scholarly journals On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Dong Qiu ◽  
Chongxia Lu
Keyword(s):  
Measurable Function ◽  
Bounded Measurable Function ◽  
Definition Of ◽  
Decomposable Measure

We mainly discussed pseudointegrals based on a pseudoaddition decomposable measure. Particularly, we give the definition of the pseudointegral for a measurable function based on a strict pseudoaddition decomposable measure by generalizing the definition of the pseudointegral of a bounded measurable function. Furthermore, we got several important properties of the pseudointegral of a measurable function based on a strict pseudoaddition decomposable measure.

On the divergence of Fourier series in the general Haar system

2021 ◽  
pp. 1-10
Author(s):  
Martin Grigoryan ◽  
Artavazd Maranjyan
Keyword(s):  
Fourier Series ◽  
Measurable Function ◽  
Haar System ◽  
Bounded Measurable Function

For any countable set $D \subset [0,1]$, we construct a bounded measurable function $f$ such that the Fourier series of $f$ with respect to the regular general Haar system is divergent on $D$ and convergent on $[0,1]\backslash D$.

scholarly journals On theA-Laplacian

2003 ◽  
Vol 2003 (13) ◽  
pp. 743-755
Author(s):  
Noureddine Aïssaoui
Keyword(s):  
Large Class ◽  
Measurable Function ◽  
Compact Support ◽  
Orlicz Spaces ◽  
Lebesgue Spaces ◽  
Laplacian Equation ◽  
Bounded Measurable Function

We prove, for Orlicz spacesLA(ℝN)such thatAsatisfies theΔ2condition, the nonresolvability of theA-Laplacian equationΔAu+h=0onℝN, where∫h≠0, ifℝNisA-parabolic. For a large class of Orlicz spaces including Lebesgue spacesLp(p>1), we also prove that the same equation, with any bounded measurable functionhwith compact support, has a solution with gradient inLA(ℝN)ifℝNisA-hyperbolic.

The control of a finite dam with penalty cost function: Markov input rate

1987 ◽  
Vol 24 (2) ◽  
pp. 457-465 ◽  
Author(s):  
F. A. Attia
Keyword(s):  
Markov Chain ◽  
Cost Function ◽  
Measurable Function ◽  
Average Cost ◽  
Input Rate ◽  
Input Process ◽  
Penalty Cost ◽  
Bounded Measurable Function ◽  
Long Run ◽  
Finite Dam

The long-run average cost per unit time of operating a finite dam controlled by a policy (Lam Yeh (1985)) is determined when the cumulative input process is the integral of a Markov chain. A penalty cost which accrues continuously at a rate g(X(t)), where g is a bounded measurable function of the content, is also introduced. An example where the input rate is a two-state Markov chain is considered in detail to illustrate the computations.

scholarly journals Nonlinear Equations of Infinite Order Defined by an Elliptic Symbol

2014 ◽  
Vol 2014 ◽  
pp. 1-7
Author(s):  
Mauricio Bravo Vera
Keyword(s):  
Hilbert Space ◽  
Laplace Operator ◽  
Nonlinear Equations ◽  
Measurable Function ◽  
Infinite Order ◽  
Formal Definition ◽  
Regularity Properties ◽  
Definition Of ◽  
The Laplace Operator ◽  
Class Of Functions

The aim of this work is to show existence and regularity properties of equations of the formf(Δ)u=U(x,u(x))onℝn, in whichfis a measurable function that satisfies some conditions of ellipticity andΔstands for the Laplace operator onℝn. Here, we define the class of functions to whichfbelongs and the Hilbert space in which we will find the solution to this equation. We also give the formal definition off(Δ)explaining how to understand this operator.

On Divergence of Fourier Series with Respect to Multiplicative Systems on the Sets of Measure Zero

2009 ◽  
Vol 16 (3) ◽  
pp. 435-448
Author(s):  
Kakha Bitsadze
Keyword(s):  
Fourier Series ◽  
Measurable Function ◽  
Measure Zero ◽  
Multiplicative System ◽  
Bounded Measurable Function ◽  
Multiplicative Systems

Abstract For any multiplicative system of bounded type and any set of measure zero there exists a bounded measurable function whose Fourier series with respect to this system diverges on this set.

scholarly journals Powers of sequences and convergence of ergodic averages

2009 ◽  
Vol 30 (5) ◽  
pp. 1431-1456 ◽  
Author(s):  
N. FRANTZIKINAKIS ◽  
M. JOHNSON ◽  
E. LESIGNE ◽  
M. WIERDL
Keyword(s):  
Ergodic Theorem ◽  
Measurable Function ◽  
Pointwise Convergence ◽  
Ergodic Averages ◽  
Bounded Measurable Function ◽  
Mean Ergodic Theorem ◽  
The Mean ◽  
Good For

AbstractA sequence (sn) of integers is good for the mean ergodic theorem if for each invertible measure-preserving system (X,ℬ,μ,T) and any bounded measurable function f, the averages (1/N)∑ Nn=1f(Tsnx) converge in the L2(μ) norm. We construct a sequence (sn) which is good for the mean ergodic theorem but such that the sequence (s2n) is not. Furthermore, we show that for any set of bad exponents B, there is a sequence (sn) where (skn) is good for the mean ergodic theorem exactly when k is not in B. We then extend this result to multiple ergodic averages of the form (1/N)∑ Nn=1f1(Tsnx)f2(T2snx)⋯fℓ(Tℓsnx). We also prove a similar result for pointwise convergence of single ergodic averages.

scholarly journals Upper integral and its geometric meaning

2007 ◽  
Vol 57 (6) ◽  
Author(s):  
Josef Bukac
Keyword(s):  
Linear Space ◽  
Metric Space ◽  
Measurable Function ◽  
Complete Metric Space ◽  
Product Measure ◽  
Outer Measure ◽  
Geometric Meaning ◽  
Outer Product ◽  
The Difference ◽  
Definition Of

AbstractThe Hahn definition of the integral is recalled, the requirement of measurability of the integrand omitted. Both the upper and lower integrals comply with this definition and so does any measurable function between them.The outer product measure of the hypograph of a nonnegative bounded nonmeasurable function is equal to the upper integral which is equal to one of the Fan integrals. The outer measure of the graph of a bounded nonmeasurable function is equal to the difference between the upper and lower integrals.A norm for not necessarily measurable functions is defined with the upper integral. The linear space with this norm is complete. The convergence in this space implies the convergence in outer measure. The distance as an outer measure of the symmetric difference of two sets gives us a complete metric space of classes of subsets.

The control of a finite dam with penalty cost function: Markov input rate

1987 ◽  
Vol 24 (02) ◽  
pp. 457-465 ◽  
Author(s):  
F. A. Attia
Keyword(s):  
Markov Chain ◽  
Cost Function ◽  
Measurable Function ◽  
Average Cost ◽  
Input Rate ◽  
Input Process ◽  
Penalty Cost ◽  
Bounded Measurable Function ◽  
Long Run ◽  
Finite Dam

The long-run average cost per unit time of operating a finite dam controlled by a policy (Lam Yeh (1985)) is determined when the cumulative input process is the integral of a Markov chain. A penalty cost which accrues continuously at a rate g(X(t)), where g is a bounded measurable function of the content, is also introduced. An example where the input rate is a two-state Markov chain is considered in detail to illustrate the computations.

scholarly journals Extension on the fuzzy integral based on⊕-decomposable measure

2001 ◽  
Vol 28 (8) ◽  
pp. 437-445
Author(s):  
Mamadou Traore
Keyword(s):  
Metric Space ◽  
Measurable Function ◽  
Fuzzy Integral ◽  
Pettis Integral ◽  
Decomposable Measure

We extend the concept of fuzzy integral based on⊕-decomposable measure from nonnegative fuzzy measurable function to extended real-valued fuzzy measurable function. Further investigations of fuzzy integrals based on pseudo-additive decomposable measure are carried out. Meanwhile, the space(S(μ),σ(⋅,⋅))of all fuzzy measurable function will be proved to be a pseudo-metric space. Finally, as an application of this extension the Pettis integral will be obtained for that kind of fuzzy integral.

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