Test the noise removal features of the Haar wavelets by adding random noise to one of the functions in Exercises 9.7.1 and 9.7.2, computing the wavelet series, and then setting the high “frequency” modes to zero. What do you observe? Is this a reasonable denoising algorithm when compared with a Fourier method?

9.7.1. Let f(x) = 1. (a) Determine its Haar wavelet coefficients cj.k. (b) Graph the partial sums s, (a) 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 || f-8,. || = max{ |f(x)-s, (a)||0 x1} for each of your partial sums? 9.7.2. Answer Exercise 9.7.1 for the functions (a) x - I, (b) cos TX, (c) e { -e 1 0