Department of Mathematics, Semnan University, Semnan, Iran
10.22091/maa.2026.15340.1053
Abstract
This paper completes the analytic foundation for the differential operator \(\ha{f}(t) = \sum_{n=0}^{\infty} \frac{(-1)^n f^{(n)}(t)}{n!} t^n\) introduced in earlier work. We establish a complete set of convergence tests for the series \(\ha{f}(t)\). These tests serve to determine the domain of convergence for specific functions and to rigorously analyze the exotic, nowhere-analytic examples for which convergence occurs only on thin sets. Our central result is a rigidity theorem: if \(\ha{f}(t)\) converges on an interval, then \(f\) must be analytic on a subinterval, and \(\ha{f}(t) \equiv f(0)\) there. This definitively answers two open questions from the precursor paper: (1) no nonanalytic function can yield a convergent \(\ha{f}\)-series on an interval, and (2) the point spectrum of the operator \(f \mapsto f(0) - \ha{f}\) on interval domains is trivial (eigenvalue zero only). The work thus settles the basic analytic properties of \(\ha{f}\) and provides the necessary tools for its further application.
Akrami, S. E. (2026). On the convergence of the operator \(\hat{f}(t)=\sum_{n=0}^\infty \frac{(-1)^n f^{(n)}(t)}{n!} t^n\) and its spectral properties. Measure Algebras and Applications, (), -. doi: 10.22091/maa.2026.15340.1053
MLA
Seyed Ebrahim Akrami. "On the convergence of the operator \(\hat{f}(t)=\sum_{n=0}^\infty \frac{(-1)^n f^{(n)}(t)}{n!} t^n\) and its spectral properties". Measure Algebras and Applications, , , 2026, -. doi: 10.22091/maa.2026.15340.1053
HARVARD
Akrami, S. E. (2026). 'On the convergence of the operator \(\hat{f}(t)=\sum_{n=0}^\infty \frac{(-1)^n f^{(n)}(t)}{n!} t^n\) and its spectral properties', Measure Algebras and Applications, (), pp. -. doi: 10.22091/maa.2026.15340.1053
VANCOUVER
Akrami, S. E. On the convergence of the operator \(\hat{f}(t)=\sum_{n=0}^\infty \frac{(-1)^n f^{(n)}(t)}{n!} t^n\) and its spectral properties. Measure Algebras and Applications, 2026; (): -. doi: 10.22091/maa.2026.15340.1053
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