On discrete and continuous norms in Paley-Wiener spaces and consequences for exponential frames

Abstract

It is well known that for certain sequences {t_n}_n∈ℤ the usual L^p norm ∥·∥_p in the Paley-Wiener space PW ^p _τ is equivalent to the discrete norm ‖f‖_p,{tn}:=(∑ ^∞_n=−∞ |f(t_n)|^p)^1/p for 1 ≤ p = < ∞ and ‖f‖_∞,{tn}:=sup _n∈ℤ|f(t_n| for p=∞). We estimate ∥f∥_p from above by C∥f∥_p, _n and give an explicit value for C depending only on p, τ, and characteristic parameters of the sequence {t_n}_n∈ℤ. This includes an explicit lower frame bound in a famous theorem of Duffin and Schaeffer.