Slow growth for universal harmonic functions (Q607947)

The authors show that, for any continuous increasing function \(\phi:[0,\infty) \to [0,\infty)\) satisfying that \[ \lim\limits_{t\to \infty}\frac{\log(\phi(t))}{\log t}=+\infty, \] there exists a Banach space \(X\) of harmonic functions in \(\mathbb R^n\) (containing the harmonic polynomials) such that the translation operator \(T_a\) is bounded and mixing for any \(a\neq 0\), and the elements in the unit ball of \(X\) satisfy that \(|H(x)|\leq \phi(\|x\|)\). In particular there are universal harmonic functions of arbitrary slow transcendental growth. The result solves a problem posed by \textit{D. H. Armitage} [J. Approximation Theory 136, No. 2, 230--243 (2005; Zbl 1077.31003)], and its proof combines ideas from Dynamical systems and Operator theory.