Riesz basis, Paley-Wiener class and tempered splines

Abstract

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ℒ2m (T, ℝ) ∩ L2 (ℝ) which is the space of polynomial splines with irregularly distributed nodesT = {t j } j ∈ℤ, where {t j }j∈ℤ is a real sequence such that {e it ξ} j }j ∈ℤ constitutes a Riesz basis for L2([ −π,π]). From these results, the asymptotic relation is proved, where B π,2 denotes the set of all functions from L2( R) which can be continued to entire functions of exponential type ⪯ ϕ, i.e. the classical Paley-Wiener class.