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 {w_jk: 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 w_jks using the inner product <^.|^.> in L²(R).

Let c_jk = <f | w_jk>, 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 |c_jk| is below the threshold, then only finitely many terms remain and the sum is an approximation to f. The finitely many c_jks 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.