In their great book "Wavelet methods for time series analysis" (2006), Percival & Walden state on p. 83 that the first-round pyramid algorithm scaling filter coefficients
$\tilde{V}_{i,t}$
can be approximated by
$\frac{1}{N} \sum_{k=-\frac{N}{4}+1}^{\frac{N}{4}}{\chi_{k}e^{i 2 \pi t k/N}}$
arguing that $2^{1/2} \tilde{V}_{i,t}$ "is formed by filtering [the time series] $\{X_{t}\}$ with the low-pass filter $\{g_l\}$ with nominal pass-band [-1/4, 1/4] ..."
Why is it $\tilde{V}_{i,t}$ and not $2^{1/2} \tilde{V}_{i,t}$ which is approximated by the above expression?
My understanding is that the approximation is due to limiting the frequency domain of the inverse DFT according to the pass-band of the filter?
