Introduction

Anticipating the formal definition below, we note that a wavelet is a function ψ ∊ L2(ℝ) whose scaled, dilated, translated copies ψj,k for j, k ∊ ℤ form an orthonormal basis for i > L2(ℝ). Given such a function, the results of Section 4.6 tell us that any function in L2(ℝ) can be expressed as a Fourier series with respect to the functions ψj,k, though we will usually call such a series a wavelet series and the corresponding Fourier coefficients the wavelet coefficients. All of this, of course, corresponds exactly to what we know from earlier chapters in the specific case of the Haar wavelet H.

Recall from Section 2.3 that for k = 0, 1, 2, … we denote by Ck(I) the space of functions on an interval I which have continuous derivatives up to and including order k. Note that the case k = 0 is included: C0(I) is just C(I), the space of continuous functions on I. Further, recall that we define C∞(I) to be the space of infinitely differentiable functions on I. Terms such as smooth and smoothness are often used in an informal way to describe the location of a function in the hierarchy of spaces Ck(ℝ); a function in Ck+1(ℝ) might be said to be smoother than one in Ck(ℝ), for example.

Applications of wavelets usually call for the use of wavelets that satisfy specific conditions of one kind or another, and the condition that a wavelet have some degree of smoothness is almost universal. The minimal requirement for a wavelet in practice is therefore usually continuity (membership of C0(ℝ)), though a higher order of smoothness might well be preferred (membership of C1(ℝ) or C2(ℝ), for example). This means that we need to have methods for finding wavelets that are smoother than the Haar wavelet, which is not continuous and so is not in Ck(ℝ) for any k ≥ 0.

For example, consider the use of wavelets for data compression. (This will often involve data in several dimensions – an image file is an obvious two-dimensional example – while we are only considering wavelets in one dimension, in L2(ℝ), but the principle is essentially the same.)