Questions on Daubechies wavelets

Question

Is the refinement equation for the orthonormal Daubechies scaling function $$\phi(x) = \sqrt{2} \sum_n h_n \phi(2x-n) \;?$$ The filter coefficients for Daubechies wavelets have been given e.g. in this page. Consider case D20. Do the indices $n$ for $h_n$ run from 0 to 19? Or do we set $h_{-n} = h_n$?

Suppose we have an arbitrary function $f \in L^2(\mathbb{R})$. Can we approximate $f$ with $$f(x) \approx 2^{j/2} \sum_k c_k \phi(2^j x - k)$$ where $$c_k = 2^{j/2} \int f(x) \phi(2^j x - k) dx$$ when $j$ is sufficiently large? Can we approximate $$f\left(\frac{k}{2^j}\right) \approx c_k \;?$$

Answer 1

The Daubechies wavelets are not symmetric, in fact, there are no continuous, orthogonal and symmetric 2-channel wavelets (but 3-channel and up works). So it is the first way.

Yes, that would be the correct way of sampling a signal. As that is unpractical, some discretized approximation is used. The single-point approximation you mentioned is the worst of these. One fast way is point-sampling with a higher density and then applying the low-pass filter of the wavelet.