For an example these in these equations for continuous time fourier series:
Does it denote that the function is periodic?
If it denotes that the function is periodic then shouldn't there be a tilde above the function in the frequency domain of this next two equations of discrete time fourier transform since its periodic in f domain?
Sometimes it is also denoted by $\hat{x}$, but that really doesn't matter. You can choose any thing you want... The definition will always remain the same. Obviously, if you denote $\tilde{x}$ by $\mathcal{E}$, that won't say that $\tilde{x}$ is no more periodic. But sometimes it is convenient to choose this notation to show that it is periodic because we can write we write $$\tilde{\tilde{\tilde{\tilde{f}}}}=f$$