ON THE DENSITY ON SOME SPECIAL FUNCTIONS IN \(L^2\) SPACE AND IN COMPLEX REGION

Abstract

In this paper we consider the system of functions \(x^n e^{-x/2}\), \(n=1,2,3,\ldots\) for every \(x\in(0,\infty)\). We shall prove that this system of functions is dense in \(L^2(0,\infty)\).

Then, let \(\lambda_1,\lambda_2,\lambda_3,\ldots\) is a sequence of positive integer numbers, such that \(\lambda_n\to\infty\). On the other hand we shall prove that for the system of functions: \(z^m e^{1/z}\), \(m=0,1,2,3,\ldots\) the sequence \(z^{\lambda_1},z^{\lambda_2},z^{\lambda_3},\ldots\) is dense, i.e. if for some analytic function \(f(z)\) about the point \(z=0\),

\[
\int_C f(z)z^{m+\lambda_p}e^{1/z}\,dz=0,
\qquad p=1,2,3,\ldots
\]

where \(C\) is a closed curve enclosing \(z=0\) of the complex region \(\Omega\) and \(f\in H(C\cup\operatorname{Int}C)\), then \(f(z)=0\) for every \(z\in\Omega\).