Lipschitz with constant greater than $\frac{1}{2}$ implies $f(t) = \sum_{n \in Z} \hat{f}(n)e^{int} $

Question

While proving Bernstein theorem, our lecteur trivially used the following fact:
$$f\in Lip_{\alpha}(\mathbb{R}),\;\; \alpha>1/2 \;\; \implies f(t) = \sum_{n \in Z} \hat{f}(n)e^{int} $$ and the series converges at $L^{2}$ and at each point.

Is this true at all? If so, should this be immediate or it is a well-known result? or it is a well-known result?