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I have two question about Daubechies wavelet

  1. are they really orthogonality? for example daub4 has the support equal $[0-3]$, and for scale function we have must: $$ \int_0^3\phi(x-n)*\phi(x-m)dx =\begin{cases} 1, & m=n \\ 0, & m\neq n \end{cases} $$ but we know $(x-m)$ and $(x-n) \in [0-3]$ let $n=0$, $m=1$, then intersection of two interval is $[2-3]$ thus $\int_0^3\phi(x-n)*\phi(x-m)dx\neq 0$ What is the problem?

  2. how to evaluated connection coefficients? and when are they equal zero

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1: As you rightly say, $(x−m)$ and $(x−n)∈[0−3]$ defines the support of $\phi_m(x)=\phi(x-m)$ and $\phi_n(x)=\phi(x-n)$. Which transforms to $x\in[m,m+3]$ resp. $x\in[n,n+3]$.
The integration goes over the whole of $\mathbb R$, but because of the supports and the multiplication, it reduces to the intersection of those intervalls. If the intersection is empty, the integral has automatically the value $0$.
The construction of the wavelet, in fact the orthogonality conditons on the coefficients of the wavelet filters, ensure that also the integrals for overlapping intervals give the value $0$ if $m\ne n$.

2: What are connection coefficients? The coefficients of the scaling sequence or the components of some linear operator in a wavelet basis?

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  • $\begingroup$ I,m trying to evaluate $\int_R\phi^{d1}(x-m)*\phi^{d2}(x-n)dx$ that $\phi^d(x)$ is dth derivative of scaling function $\endgroup$ – sorush Dec 15 '13 at 9:24

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