Let f(x)=x. (a) Determine its Haar wavelet coefficients cj,...

Let $f(x)=x$. (a) Determine its Haar wavelet coefficients $c_{j, k^*}$ (b) Graph the partial sums $s_r(x)$ of the Haar wavelet series (9.136) where $j$ goes from 0 to $r=2,5$, and 10. Compare your graphs with that of $f$ and discuss what you observe. Is the series converging to the function? Can you prove this? (c) What is the maximal deviation $\left\|f-s_r\right\|_{\infty}=\max \left\{\left|f(x)-s_r(x)\right| \mid 0 \leq x \leq 1\right\}$ for each of your partial sums?

Key Concepts

Uniform Norm (Maximum Deviation)

The uniform norm, also known as the L? norm, measures the maximum deviation between a function and its approximation over a given interval. In the context of Haar wavelet approximations, it is used to assess the worst-case error between the function and its partial sum series, thus providing a measure of the approximation quality in a maximum difference sense.

Convergence of Wavelet Series

The convergence of a wavelet series, such as the Haar wavelet series, refers to the manner and conditions under which the partial sums approach the original function. This involves considering different modes of convergence (pointwise, uniform, or in L² norm) and typically requires understanding how the approximation error diminishes as finer scales (higher resolution terms) are included in the series.

Haar Wavelets

Haar wavelets constitute one of the simplest families of wavelets and serve as the building blocks for constructing orthonormal bases in function spaces. They are characterized by their simple, piecewise constant structure and are used to decompose and analyze functions at various scales and resolutions, forming the basis for multiresolution analysis.

Wavelet Coefficients

Wavelet coefficients are the inner products of a function with a set of basis (wavelet) functions. For the Haar wavelet, these coefficients quantify how much the function correlates with each wavelet at different scales and locations, providing a hierarchical representation of the function's behavior and enabling efficient approximation and reconstruction.

Partial Sum Approximations (Haar Wavelet Series)

A Haar wavelet series representation of a function is obtained by summing up the scaled and translated Haar wavelets multiplied by their corresponding coefficients. Partial sum approximations involve summing only a finite number of terms, which effectively creates a piecewise constant approximation of the function. As more terms (finer scales) are added, the approximation typically becomes more refined.

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