Riesz basis, Paley-Wiener class and tempered splines

The Marcinkiewicz-Zygmund inequality and the Bernstein inequality are established on ℒ_2m (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.