We derive several new properties concerning both universal Taylor series and Fekete universal series from classical polynomial inequalities. In particular, we study some density properties of their approximating subsequences. Moreover we exhibit summability methods which preserve or imply the universality of Taylor series in the complex plane. Likewise we show that the partial sums of the Taylor expansion around zero of a C ∞ function is universal if and only if the sequence of its Cesàro means satisfies the same universal approximation property.
We prove that the classical universal Taylor series in the complex plane are never frequently universal. On the other hand, we prove the 1-upper frequent universality of all these universal Taylor series.
We unify the recently developed abstract theories of universal series and extended universal series to include sums of the form \sum_k ak x_n,k for given sequences of vectors (x_n,k)n≥k≥0 in a topological vector space X. The algebraic and topological genericity as well as the spaceability are discussed. Then we provide various examples of such generalized universal series which do not proceed from the classical theory. In particular, we build universal series involving Bernstein's polynomials, we obtain a universal series version of MacLane's Theorem, and we extend a result of Tsirivas concerning universal Taylor series on simply connected domains, exploiting Bernstein- Walsh quantitative approximation theorem
For any
$\alpha \in \mathbb {R},$
we consider the weighted Taylor shift operators
$T_{\alpha }$
acting on the space of analytic functions in the unit disc given by
$T_{\alpha }:H(\mathbb {D})\rightarrow H(\mathbb {D}),$
$ \begin{align*}f(z)=\sum_{k\geq 0}a_{k}z^{k}\mapsto T_{\alpha}(f)(z)=a_1+\sum_{k\geq 1}\Big(1+\frac{1}{k}\Big)^{\alpha}a_{k+1}z^{k}.\end{align*}$
We establish the optimal growth of frequently hypercyclic functions for
$T_\alpha $
in terms of
$L^p$
averages,
$1\leq p\leq +\infty $
. This allows us to highlight a critical exponent.
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