The method of Bellman functions is a powerful method that can be used to bound sums indexed by dyadic intervals. Many discrete results in Harmonic Analysis have been obtained with this method in the last few years. To date it can be used on sums involving inner products of a function with the scaling or wavelet functions from the Haar wavelet system, but not the scaling or wavelet functions of general wavelet systems. Many of the discrete results obtained with the current Bellman function method could be expanded to the continuous setting if the Bellman function method was available for more general wavelets. The PI proposes to expand the method of Bellman functions from its current scope to sums involving any MRA wavelet.
Wavelets have been useful in many areas such as mathematics, physics, engineering and signal processing. They are the basis of the new world wide web image compression standard JPEG2000. They prove to also be a useful tool in harmonic analysis. It is frequently necessary to estimate the size of sums involving wavelets. The Bellman function method is a very useful tool in doing so, but currently only works for a very special wavelet called the Haar wavelet. The PI is interested in generalizing the method where possible to more general classes of wavelets.
