According to Theorem 16 in this document, if $f \in C^{k + \lceil\frac{d+1}{2}\rceil}(\mathbb{T}^d, \mathbb{C})$ then $$ f(x) = \sum_{k \in \mathbb{Z}^d} \hat{f}(k)e^{i \langle k, x \rangle}, $$ in the $C^k(\mathbb{T}^d, \mathbb{C})$ norm.
Does anyone know of a more permanent reference for this fact? Specifically, something that isn't just hosted on the authors website, such as a book or paper.
I contacted the author of the lecture notes in my original post, Oliver Bräunling, and he rightly directed me to Proposition 5.6 and Theorem 5.7 in the book "Elliptic operators, topology and asymptotic methods" by John Roe. As Oliver pointed out, the result is a corollary of the proof of the Sobolev embedding theorem on the $d$-dimensional torus.