On the divergence of Fourier series in the general Haar system

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 $[0,1]\backslash D$.

On Properties of Pseudointegrals Based on Pseudoaddition Decomposable Measures

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

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.

Powers of sequences and convergence of ergodic averages

A 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.

On theA-Laplacian

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

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.

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

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.