Let $x_{0},\dots,x_{n-1}$ be some complex numbers.
Let $\hat{x}_0,\dots,\hat{x}_{n-1}$ denote the discrete Fourier coefficients, so for $0 \leq k \leq n-1$ $$\hat{x}_k = \sum_{j=0}^{n-1} x_j \exp - \dfrac{2\pi \mathbb{i} j k}{n}$$ where $\mathbb{i}$ is the imaginary unit.
Now for any real $t$, define $$ u(t) = \dfrac{1}{n} \sum_{j=0}^{n-1} \hat{x}_j\exp\dfrac{2\pi\mathbb{i}jt}{n}. $$
$u(t)$ is function with $n$ as a period such that $u(k) = x_k$ for $0 \leq k \leq n-1$.
Does this kind of interpolation have a name? Can anyone point me to references about this?
