Abstract. Wavelet bases are thought to provide, for functions with local singularities in arbitrary dimension, a representation that is optimally local in space and frequency. In reality, even in moderate dimension, standard wavelet constructions fail miserably. For example, when using the tensor-product of Daubechies' wavelets with 4 vanishing moments, a single local singularity in 5D is represented by about 500,000 non-zero coefficients in each scale, and in 10D that number reaches 300,000,000,000. That means that such constructs are, in reality, non-local at all.

We provide a theory and corresponding constructions of wavelet bases that are truly local in space and frequency in any spatial dimension. The theory begins with the fundamental principles of frame theory and continues with more recent developments. The resulting constructs are really local: a system of ours that performs in frequency on par with or better than the above Daubechies' system yields at each scale 5 non-zero coefficients for each local singularity, and this number is independent of the spatial dimension.

The talk is based on joint work with Youngmi Hur (JHU).