On the sequence of even order moments of the function \({exp}(-t^2)\) on compact intervals

First, this paper provides some approximation and estimation type results for some moments of the Gauss function, motivated by the fact that the moments of even orders \(n=2l,\ l∈ N={\{}0,1, \}\) of the function \(exp(-t^2)\) on bounded intervals . Second, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \([0,b]⊆ [0,{1}/{√2}]\) is studied and solved. Here the point is using Jensen inequality. Third, the problem of asymptotic behavior of the sequence of all orders for the same function on any interval \([0,b]⊂ [0,+∞ )\) is deduced, via elements of complex analysis (Vitali’s theorem). The convergence holds uniformly on compact subsets of the complex plane. Fourth, the asymptotic behavior of the sequence of all moments on \([0,1],\ \)as \(n→ ∞ ,\) for an arbitrary function \(f∈ C([0,1])\) is determined precisely, by means of Korovkin’s approximation theorem. Consequently, a similar result for complex analytic functions is deduced, using Vitali’s theorem. This is the fifth aim of the paper.