The first approximation - cutting off in frequency domain

Switching from the definition (2.7) to the frequency domain, one obtains

$$s_1\left(x\right) =\sum_{k \in I_n} \hat g_k \, c_k\left(\tilde... ...(\tilde \varphi\right) \, {\rm e}^{-2\pi{\mbox{\scriptsize {i}}} (k + n r)x }$$ (2.8)

with the discrete Fourier coefficients

$$\hat g_k := \sum_{l \in I_n} g_l \, {\rm e}^{ 2\pi{\mbox{\scriptsize {i}}} \frac{k l}{n}}.$$ (2.9)

Comparing (2.6) to (2.7) and assuming $c_k\left(\tilde\varphi\right)$ small for $\vert k\vert\ge n-\frac{N}{2}$ suggests to set

$$\hat g_k := \left\{ \begin{array}{ll} \frac{\hat f_k}{c_k \left... ...\\ [1ex] 0 & \text{for } k \in I_n \backslash I_N . \\ \end{array} \right.$$ (2.10)

Then the values $g_l$ can be obtained from (2.9) by

$$g_l = \frac{1}{n}\sum\limits_{k \in I_N} \hat g_k \, {\rm e}^{-2\pi{\mbox{\scriptsize {i}}} \frac{k l}{n}} \qquad (l \in I_n),$$

a FFT of size $n$.

This approximation causes an aliasing error.