Bounded Power Series on the Real Line

• DOI: 10.48550/arxiv.2312.09267
• Publication Date: 2024-10-01
• Authors: Davide Sclosa
• Keywords: Entire; Turán; Mathematics - Classical Analysis and ODEs; 30H05 (Primary), 30B10, 13J05 (Secondary); Classical Analysis and ODEs (math.CA); FOS: Mathematics; Power series; Shift; Convex; Bounded

this version: added conjectures 2.6 and 2.7; improved introduction and abstract; minor reorganization of the content<br/>We investigate power series that converge to a bounded function on the real line. First, we establish relations between coefficients of a power series and boundedness of the resulting function; in particular, we show that boundedness can be prevented by certain Turán inequalities and, in the case of real coefficients, by certain sign patterns. Second, we show that the set of bounded power series naturally supports three topologies and that these topologies are inequivalent and incomplete. In each case, we determine the topological completion. Third, we study the algebra of bounded power series, revealing the key role of the backward shift operator.<br/>