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

Abstract

UDC 517.5

Let $(\mathbb{X},\|.\|_{\mathbb{X}})$ denote a complex Banach space and $L(\mathbb{X})=BC (\mathbb{R}\to\mathbb{X})$ be the set of all $\mathbb{X}$-valued bounded continuous functions $f\colon\mathbb{R}\to\mathbb{X}.$
For $f\in L(\mathbb{X})$ we define $\|f\|_{L(\mathbb{X})}=\sup\{ \|f(x)\|_{\mathbb{X}}\colon x\in\mathbb{R}\}.$ Then $(L(\mathbb{X}),\|.\|_{L(\mathbb{X})})$ itself is a Banach space. The Beurling spectrum $\mathrm{Spec}(f)$ of a function $f\in L(\mathbb{X})$ is defined by
$$\mathrm{Spec}(f)=\left\{\zeta\in\mathbb{R}\colon\forall\epsilon>0 \exists \varphi\in\mathcal{S}(\mathbb{R})\colon\mbox{supp}\,\widehat{\varphi}\subset(\zeta-\epsilon,\zeta+\epsilon),\varphi*f\nequiv 0\right\}.\right\}.$$ We obtain the following Paley-Wiener type theorem for functions with values in Banach spaces:

Let $f\in L(\mathbb{X})$ and $K$ be an arbitrary compact set in $\mathbb{R}.$ Then $\mbox{Spec}(f)\subset K$ if and only if for any $\tau > 0$ there exists a constant $C_\tau < \infty$ such that
$$
\|P(D)f\|_{L(\mathbb{X})}\leq C_\tau
\|f\|_{L(\mathbb{X})}\sup\limits_{x\in K^{(\tau)}} |P(x)|
$$
for all polynomials with complex coefficients $P(x),$ where the differential operator $P(D)$ is obtained from $P(x)$ by substituting $x \to -i\,\dfrac{d}{dx},$ $\dfrac{d}{dx}$ is the usual derivative in $L(\mathbb{X})$ and $K^{(\tau)}$ is the $\tau$-neighborhood in $\mathbb{C}$ of $K.$

Moreover, Paley-Wiener type theorem for integral operators and one for some special compacts $K$ are also given.