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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?

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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

Unless I'm missing something, your question does not make sense. The cascade algorithm is an iterative solution to

$\phi(x)=\sum_{k=0}^{N-1} c_k \phi(2 x - k)$,

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 $c_k$.

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