Suppose that we know that signal $f(t)$ is $T_1$-periodic. Let $f_1 = 1/T_1$. But we want to know whether signal is $T_2$-periodic also. Let $f_2 = 1/T_2$, and $f_2$ is positive integer multiples of $f_1$. We know that there is no frequency-domain signals between $f_1$ and $f_2$ in Fourier transform. But we want to check periodicity without using fourier transform and just using samples, without having to sample at full Nyquist rate.
Is there any way to do this using samples? $f(t)$ is assumed to be from $\mathbb{R} \to \mathbb{C}$.