# Daubechies scaling function

In Mathematica in wavelet package with the following codes

Phi[x_, j_, k_] := 2^(j/2) WaveletPhi[DaubechiesWavelet[3], 2^(j) x - k]


Mathematica computes Phi for various values of $x,j,k$. How does Mathematica do this computation? My guess is cascade algorithm. If it is cascade algorithm, again how Mathematica does this computation because until now i have seen cascade algorithm just for $j=0$ and $k=0$. If Mathematica does not use cascade algorithm what is the algorithm?

• Trace[Phi[x, 3, 4]] might give some indication. Not that I really follow the details of what it shows, but it shows a lot. – Daniel Lichtblau Jun 11 '13 at 21:07

$\phi(x)=\sum_{k=0}^{N-1} c_k \phi(2 x - k)$,
which computes $\phi(x)$ approximately at dyadic points $x=k\times2^{-j}$, to whatever resolution you choose for the initial approximation. The definition
$\phi_{j,k}(x)=2^{j/2} \phi(2^{j} x - k)$
only requires the ability to compute $\phi(x)$ for at these dyadic points.
Indeed, you can recursively compute $\phi(x)$ for any rational $x=p/q$ exactly in terms of the filter coefficients $c_k$.