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

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.