ON THE DENSITY ON SOME SPECIAL FUNCTIONS IN \(L^2\) SPACE AND IN COMPLEX REGION - Tomovski, Živorad (Matematichki Bilten, 1997)
In this paper we consider the system of functions \(x^n e^-x/2\), \(n=1,2,3,…\) for every \(x(0,∞)\). We shall prove that this system of functions is dense in \(L^2(0,∞)\). Then, let \(λ_1,λ_2,λ_3,…\) is a sequence of positive integer numbers, such that \(λ_n→∞\). On the other hand we shall prove that for the system of functions: \(z^m e^1/z\), \(m=0,1,2,3,…\) the sequence \(z^λ_1,z^λ_2,z^λ_3,…\) is dense, i.e. if for some analytic function \(f(z)\) about the point \(z=0\), \[ _C f(z)z^m+λ_pe^1/z dz=0, p=1,2,3,… \] where \(C\) is a closed curve enclosing \(z=0\) of the complex region \(Ω\) and \(fH(CIntC)\), then \(f(z)=0\) for every \(zΩ\).