The Haar wavelets

Introduction

We use dilates of the complex exponential wave

w(x) ≔ e2πix, -∈ < x < ∈

when we write the familiar Fourier synthesis equation

for a suitably regular function f on ℝ. (The function F is the Fourier transform of f.) For example, the identity

shows how to synthesize a normal density by combining waves of constant amplitude,

which stretch from x = -∞ to x = +∞. When 0 < σ ≪ 1, this density is an approximate delta with 99.7% of its integral in the tiny interval -3σ ≤ x ≤ 3σ. Almost perfect destructive interference must occur at every point |x| > 3σ. Such a synthesis has mathematical validity (a proof is given in Section 1.5), but it is physically unrealistic. In practice, we cannot produce the audio signal for a 1-ms “click” by having tubas, trombones, …, piccolos play sinusoidal tones of constant amplitude for all eternity!

In this chapter you will learn to synthesize a function f with localized, oscillatory basis functions called wavelets. We use the prototype Haar wavelet

to introduce the fundamental concepts from an exciting new branch of analysis created by mathematicians, electrical engineers, physicists, … during the last two decades of the 20th century. More sophisticated wavelets for audio signal processing, image compression, etc. are described in the following sections.

You will observe that the wavelet (1) oscillates. The positive and negative parts of ψ have the same size in the sense that

It is easy to produce a function y on R that interpolates f at the sample points, i.e., …