The second approximation - cutting off in time/spatial domain

If $\varphi$ is well localised in time/space domain $\mathbb{R}$ it can be approximated by a function

$$\psi\left(x\right)=\varphi\left(x\right) \chi_{[-{m \over n},{m \over n}]}\left(x\right)$$

with ${\rm supp} \, \psi \, \left[-{m \over n},{m \over n}\right], \;m \ll n,\; m\in \mathbb{N}$. Again, one defines its one periodic version $\tilde \psi$ with compact support in $\mathbb{T}$ as

$$\tilde \psi\left(x\right)=\sum_{r \in \mathbb{Z}} \psi\left(x+r\right).$$

With the help of the index set

$$I_{n,m} \left(x_j\right) := \left\{ l \in I_n : n x_j - m \le l \le n x_j + m \right\}$$

an approximation to $s_1$ is defined by

$$s\left(x_j\right) := \sum_{l \in I_{n,m}\left(x_j\right) } g_l \, \tilde\psi\left(x_j - {l \over n}\right) \, .$$ (2.11)

Note, that for fixed $x_j\in \mathbb{T}$, the above sum contains at most $(2m+1)$ nonzero summands.

This approximation causes a truncation error.