Affiliations: [a] Department of Mathematics, Faculty of Science, University of Zagreb, Bijenička cesta 30, Zagreb, Croatia. E-mail: [email protected] | [b] University of Dubrovnik, Ćira Carića 4, Dubrovnik, Croatia. E-mail: [email protected]
Abstract: Practical applications of semiclassical measures are tightly connected with a so called oscillatory property, prevailing leakage of information related to high frequencies. In this paper we propose a complementary, concentratory property which prevents loss of information related to low frequencies. We demonstrate that semiclassical measures attain the best performance level if both the properties are satisfied simultaneously, and address a question if this is possible to achieve for an arbitrary bounded L2 sequence, providing a negative answer. Comparison of H-measures with semiclassical ones is presented, showing precedence of the latter for problems exhibiting just a single frequency scale. Finally, we present some (strong) compactness results based on the above properties.
