Interpolation by weighted Paley–Wiener spaces associated with the Dunkl transform

Given α > − 1 2 , σ > 0 and 1 ⩽ p < ∞ , we study the interpolation problem in the space PW σ p , α of entire functions f : C → C of exponential type ⩽σ for which ∫ R | f ( x ) | p × | x | 2 α + 1 d x < ∞ , with nodes of interpolation at s j / σ , j ∈ Z , where { s j : j ∈ N } is the increasing sequence of all positive roots of the Bessel function J α + 1 ( z ) of order α + 1 , and s j = − s − j for all j ∈ Z . We prove that if 4 ( α + 1 ) 2 α + 3 : = p 1 ( α ) < p < p 2 ( α ) : = 4 ( α + 1 ) 2 α + 1 , the interpolation problem f ∈ PW σ p , α and f ( σ − 1 s j ) = c j for all j ∈ Z has a unique solution for every sequence { c j } of complex numbers satisfying ∑ j ∈ Z | c j | p ( 1 + | j | ) 2 α + 1 < ∞ , and that if p ⩽ p 1 ( α ) , the corresponding interpolation problem may not have a solution, and that the solution, if exists, is unique if and only if p ⩽ p 2 ( α ) . Finally, we show that ∫ R | f ( x ) | p | x | 2 α + 1 d x ∼ σ − 2 α − 2 ∑ j ∈ Z | f ( s j σ − 1 ) | p ( 1 + | j | ) 2 α + 1 , with the constant of equivalence depending only on p and α, holds for all entire functions f of exponential type ⩽σ if and only if p 1 ( α ) < p < p 2 ( α ) .