Let $ℬ=f:ℝ\to ℝ:fisbounded$, and $ℳ=f:ℝ\to ℝ:fisLebesguemeasurable$. We show that there is a linear operator $\Phi :ℬ\to ℳ$ such that Φ(f)=f a.e. for every $f\in ℬ\cap ℳ$, and Φ commutes with all translations. On the other hand, if $\Phi :ℬ\to ℳ$ is a linear operator such that Φ(f)=f for every $f\in ℬ\cap ℳ$, then the group $G_{\Phi }$ = a ∈ ℝ:Φ commutes with the translation by a is of measure zero and, assuming Martin’s axiom, is of cardinality less than continuum. Let Φ be a linear operator from ${ℂ}^{ℝ}$ into the space of complex-valued measurable functions. We show that if Φ(f) is non-zero for every $f\left(x\right)=e^{cx}$, then $G_{\Phi }$ must...

A perfect (exact) fractional observer of discrete-time singular linear control system of fractional order is studied. Conditions for its existence are given. The obtained results are applied to the detectability problem of the class of systems under consideration.

The aim of this paper is to give a necessary and sufficient condition for a set-valued function to be a polynomial s.v. function of order at most 2.

A method is given to find a recurrence relation for the coefficients of the series expansion of a function f with respect to classical orthogonal polynomials of a discrete variable, which follows from a linear difference equation satisfied by f.

On s’intéresse aux solutions méromorphes sur $ℂ$ d’un système de deux équations aux différences à coefficients constants et à deux pas récurrents. Lorsqu’on fait varier ce système, les solutions décrivent une certaine algèbre $𝒟[s,t]$ en rapport avec les fonctions elliptiques habituelles et celles de deuxième espèce de Hermite, ainsi que la fonction $Z$ de Jacobi. Pour un système donné, les solutions trouvées forment sur le corps des fonctions elliptiques un espace vectoriel de dimension finie, en rapport...

For the difference equation $\left(ϵ\right)x_{n+1}=A{x}_n+ϵ{\sum }_{k=-\infty }^n{R}_{n-k}{x}_k$,where $x_n\in Y,Y$  is a Banach space, $ϵ$ is a parameter and  $A$  is a linear, bounded operator. A sufficient condition for the existence of a unique special solution  $y={\left\{y_n\right\}}_{n=-\infty }^{\infty }$  passing through the point  $x_0\in Y$  is proved. This special solution converges to the solution of the equation (0) as  $ϵ\to 0$.

We describe the spectra of Jacobi operators J with some irregular entries. We divide ℝ into three “spectral regions” for J and using the subordinacy method and asymptotic methods based on some particular discrete versions of Levinson’s theorem we prove the absolute continuity in the first region and the pure pointness in the second. In the third region no information is given by the above methods, and we call it the “uncertainty region”. As an illustration, we introduce and analyse the OP family...

We discuss spectral and scattering theory of the discrete laplacian limited to a half-space. The interesting properties of such operators stem from the imposed boundary condition and are related to certain phenomena in surface physics.

In this work, we consider the singular Hahn difference equation of the Sturm-Liouville type. We prove the existence of the spectral function for this equation. We establish Parseval equality and an expansion formula for this equation on a semi-unbounded interval.

In this paper we use the fixed point method to prove asymptotic stability results of the zero solution of a generalized linear neutral difference equation with variable delays. An asymptotic stability theorem with a sufficient condition is proved, which improves and generalizes some results due to Y. N. Raffoul (2006), E. Yankson (2009), M. Islam and E. Yankson (2005).

Let X be an arbitrary Abelian group and E a Banach space. We consider the difference-operators ∆n defined by induction:(∆f)(x;y) = f(x+y) - f(x), (∆nf)(x;y1,...,yn) = (∆n-1(∆f)(.;y1)) (x;y2,...,yn)(n = 2,3,4,..., ∆1=∆, x,yi belonging to X, i = 1,2,...,n; f: X --> E).Considering the difference equation (∆nf)(x;y1,y2,...,yn) = d(x;y1,y2,...,yn) with independent variable increments, the most general solution is given explicitly if d: X x Xn --> E is a given bounded function. Also the...