## Wavelet AnalysisHere is the purest and simplest version of what a wavelet is. For a
square integrable function w on
^{2}(R), the Hilbert space of square
integrable measurable functions on R, such that _{jk}:
j, k in Z}^{2}(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 Let
c_{jk}|
is below the threshold, then only finitely many terms remain and the sum
is an approximation to f. The finitely many
c_{jk}s 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. |

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