Here is the purest and simplest version of what a wavelet is. For a square integrable function w on R and integers j and k, define
If you fix a nice wavelet w, then you can use it to analyse an arbitrary function f in L2(R) by comparing it to the wjks using the inner product <.|.> in L2(R).
Let cjk = <f | wjk>, for all j, k in Z. Then
My personal interest comes from continuous versions of (2), where the
double summation is replaced by a certain weighted double integral. In fact,
there is a locally compact group in the background and a unitary representation
of that group whose special properties "allow" (2) to occur. See
[D. Bernier and K. Taylor: Wavelets from Square-Integrable Representations. SIAM J. Math. Anal., 27 594-608(1996)] for a detailed discussion.
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