3. Function Approximation

Imagine a sequence of spaces

\begin{displaymath}
V_j=\textrm{span}\{\varphi_{jk}, k\in Z, j\in Z,\varphi_{jk}\in L_2(R)\}.
\end{displaymath}

This sequence $\{V_j,j\in Z\}$ is called multiresolution analysis of L2(R) if $V_j\subset V_{j+1}$ and $\bigcup_{j\geq 0} V_j$ is dense in L2(R). Further, let us have $\psi_{jk}$ such that for

\begin{displaymath}W_j=\textrm{span}\{\psi_{jk}\},\end{displaymath}

we have

\begin{displaymath}V_{j+1}=V_j\oplus W_j,\end{displaymath}

where the circle around plus sign denotes direct sum. Then we can decompose the space L2(R) in the following way:

\begin{displaymath}L_2(R)=V_{j_0}\oplus W_{j_0}\oplus W_{j_0+1}+\dots\end{displaymath}

and we call $\varphi_{jk}$ father and $\psi_{jk}$ mother wavelets. This means that any $f\in L_2(R)$ can be represented as a series

\begin{displaymath}
f(x)=\sum_k\alpha_k\varphi_{j_0k}(x)+\sum_{j=j_0}^\infty\sum_k
\beta_{jk}\psi_{jk}(x).
\end{displaymath}

According to a given multiresolution analysis, we can approximate a function with arbitrary accuracy. Under smoothness conditions on f, we can derive upper bounds for the approximation error. Smoothness classes which are particularly well suited to the study of approximation properties of wavelet bases are given by the scale of Besov spaces Bp,qm. Here m is the degree of smoothness while p and q characterize the norm in which smoothness is measured. These classes contain traditional Hölder and L2-Sobolev smoothness classes, by setting $p=q=\infty$ and p=q=2, respectively.

For a given Besov class Bp,qm(C) there exists the following upper bound for the approximation error measured in L2:

\begin{displaymath}
\sup_{f\in B_{p,q}^m(C)}\left\{\left\Vert\sum_{k\in Z}
\lang...
...rt _{L_2}\right\} = O\left(2^{-2J(m+1/2-1/\min\{p,2\}}\right).
\end{displaymath}

The decay of this quantity as $J \rightarrow \infty$ provides a characterization of the quality of approximation of a certain functional class by a given wavelet basis. A fast decay is favorable for the purposes of data compression and statistical estimation.

The following display provides an impression of how a discontinuous function is represented in the domain of coefficients.


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The display contains two plots: the upper shows a jump function, the lower the mother wavelet coefficients corresponding to their position in scale and time. The large coefficients are caused by the discontinuity (center) and by boundary effects since we use a periodic wavelet transform for a nonperiodic function.

You can examine the effects of approximating a wide range of functions using various wavelet bases with the following interactive menu.


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