Volume 6, Issue 3

We consider functions satisfying the subcritical Beurling's condition, viz., ∫ℝn∫ℝn|f(x)||f^(y)|ea|x·y|dxdy<∞$ \int _{\mathbb {R}^n}\int _{\mathbb {R}^n} \vert f(x)\vert \vert \widehat{f}(y)\vert e^{a \vert x \cdot y\vert } \, dx \, dy &lt; \infty $ for some 0<a<1.${ 0 &lt; a &lt; 1.}$ We show that such functions are entire vectors for the Schrödinger representations of the Heisenberg group. If an eigenfunction f of the Fourier transform satisfies the above condition, we show that the Hermite coefficients of f have certain exponential decay which depends on a .

We establish a necessary and sufficient condition on a continuous function on [-1,1] under which the family of functions on the unit sphere 𝕊 d -1 constructed in the described manner is fundamental in C (𝕊 d -1 ). In our construction of functions and proof of the result, we essentially use Dunkl harmonic analysis.

The purpose of the paper is to describe the boundary behavior of the Green function of the parabolic equation Δ u + V u + 2((∇ϕ)/ϕ)·∇ u - u t = 0 in D ×ℝ + , where D is a bounded C 1,1 domain in ℝ n ( n ≥ 3), ϕ is the positive eigenfunction corresponding to the first eigenvalue of -Δ, u 0 is the initial value such that ϕ u 0 ∈ L 2 ( D ) and V belongs to a class of time dependent potentials which can be written as a nonlinear combination of derivatives of a function.

The paper is devoted to investigation questions about constructing the explicit form of the Green's function of the Robin problem in the unit ball of ℝ 2 . In constructing this function we use the representation of the fundamental solution of the Laplace equation in the form of a series. An integral representation of the Green function is obtained and for some values of the parameters the Green function is given in terms of elementary functions.

In this paper, we define chain recurrence and study properties of chain recurrent sets in a nonautonomous discrete dynamical system induced by a sequence of homeomorphisms on a compact metric space. We also study chain recurrent sets in a nonautonomous discrete system having shadowing property.

Let G be a group, and let σ be an involutive automorphism on G . We determine the complex-valued solutions (F1,F2,f)$(F_{1},F_{2},f)$ of the functional equation F1(xy)+F2(σ(y)x)=f(x)f(y)$F_{1}(xy)+F_{2}(\sigma (y)x)=f(x)f(y)$ (x,y∈G$x,y\in G$) in terms of characters and additive functions. From this we find the complex-valued solutions ( f , g ) of the equation f(σ(y)xz0)+g(xyz0)=2f(x)f(y)$f(\sigma (y)xz_{0})+g(xyz_{0})=2f(x)f(y)$, where z 0 is a fixed element in G . This equation provides a joint generalization of many functional equations such as d'Alembert's, Kannappan's or Van Vleck's equations.