14.3 Function Approximation

Imagine a sequence of spaces

$\displaystyle V_j=\textrm{span}\{\varphi_{jk}, k\in Z, j\in Z,\varphi_{jk}\in L_2(R)\}.
$

This sequence $ \{V_j,j\in Z\}$ is called multiresolution analysis of $ L_2(R)$ if $ V_j\subset V_{j+1}$ and $ \bigcup_{j\geq 0} V_j$ is dense in $ L_2$(R). Further, let us have $ \psi_{jk}$ such that for

$\displaystyle W_j=\textrm{span}\{\psi_{jk}\},$

we have

$\displaystyle V_{j+1}=V_j\oplus W_j,$

where the circle around plus sign denotes direct sum. Then we can decompose the space $ L_2(R)$ in the following way:

$\displaystyle L_2(R)=V_{j_0}\oplus W_{j_0}\oplus W_{j_0+1}+\dots$

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

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

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 $ B_{p,q}^m$. 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 $ L_2$-Sobolev smoothness classes, by setting $ p=q=\infty$ and $ p=q=2$, respectively.

For a given Besov class $ B_{p,q}^m(C)$ there exists the following upper bound for the approximation error measured in $ L_2$:

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

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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