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.

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 LEBESGUE’S CRITERION FOR INTEGRABILITY

EDUPEDIA ◽  
2018 ◽  
Vol 2 (1) ◽  
pp. 12
Author(s):  
Bayu Riyadhus Saputro ◽  
Sumaji .
Keyword(s):  
Lebesgue Measure ◽  
Measurable Function ◽  
Integrable Function ◽  
Measure Zero ◽  
Bounded Function ◽  
Simple Function ◽  
Outer Measure ◽  
Lebesgue Integral ◽  
Lebesgue Integrable Function ◽  
Measurable Set

Lebesgue integral was defined constructively by using outer measure, measurable set, measurable function and Lebesgue measure concept. Then it was constructed by using simple function. The requirement of a Lebesgue integrable function to be Riemann integrable is bounded function and the set of discontinuities of that function has measure zero.

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