Moments of Real, Respectively of Complex Valued Functions, with Applications

The first aim of this study is to point out new aspects of approximation theory applied to a few classes of holomorphic functions, via Vitali’s theorem. The approximation is made with the aid of the complex moments of the involved functions, that are defined similarly to the moments of a real valued continuous function. Applying uniform approximation of continuous functions on compact intervals via Korovkin’s theorem, the hard part concerning uniform approximation on compact subsets of the complex plane follows according to Vitali’s theorem. The theorem on the set of zeros of a holomorphic function is also applied. In the end, existence and uniqueness of solution for a mul-tidimensional moment problem is characterized in terms of limits of sums of quadratic expressions. This is an application appearing at the end of the title. Consequences resulting from the first part of the paper are pointed out with the aid of functional calculus for self-adjoint operators.