δ-Universal Taylor Series

Abstract

We present a new type of universal series, termed $\delta$-universal series, for which the sum of squared coefficients satisfies $$\sum\limits_{k=0}^{\infty} |a_k|^2 < \delta$$ for an arbitrarily small $\delta > 0$. We establish a version of Seleznev's theorem within this framework. To construct such $\delta$-universal series, we develop a variation of Mergelyan's theorem.