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Marcinkiewicz-Zygmund inequalities
We study a generalization of the classical Marcinkiewicz-Zygmund
inequalities. We relate this problem to the sampling sequences in the
Paley-Wiener space and by using this analogy we give sharp necessary and
sufficient computable conditions for a family of points to satisfy the
Marcinkiewicz-Zygmund inequalities
Smooth Parametrizations in Dynamics, Analysis, Diophantine and Computational Geometry
Smooth parametrization consists in a subdivision of the mathematical objects
under consideration into simple pieces, and then parametric representation of
each piece, while keeping control of high order derivatives. The main goal of
the present paper is to provide a short overview of some results and open
problems on smooth parametrization and its applications in several apparently
rather separated domains: Smooth Dynamics, Diophantine Geometry, Approximation
Theory, and Computational Geometry.
The structure of the results, open problems, and conjectures in each of these
domains shows in many cases a remarkable similarity, which we try to stress.
Sometimes this similarity can be easily explained, sometimes the reasons remain
somewhat obscure, and it motivates some natural questions discussed in the
paper. We present also some new results, stressing interconnection between
various types and various applications of smooth parametrization
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