity in density of a function f:[0,∞)→R is zero if each of the sets {t:|f(t)|≥ε} has zero density whenever ε>0. It is proved that every Lebesgue integrable function f:[0,∞)→R verifies this type of behavior at infinity with respect to a scale of densities including the usual one, d(A)=limr→∞m(A∩[0,r))r.

Mean Value Integral Inequalities

Let \(F:[a,b]\longrightarrow \R\) have zero derivative in a dense subset of \([a,b]\). What else we need to conclude that \(F\) is constant in \([a,b]\)? We prove a result in this direction using some new Mean Value Theorems for integrals which are the real core of this paper. These Mean Value Theorems are proven easily and concisely using Lebesgue integration, but we also provide alternative and elementary proofs to some of them which keep inside the scope of the Riemann integral.

Second Order Periodic Boundary Value Problems Involving the Distributional Henstock-Kurzweil Integral

We apply the distributional derivative to study the existence of solutions of the second order periodic boundary value problems involving the distributional Henstock-Kurzweil integral. The distributional Henstock-Kurzweil integral is a general intergral, which contains the Lebesgue and Henstock-Kurzweil integrals. And the distributional derivative includes ordinary derivatives and approximate derivatives. By using the method of upper and lower solutions and a fixed point theorem, we achieve some results which are the generalizations of some previous results in the literatures.

On Generalized Stancu’s Polynomials

We have tested the convergence of the Generalized Stancu’s Polynomial R α n f, x and have also tested the degree of approximation of Lebesgue integrable functions by R α n f, x

Negligible variation and the change of variables theorem

We prove a necessary and sufficient condition for the change of variables formula for the HK integral, with implications for the change of variables formula for the Lebesgue integral. As a corollary, we obtain a necessary and sufficient condition for the Fundamental Theorem of Calculus to hold for the HK integral. The paper on which this talk was based will be published in a forthcoming issue of the Indiana University Mathematical Journal.

Further randomization of Riemann sums leading to the Lebesgue integral

In this paper we randomize in a particular way the sequence of partitions based on which the random Riemann sums are defined for a Lebesgue integrable function f on (0, 1). Convergence of such sums to the Lebesgue integral of f is investigated.

A note on the Kluvánek integral

ABSTRACT For real valued functions, Igor Kluv´anek has introduced an integral, called Archimedean, which is equivalent to the Lebesgue integral. It is based on the summation of infinite series and it avoids measure. Modifying the definition by Kluv´anek, we introduce an integral which has some good and some bad properties and we compare it with the Lebesgue integral.

Generality of Henstock-Kurzweil type integral on a compact zero-dimensional metric space

ABSTRACT A Henstock-Kurzweil type integral on a compact zero-dimensional metric space is investigated. It is compared with two Perron type integrals. It is also proved that it covers the Lebesgue integral.

The stochastic Fubini theorem revisited

In this paper, we present an elementary and self-contained proof of the stochastic Fubini theorem, which states that one can interchange a Lebesgue integral and a stochastic integral. The integrability conditions we use are weaker and more natural than the usual conditions in the literature. In particular, we do not need integrability in , and we use -integrability instead of -integrability in the additional parameter.

On the degree of continuity and smoothness of sine and cosine Fourier transforms of Lebesgue integrable functions

We consider complex-valued functions f∈L1(ℝ+), where ℝ+:=[0,∞), and prove sufficient conditions under which the sine Fourier transform \(\hat{f}_{s}\) and the cosine Fourier transform \(\hat{f}_{c}\) belong to one of the Lipschitz classes Lip (α) and lip (α) for some 0<α≦1, or to one of the Zygmund classes Zyg (α) and zyg (α) for some 0<α≦2. These sufficient conditions are best possible in the sense that they are also necessary if f(x)≧0 almost everywhere.

A Liapunov type inequality for Sugeno integrals

The classical Liapunov inequality shows an interesting upper bound for the Lebesgue integral of the product of two functions. This paper proposes a Liapunov type inequality for Sugeno integrals. That is, we show that H s , t , r ( ( S ) ∫ 0 1 f ( x ) s d μ ) r − t ≤ ( ( S ) ∫ 0 1 f ( x ) t d μ ) r − s ( ( S ) ∫ 0 1 f ( x ) r d μ ) s − t holds for some constant H s , t , r where 0 < t < s < r , f : [ 0 , 1 ] → [ 0 , ∞ ) is a non-increasing concave function, and μ is the Lebesgue measure on R . We also present two interesting classes of functions for which the classical Liapunov type inequality for Sugeno integrals with H s , t , r = 1 holds. Some examples are provided to illustrate the validity of the proposed inequality.

On some properties of double conjugate trigonometric Fourier series

A theorem of Ferenc Lukacs [4] states that the partial sums of conjugate Fourier series of periodic Lebesgue integrable functions f diverge at logarithmic rate at the points of discontinuity of first kind of f. F. Moricz [5] proved an analogous theorem for the rectangular partial sums of bivariate functions. The present paper proves analogues of Moricz’s theorem for generalized Cesaro means and for positive linear means.

Discrete phase retrieval in musical structures

This paper describes phase-retrieval approaches in music by focusing on the particular case of the cyclic groups (beltway problem). After presenting some old and new results on phase retrieval, we introduce the extended phase retrieval for a generalized musical Z-relation. This concept is accompanied by mathematical definitions and motivations from computer-aided composition. We assume from the reader basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Z-relation and homometry in musical distributions

This paper defines homometry in the rather general case of locally-compact topological groups, and proposes new cases of its musical use. For several decades, homometry has raised interest in computational musicology and especially set-theoretical methods, and in an independent way and with different vocabulary in crystallography and other scientific areas. The link between these two approaches was only made recently, suggesting new interesting musical applications and opening new theoretical problems. We present some old and new results on homometry, and give perspective on future research assisted by computational methods. We assume from the reader's basic knowledge of groups, topological groups, group algebras, group actions, Lebesgue integration, convolution products, and Fourier transform.

Kurzweil-Henstock Integration on Manifolds

In this paper, we give an alternative proof that the Kurzweil-Henstock integral using partition of unity is equivalent to the Lebesgue integral in the $n$-dimensional Euclidean space. We also define and discuss the Kurzweil-Henstock integral on manifolds.

Fusion integral

AbstractIn this paper we shall introduce an operator‐valued integral over a Hilbert space, called the fusion integral. We shall prove that it is additive (finitely or infinitely) but satisfies no additional condition. We shall show that it has the retrieval property and establish a useful connection between the fusion and the Lebesgue integrals. © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim © 2011 WILEY‐VCH Verlag GmbH &amp; Co. KGaA, Weinheim

Anatomy of the failure rate: A mathematical dissection

AbstractThe failure rate function is one of the most commonly used notion in reliability engineering and in survival analysis. The purpose of this expository paper is to point out some mathematical issues connected with this notion, a consequence of which is that under certain circumstances the famous exponentiation formula overestimates the probability of survival. To demonstrate this we point out a scenario that results in Volterra's product integrals, instead of the usual Lebesgue integral for which the exponentiation formula is exact. Copyright © 2011 John Wiley &amp; Sons, Ltd.

Stability under integration of sums of products of real globally subanalytic functions and their logarithms

We study Lebesgue integration of sums of products of globally subanalytic functions and their logarithms, called constructible functions. Our first theorem states that the class of constructible functions is stable under integration. The second theorem treats integrability conditions in Fubini-type settings, and the third result gives decay rates at infinity for constructible functions. Further, we give preparation results for constructible functions related to integrability conditions.

Fatou’s lemma for Sugeno integral

Fatou’s lemma plays an important role in classical probability and measure theory. Non-additive measure is a generalization of additive probability measure. Sugeno’s integral is a useful tool in several theoretical and applied statistics which have been built on non-additive measure. In this paper, a Fatou-type lemma for Sugeno integral is shown. The studied inequality is based on the classical Fatou lemma for Lebesgue integral. To illustrate the proposed inequalities some examples are given.