Wavelets satisfy the property of MRA. However, to construct wavelet basis you need to obtain a scaling function by solving corresponding refinement equation ( scaling equation). From their obtain corresponding mother wavelet.
no, this is not necessary. However, for 1D compactly supported wavelets, they are always provided by a MRA, and they are more easily constructed through that MRA
Thus, the gain of using another method should be questionable...
Thank you so much.
Does the traditional method in Legendre and Chebyshev wavelets use the MRA ? If the Chebyshev wavelets use it, if possible, I would like to see a detailed explanation, because I searched in many references but there were no explanations. Thank you in advanced.
Good question and thank you Pierre-Gilles for your interesting answer. I did not know that. WT is by essence built on the property of a mother wave dilation, But this may not be the only condition for an MRA. I think there is another condition concerning the initialization of the MRA at level 0, but I do not remember exactly which one. Maybe you can precise it to us.
Actually the idea of multi resolution analysis in terms of Legendre polynomials is not the same as other family of wavelets like Haar or db3,db4 , etc.
In legender wavelet system there is nothing but the domain is divide into m-parts, (known as resolution level of Legendre polynomials) then on each part finite terms of legendre polynomials are derived. All the work you have to done is to shift and scale the domain of L W from [-1,1] to [x_i, x_i+1], derive the relations of orthogonality etc and your wavelets are ready to implement.
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