The ansatz

One wants to approximate the trigonometric polynomial $f$ in (2.6) by a linear combination of shifted 1-periodic window functions $\tilde \varphi$ as

$$s_1\left(x\right) := \sum_{l \in I_n} g_{l} \; \tilde \varphi\left(x - \frac{l}{n}\right) \, .$$ (2.7)

With the help of an oversampling factor $\sigma >1$ the fft-length is given by $n:=\sigma N$.