Define
![\begin{displaymath}
\psi(x)\equiv\cases{
1 & $0\leq x\leq {\textstyle{1\over 2}...
... & ${\textstyle{1\over 2}}\leq x \leq 1$\cr
0 & otherwise\cr}
\end{displaymath}](h_39.gif) |
(1) |
and
![\begin{displaymath}
\psi_{jk}(x)\equiv\psi(2^jx-k),
\end{displaymath}](h_40.gif) |
(2) |
where the Functions plotted above are
Then a Function
can be written as a series expansion by
![\begin{displaymath}
f(x)=c_0+\sum_{j=0}^\infty \sum_{k=0}^{2^j-1} c_{jk}\psi_{jk}(x).
\end{displaymath}](h_57.gif) |
(3) |
The Functions
and
are all Orthogonal in
,
with
![\begin{displaymath}
\int_0^1 \phi(x)\phi_{jk}(x)\,dx=0
\end{displaymath}](h_61.gif) |
(4) |
![\begin{displaymath}
\int_0^1 \phi_{jk}(x)\phi_{lm}(x)\,dx=0.
\end{displaymath}](h_62.gif) |
(5) |
These functions can be used to define Wavelets. Let a Function be defined on
intervals,
with
a Power of 2. Then an arbitrary function can be considered as an
-Vector
, and the
Coefficients in the expansion
can be determined by solving the Matrix equation
![\begin{displaymath}
{\bf f}={\hbox{\sf W}}_n {\bf b}
\end{displaymath}](h_66.gif) |
(6) |
for
, where
is the Matrix of
basis functions. For example,
![\begin{displaymath}
{\hbox{\sf W}}_4 =\left[{\matrix{ 1 & \hfil 1 & \hfil 1 & \h...
...cr \hfil 1 & \hfil -1 & & \cr & & 1 & \cr & & & 1\cr}}\right].
\end{displaymath}](h_68.gif) |
(7) |
The Wavelet Matrix can be computed in
steps, compared to
for the
Fourier Matrix.
See also Wavelet, Wavelet Transform
References
Haar, A. ``Zur Theorie der orthogonalen Funktionensysteme.'' Math. Ann. 69, 331-371, 1910.
Strang, G. ``Wavelet Transforms Versus Fourier Transforms.'' Bull. Amer. Math. Soc. 28, 288-305, 1993.
© 1996-9 Eric W. Weisstein
1999-05-25