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

Let X.X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left(\mathbbm{X},{\left\Vert .\right\Vert}_{\mathbbm{X}}\right) $$\end{document} denote a complex Banach space and let LX=BCℝ→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ L\left(\mathbbm{X}\right)= BC\left(\mathbb{R}\to \mathbbm{X}\right) $$\end{document} be the set of all X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbbm{X} $$\end{document} –valued bounded continuous functions f : ℝ→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbb{R}\to \mathbbm{X} $$\end{document}. For f ∈ L (X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbbm{X} $$\end{document}), we define fLX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\left\Vert f\right\Vert}_{L\left(\mathbbm{X}\right)} $$\end{document} = sup fxX:x∈ℝ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{{\left\Vert f(x)\right\Vert}_{\mathbbm{X}}:x\in \mathbb{R}\right\} $$\end{document}. Then LX.LX\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left(L\left(\mathbbm{X}\right),{\left\Vert .\right\Vert}_{L\left(\mathbbm{X}\right)}\right) $$\end{document} is itself a Banach space. The Beurling spectrum Spec(f) of a function f ∈ L (X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \mathbbm{X} $$\end{document}) is defined by Spec(f) = ζ∈ℝ:∀ϵ>0∃φ∈Sℝ:suppφ̂⊂ζ−ϵζ+ϵφ∗f≢0.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ \left\{\zeta \in \mathbb{R}:\forall \epsilon >0\exists \varphi \in \mathcal{S}\left(\mathbb{R}\right):\operatorname{supp}\hat{\varphi}\subset \left(\zeta -\epsilon, \zeta +\epsilon \right),\varphi \ast f\not\equiv 0\right\}. $$\end{document} We obtain the following Paley–Wiener type theorem for functions with values in Banach spaces: