Wavelet Analysis

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
(1) wjk(t) = 2-j/2w(2-jt-k), for all t in R.
There actually exist functions w in L2(R), the Hilbert space of square integrable measurable functions on R, such that {wjk: j, k in Z} is an orthonormal basis of L2(R). Such an w is called a wavelet. There are many variations on this basic idea, but we will stick to the simple version for this discussion.

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
(2) f = Sj, k cjk wjk.
If one fixes a threshold and throws away those terms in the sum in (2) for which |cjk| is below the threshold, then only finitely many terms remain and the sum is an approximation to f. The finitely many cjks can be stored, transmited or manipulated in various ways.

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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Last update: July 8, 1998