Multivariate Haar wavelets and B-splines
Tanya Zaitseva

Meeting • 13 August 2019

Abstract
Every day we need to store lots of information: images, audio, video, etc. One efficient way to do this is to use wavelets. The Haar system on the real line is the simplest example of wavelets. Multidimensional Haar system can be constructed as a direct product of one-dimensional systems, although, these systems have a big amount of corresponding generating mother wavelet functions. That is why it is more effective to construct Haar multivariate functions using an arbitrary integer dilation matrix M. In this case the support of wavelet scaling function is some special fractal set. I will present a classification of all Haar systems with only one generating function and discuss their Holder regularity, it is a useful characteristic of wavelet systems. The big Holder regularity implies the fast convergence of the corresponding partial sums of wavelet expansions, that is also important in practice. If we convolve several wavelet scaling functions, we will get multivariate B-splines. It turns out that multivariate B-splines which are based on these fractal sets have higher Holder regularity than, for example, famous rectangular B-splines. So, they approximate functions better. These splines can be also used to construct the Battle-Lemarie spline wavelets.