The uniqueness is Exponential space.
If: \(f(s) = O(e^{\rho |\Re(s)| +\tau|\Im(s)|})\) for \(0 < \tau < \pi/2\) then there is a unique continuum sum \(F(s)\) that also belongs to this space. That's the central thesis of my paper. No 1-periodic function is in this space. No function which satisfies \(g(s+1) - g(s) = 0\) in this space other than a constant... And yes, this is Tommy's continuum sum, sending polynomials to polynomials.
If: \(f(s) = O(e^{\rho |\Re(s)| +\tau|\Im(s)|})\) for \(0 < \tau < \pi/2\) then there is a unique continuum sum \(F(s)\) that also belongs to this space. That's the central thesis of my paper. No 1-periodic function is in this space. No function which satisfies \(g(s+1) - g(s) = 0\) in this space other than a constant... And yes, this is Tommy's continuum sum, sending polynomials to polynomials.