Marcinkiewicz–Zygmund inequalities

Abstract

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.

The Kadets $1/4$ theorem for polynomials

“…In particular, the theorem of Chui and Zhong to be used in this note can be obtained from a theorem given in [6]. We refer to [9] for the details of this link and to [12], where the connection between Marcinkiewicz-Zygmund inequalities and model spaces was first mentioned explicitly.…”

Marcinkiewicz–Zygmund inequalities and interpolation by spherical harmonics

“…This leads to the following definition. The precise formulation of the sparsity requirement is expressed in terms of the following Beurling type densities [18]. Definition 1.5.…”

“…Indeed, it is shown in [18] that if a triangular family is L p -MZ then its lower density has to be greater or equal to 1/2π, and that the converse holds for families with densities greater to 1/2π. The corresponding result for interpolation can be proved without a lot of effort.…”

Integral concentration of idempotent trigonometric polynomials with gaps

“…This is known as the Marcinkiewicz-Zygmund Theorem: the implied constants depend only on p but tend to ∞ for p tending to 1 or ∞. For the exact form fitting to our Taylor polynomials see Theorem (7.10), p. 30 chapter X in [34]; see also [25] for recent extensions. Inequality (66) then follows using the Marcinkiewicz-Zygmund Theorem twice, and invariance by translation of the L p norm.…”