On the sequence of even order moments of the function \ (exp (-t²) \) 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 (-t²) \) 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.