In this paper, we study norm almost periodic measures on locally compact Abelian groups. First, show that the periodicity of $$\mu $$ is equivalent to equi-Bohr *g$$ for all g in a fixed family functions. Then, that, absolutely continuous measures, Stepanov Radon–Nikodym density.

The points of E are now enclosed in the open intervals ĉ y, so that each point is inside of an infinite number of intervals, and (x) is defined to be the sum of all the a-intervals or parts thereof which lie to the left of x. Thus cj>(x) is monotone and can easily be shown to be absolutely continuous as follows: If i + j = N is chosen sufficiently large, the N(x) formed for this finite se...

We develop a version of Herbrand’s theorem for continuous logic and use it to prove that definable functions in infinite-dimensional Hilbert spaces are piecewise approximable by affine functions. We obtain similar results for definable functions in Hilbert spaces expanded by a group of generic unitary operators and Hilbert spaces expanded by a generic subspace. We also show how Herbrand’s theor...

This paper investigates the absolute exponential stability of a general class of delayed neural networks, which require the activation functions to be partially Lipschitz continuous and monotone nondecreasing only, but not necessarily differentiable or bounded. Three new sufficient conditions are derived to ascertain whether or not the equilibrium points of the delayed neural networks with addi...

We investigate the spectral properties of Schrödinger operators in `2(Z) with limit-periodic potentials. The perspective we take was recently proposed by Avila and is based on regarding such potentials as generated by continuous sampling along the orbits of a minimal translation of a Cantor group. This point of view allows one to separate the base dynamics and the sampling function. We show tha...

It is shown that there are no consistent decision rules for the hypothesis testing problem of distinguishing between absolutely continuous and purely singular probability distributions on the real line. In fact, there are no consistent decision rules for distinguishing between absolutely continuous distributions and distributions supported by Borel sets of Hausdorff dimension 0. It follows that...

I explore some consequences of a groundbreaking result of Breimesser and Pearson on the absolutely continuous spectrum of one-dimensional Schrödinger operators. These include an Oracle Theorem that predicts the potential and rather general results on the approach to certain limit potentials. In particular, we prove a Denisov-Rakhmanov type theorem for the general finite gap case. The main theme...

In this paper we study some absolutely continuous representations of function algebras, which are weak -spectral in the sense of [5] and [6], for a scalar > 0. Precisely we investigate certain conditions for the existence of a spectral -dilation of such representation. Among others we obtain di¤erent results which generalize the corresponding theorems of D. Gaşpar [3]. Full text

We characterize the subexponential densities on $(0,\infty)$ for compound Poisson distributions $[0,\infty)$ with absolutely continuous L\'evy measures. As a corollary, we show that class of all probability density functions $\mathbb R_+$ is closed under generalized convolution roots sums. Moreover, give an application to distribution supremum random walk.