Relations between functional norms of a non-negative function and its square root on the positive cone of Besov and Triebel-Lizorkin spaces

In this communication we study in detail the relations between the smoothness of f and root f in the case when the smoothness of the univariate non-negative functions f is measured via Besov and Triebel-Lizorkin space scales. The results obtained can be considered also as embedding theorems for usual Besov and Triebel-Lizorkin spaces and their analogues in Hellinger metric. These results can be used in constrained approximation using wavelets, with applications to probability density estimation in speech recognition, non-negative non-parametric regression-function estimation in positron-emission tomography (PET) imaging, shape/order-preserving and/or one-sided approximation and many others.