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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 $L^{2}$ 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.

Keywords

Semiclassical measures H-measures oscillatory sequence concentratory sequence characteristic scale Kolmogorov–Riesz compactness theorem

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Characteristic scales of bounded L 2 sequences

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