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

Document Type : Original Article

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Department of Mathematics, Semnan University, Semnan, Iran

Abstract

This paper completes the analytic foundation for the infinite-order differential operator \(\hat{f}(t) = \sum_{n=0}^{\infty} \frac{(-1)^n f^{(n)}(t)}{n!} t^n\) introduced in the earlier work‎. ‎We establish a complete set of convergence tests (ratio‎, ‎growth‎, ‎bounded variation‎, ‎root) for the series \(\hat{f}(t)\)‎. ‎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 \(\hat{f}(t)\) converges on an interval‎, ‎then \(f\) must be analytic on a subinterval‎, ‎and \(\hat{f}(t) \equiv f(0)\) there‎. ‎This definitively answers two open questions from the precursor paper‎: ‎(1) no non-analytic function can yield a convergent \(\hat{f}\)-series on an interval‎, ‎and (2) the point spectrum of the operator \(f \mapsto f(0)‎ - ‎\hat{f}\) on interval domains is trivial (eigenvalue zero only)‎. ‎The work thus settles the basic analytic properties of \(\hat{f}\) and provides the necessary tools for its further application‎.

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References

[1] Akrami, S.E. (2019). On the spectrum of an infinite-order differential operator and its relation to Hamiltonian mechanics. J. Appl. Anal., 25, 131–139. DOI: https://doi.org/10.1515/jaa-2019-0014.
[2] Apostol, T.M. (1975). Mathematical Analysis, 2nd ed. Addison-Wesley.
[3] Balser, W. (2000). Formal Power Series and Linear Systems of Meromorphic Ordinary Differential Equations. Springer, New York. DOI: https://doi.org/10.1007/b97608.
[4] Boas, R.P. (1996). A Primer of Real Functions, 4th ed., Carus Math. Monogr. No. 13. Mathematical Association of America.
[5] Loday-Richaud, M. (2016). Divergent Series, Summability and Resurgence II, Lecture Notes in Math, 2154. Springer Cham. DOI: https://doi.org/10.1007/978-3-319-29075-1.
Measure Algebras and Applications
Volume 3, Issue 2 - Serial Number 5
December 2025
Pages 46-51

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