 Solution

Use the printed unnormalized inverse discrete Fourier transform, with positive exponent: $x_{\ell }={\sum }_j{\omega }_n^{j\ell }{y}_j$. Its natural domain and codomain are ${\mathbb{C}}^n$; the printed real-to-real designation is not literal, as the first example below has complex output. There is no factor $1/n$ in the convention of this question.