Mixed Hölder matrix discovery via wavelet shrinkage and Calderón-Zygmund decompositions

The second part of this paper develops a theory of Besov spaces on products of tree geometries. We show that matrices with small Besov norm can be written as a sum of a mixed Hölder matrix and a matrix with small support. Such decompositions are known as Calderón-Zygmund decompositions and are of general interest in harmonic analysis. The decompositions we establish impose fewer conditions on the function with small support than previous decompositions of this type while maintaining the same guarantees on the mixed Hölder matrix. As such, they are applicable to a greater variety of matrices and should find use in many data organization problems. As part of our analysis, we provide characterizations of the underlying Besov spaces using wavelets and other multiscale difference operators that are analogous to those from the classical Euclidean theory.