This talk is about the discrete approximation of piecewise smooth functions, and investigates two issues with reference to a random piecewise smooth model (Cohen/d'Ales 1997).
(i) It is well known that piecewise smooth functions are approximately sparse in wavelet bases. However, other sparse representations are possible, such as the discrete gradient basis. I will give a result showing that signals drawn from a random piecewise constant model have sparser representations in the discrete gradient basis than in a Haar wavelet basis (with high probability).
(ii) Suppose we wish to obtain information about a piecewise smooth function in a discrete Fourier transform (DFT) basis, with the restriction that we only sample an incomplete set of frequencies. Which frequencies should we sample? I will argue that one would expect to extract the most information about the function by sampling the lowest frequencies.
These two issues offer some intriguing perspectives on recent advances in compressive imaging (in which signals/images are reconstructed from incomplete Fourier samples), challenging some of the typically assumed best practice.