Continuum sums -- a big problem and some interesting observations

[mike3]

I believe in such cases the periodic component can be easily excluded (i.e. by requireing monotonous derivatives of higher orders). This is related to positive real axis. So to obtain the "natural solution" to the continous sum, just purify it of the periodic component on the positive real axis. Of course in any case the periodic component on the complex plane still will remain.

You can also use checks by other methods such as Mueller's formula (this formula is well-applicable to the function you presented as an example).

If
\( \lim_{x\to{+\infty}}f(x)=0 \)

then

\( \sum _x f(x)=\sum_{n=0}^\infty\left(f(n)-f(n+x)\right)+ C \)

The problem with the Mueller formula, and part of the reason for the introduction of the Fourier continuum sum, is that it does not preserve analyticity.

Consider \( f(x) = \frac{\sin\left(x^2\right)}{x^2} \). If we apply Mueller's formula to this at the real axis we get a reals-only continuum sum that is not analytic anywhere, heck, it may not even be real-differentiable anywhere(!).

The Fourier method also preserves the result from Faulhaber's formula. Canceling the wobble would mean that we no longer have agreement with Faulhaber's formula.