Rademacher Function

In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and extensively used as a teaching example.

The Haar sequence was proposed in 1909 by Alfréd Haar.
Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval  , 1 The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1.

The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage for the analysis of signals with sudden transitions ( discrete signals), such as monitoring of tool failure in machines.

The Haar wavelet's mother wavelet function $\psi(t)$ can be described as
: $\psi(t) = \begin 1 \quad & 0 \leq t < \frac,\\ -1 & \frac \leq t < 1,\\ 0 &\mbox \end$

Its scaling function $\varphi(t)$ can be described as
: $\varphi(t) = \begin1 \quad & 0 \leq t < 1,\\0 &\mbox\end$

Haar functions and Haar system

For every pair ''n'', ''k'' of integers in $\mathbb$, the Haar function ''ψ''_''n'',''k'' is defined on the real line $\mathbb$ by the formula
:$\psi_(t) = 2^ \psi(2^n t-k), \quad t \in \mathbb.$
This function is supported on the right-open interval , ''i.e.'', it vanishes outside that interval. It has integral 0 and norm 1 in the Hilbert space  ''L''²($\mathbb$),
:$\int_ \psi_(t) \, d t = 0, \quad \, \psi_\, ^2_ = \int_ \psi_(t)^2 \, d t = 1.$
The Haar functions are pairwise orthogonal,
:$\int_ \psi_(t) \psi_(t) \, d t = \delta_ \delta_,$
where $\delta_$ represents the Kronecker delta. Here is the reason for orthogonality: when the two supporting intervals $I_$ and $I_$ are not equal, then they are either disjoint, or else the smaller of the two supports, say $I_$, is contained in the lower or in the upper half of the other interval, on which the function $\psi_$ remains constant. It follows in this case that the product of these two Haar functions is a multiple of the first Haar function, hence the product has integral 0.

The Haar system on the real line is the set of functions
:$\ \cup \.$
It is complete in ''L''²($\mathbb$): ''The Haar system on the line is an orthonormal basis in'' ''L''²($\mathbb$).

Haar wavelet properties

The Haar wavelet has several notable properties:

#Any continuous real function with compact support can be approximated uniformly by linear combinations of $\varphi(t),\varphi(2t),\varphi(4t),\dots,\varphi(2^n t),\dots$ and their shifted functions. This extends to those function spaces where any function therein can be approximated by continuous functions.
#Any continuous real function on , 1can be approximated uniformly on , 1by linear combinations of the constant function 1, $\psi(t),\psi(2t),\psi(4t),\dots,\psi(2^n t),\dots$ and their shifted functions.
#Orthogonality in the form

    $\int_^2^\psi(2^n t-k)\psi(2^ t - k_1)\, dt = \delta_\delta_.$

Here, $\delta_$ represents the Kronecker delta. The dual function of ψ(''t'') is ψ(''t'') itself.
# Wavelet/scaling functions with different scale ''n'' have a functional relationship: since
::
\begin
\varphi(t) &= \varphi(2t)+\varphi(2t-1)\\ 2em \psi(t) &= \varphi(2t)-\varphi(2t-1),
\end
:it follows that coefficients of scale ''n'' can be calculated by coefficients of scale ''n+1'':
:If $\chi_w(k, n)= 2^\int_^\infty x(t)\varphi(2^nt-k)\, dt$
:and $\Chi_w(k, n)= 2^\int_^\infty x(t)\psi(2^nt-k)\, dt$
:then
::$\chi_w(k,n)= 2^ \bigl( \chi_w(2k,n+1)+\chi_w(2k+1,n+1) \bigr)$
::$\Chi_w(k,n)= 2^ \bigl( \chi_w(2k,n+1)-\chi_w(2k+1,n+1) \bigr).$

Haar system on the unit interval and related systems

In this section, the discussion is restricted to the unit interval , 1and to the Haar functions that are supported on , 1 The system of functions considered by Haar in 1910,
called the Haar system on , 1'' in this article, consists of the subset of Haar wavelets defined as
:$\,$
with the addition of the constant function 1 on , 1

In Hilbert space terms, this Haar system on , 1is a complete orthonormal system, ''i.e.'', an orthonormal basis, for the space ''L''²( , 1 of square integrable functions on the unit interval.

The Haar system on , 1—with the constant function 1 as first element, followed with the Haar functions ordered according to the lexicographic ordering of couples — is further a monotone Schauder basis for the space ''L''^''p''( , 1 when .see p. 3 in J. Lindenstrauss, L. Tzafriri, (1977), "Classical Banach Spaces I, Sequence Spaces", Ergebnisse der Mathematik und ihrer Grenzgebiete 92, Berlin: Springer-Verlag, .
This basis is unconditional when .

There is a related Rademacher system consisting of sums of Haar functions,
:$r_n(t) = 2^ \sum_^ \psi_(t), \quad t \in$, 1 \ n \ge 0.
Notice that , ''r''_''n''(''t''),  = 1 on , 1). This is an orthonormal system but it is not complete.
In the language of independent