Paley–Wiener-Type Theorem for Functions with Values in Banach Spaces

Let X . X denote a complex Banach space and let L X = BC ℝ → X be the set of all X –valued bounded continuous functions f : ℝ → X . For f ∈ L ( X ) , we define f L X = sup f x X : x ∈ ℝ . Then L X . L X is itself a Banach space. The Beurling spectrum Spec( f ) of a function f ∈ L ( X ) is defined by Spec( f ) = ζ ∈ ℝ : ∀ ϵ > 0 ∃ φ ∈ S ℝ : supp φ ̂ ⊂ ζ − ϵ ζ + ϵ φ ∗ f ≢ 0 . We obtain the following Paley–Wiener type theorem for functions with values in Banach spaces: Let f ∈ L ( X ) and let K be an arbitrary compact set in ℝ. Then Spec( f ) ⊂ K if and only if, for any τ > 0 , there exists a constant C τ < ∞ such that P D f L X ≤ C τ f L X sup x ∈ K τ P x for all polynomials with complex coefficients P ( x ) , where the differential operator P ( D ) is obtained from P ( x ) by the substitution x → − i d dx , d dx is the ordinary derivative in L X , and K ( τ ) is a τ -neighborhood of K in ℂ . Moreover, we also present the Paley–Wiener-type theorem for integral operators and some special compact sets K .