aar functions are a simple motivational
example for developments of this chapter.

Let
We
introduce
We
have

Proposition

(Main property of Haar functions
1) The set
(set of Haar functions) is a complete orthonormal system of functions in
.
The set
spans the class
for every

Proof

The fact that
is an orthonormal system is a direct verification.

Observe that any function
is defined by set of
values
,
and the set
is linearly independent (because it is orthonormal) and
has
members. Hence,
is a basis for
.