δ-Universal Taylor Series (Analysis in Theory and Applications, 2026)
We present a new type of universal series, termed $δ$-universal series, for which the sum of squared coefficients satisfies $$\sum\limits_{k=0}^{∞} |a_k|^2 < δ$$ for an arbitrarily small $δ > 0$. We establish a version of Seleznev's theorem within this framework. To construct such $δ$-universal series, we develop a variation of Mergelyan's theorem.
Keywords: Universal series; Seleznev’s Theorem; Mergelyan’s theorem; hardy space