On universal formal power series

The point source of this work is Seleznev's theorem which asserts the existence of a power series which satisfies universal approximation properties in C^∗. The paper deals with a strengthened version of this result. We establish a double approximation theorem on formal power series using a weighted backward shift operator. Moreover we give strong conditions that guarantee the existence of common universal series of an uncountable family of weighted backward shift with respect to the simultaneous approximation. Finally we obtain results on admissible growth of universal formal power series. We especially prove that you cannot control the defect of analyticity of such a series even if there exist universal series in the well-known intersection of formal Gevrey classes.     

Polynomial Representations via Spectral Decompositions

We show using functional analysis methods, that under some conditions, such as the positivity of a homogeneous polynomial, one can give a representation of polynomials which are positive on semi-algebraic sets, allowing irrational fractions with denominators of the form (1+t² ₁) _1/2 ...(1+t ² _n) _n/2, _iZ₊, i = 1, ..., n