Paley – Wiener type theorem for functions with values in Banach spaces

Abstract

UDC 517.5 Let ( 𝕏 , ‖ . ‖ 𝕏 ) denote a complex Banach space and L ( 𝕏 ) = B C ( 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 ) ) itself is a Banach space. The Beurling spectrum S p e c ( f ) of a function f ∈ L ( 𝕏 ) is defined by S p e c ( 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 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 < b r > ‖ P ( D ) f ‖ L ( 𝕏 ) ≤ C τ < b r > ‖ f ‖ L ( 𝕏 ) sup x ∈ K ( τ ) | P ( x ) | < b r > for all polynomials with complex coefficients P ( x ) , where the differential operator P ( D ) is obtained from P ( x ) by substituting x → - i d ⅆ x , d ⅆ x is the usual derivative in L ( 𝕏 ) and K ( τ ) is the τ -neighborhood in ℂ of K . Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts K are also given.