Vtyjkyuk

I am having problems either finding such a sequence or showing that one does not exist. Here is the situation.

Take $f$ to be some positive real number which is not an integer.

Recall that $operatornamesinc(x)$ is $sin(x) / x$ if $x neq 0$, and $operatornamesinc(x) := 1$.

I would like to show that no non-zero series $a_f(k)_k=1^infty subset mathbbR$ exists which satisfies
$$sum_k=1^infty a_f(k) operatornamesinc (pi k f) = 0.$$
(I use non-zero to simply mean that there is at least one non-zero element of the sequence.)

If it is possible to do this, then I would perhaps like to say that for $N$ a large positive integer, there is no sequence of numbers $a_f(k)_k=1^N subset mathbbR$ which satisfies
$$sum_k=1^N a_f(k) operatornamesinc (pi k f) = 0.$$
If this is also possible, then I would maybe add some conditions like $f$ being irrational so that no such sequence would exist.

As a consequence of the Poisson summation formula, for any $alphainmathbbRsetminusmathbbZ$ we have
$$ sum_kgeq 1textsinc(pi k alpha)=frac12left[-1+sum_kinmathbbZtextsinc(pi k alpha)right]=frac12left[-1+sum_sinmathbbZfractextsign(alpha-2s)+textsign(alpha+2s)2alpharight] $$
such that for any $alphainmathbbR^+setminusmathbbZ$ we get
$$ sum_kgeq 1textsinc(pi kalpha) = frac12left[-1+frac2lfloor alpha/2rfloor +1alpharight]neq 0. $$
Similarly you may compute in explicit terms $sum_kgeq 1textsinc^2(pi k alpha)$. This disproves your claim for
$$ a_f(k) = 1+ C_f,textsinc(pi k f).$$