Nikol’skii Inequality Between the Uniform Norm and Integral Norm with Bessel Weight for Entire Functions of Exponential Type on the Half-Line

Abstract

We study a Nikol’skii type inequality for even entire functions of given exponential type between the uniform norm on the half-line [0,∞) and the norm (∫ 0 ∞ |f(x)| q x2α+1dx)1/q of the space L q ((0,∞), x2α+1) with the Bessel weight for 1 ≤ q −1/2. An extremal function is characterized. In particular, we prove that the uniform norm of an extremal function is attained only at the end point x = 0 of the half-line. To prove these results, we use the Bessel generalized translation.