A 2-dimensional real Banach space with constant of analyticity less than one

• Published in: Journal of mathematical analysis and applications Vol. 541; no. 1; p. 128718
• Main Author: Rodríguez, Jorge Tomás
• Format: Journal Article
• Language: English
• Published: Elsevier Inc 01-01-2025
• Subjects: Constant of analyticity; Radius of analyticity; Radius of convergence; Real analytic function; Real analytic function; Radius of analyticity; Radius of convergence; Constant of analyticity
• ISSN: 0022-247X
• DOI: 10.1016/j.jmaa.2024.128718

We show that on the real 2-dimensional Banach space ℓ12 there is an analytic function f:Bℓ12→R such that its power series at origin has radius of uniform convergence one, but for some a∈Bℓ12 the power series centred at that point has radius of uniform convergence strictly less than 1−‖a‖. This result highlights a fundamental distinction in real analytic functions (compared to complex analytic functions), where the radius of analyticity can differ from the radius of uniform convergence. Moreover, this example provides the first non-trivial upper bound for the constant of analyticity.