Journal of Inequalities and Applications 
Volume 1 (1997), Issue 2, Pages 149-164
doi:10.1155/S1025583497000106

Hermite interpolation and an inequality for entire functions of exponential type

Georgi R. Grozev1 and Qazi I. Rahman2

 1Numetrix Ltd., 655 Bay Street, Suite 1200, Ontario, Toronto M5G 2K4, Canada
2Département de Mathématiques et de Statistique, Université de Montréal, Québec, Montréal H3C 3J7, Canada
Received 11 September 1996

Abstract

Let c∈[0,1),p>0. It is shown that if f is an entire function of exponential type cmπ and ∑n=∞∞∑μ=0m−1|f(μ)(λn)|p<∞, where {λn}n∈ℤ is a sequence of real numbers satisfying |λn−n|≤Δ<∞, |λn+u−λn|≥δ>0 for u≠0, then ∫−∞∞|f(x)|pdx<B∑n=−∞∞∑μ=0m−1|f(μ)(λn)|p, where B depends only on c,p,Δ, and δ. A sampling theorem for irregularly spaced sample points is obtained as a corollary. Our proof of the main result contains ideas which help us to obtain an extension of a theorem of R.J. Duffin and A.C. Schaeffer concerning entire functions of exponential type bounded at the points of the above sequence {λn}n∈ℤ .