The Paley-Wiener theorem with general weights
The authors consider weight functions u(x), v(x) in \({\mathbb{R}}^ n\), \(p\geq 1\), \(q\geq 1\) that satisfy the condition \(F^ p_ q: \| \hat f\|_{q,u}\leq C\| f\|_{p,v}\) for any simple function on \({\mathbb{R}}^ n\). Here \(\hat f\) is the Fourier transform of f, \(\| f\|_{p,w}=\| w^{1/p}f\|_ p\) and \(\| \cdot \|_ p\) is the usual \(L_ p\)-norm on \({\mathbb{R}}^ n.\) The main result of the paper is the following statement (Theorem 3): Let the weight functions u,v satisfy the condition \(F^ p_ q\), u(x) be even, v(x) be such that \(v(x)\geq c>0\) for \(| x| \geq N>0\). Let also an entire function f(z), \(z\in {\mathbb{C}}^ n\), be such that \(\| f\|_{p,v}<\infty\) and \(| f(z)| <A_{\epsilon}e^{(\sigma +\epsilon)| z|}\), \(\epsilon >0\). Then a measurable function g(x), \(x\in {\mathbb{R}}^ n\), exists such that \(\| g\|_{q,u}<\infty\), supp \(g\subset \{| t| \leq \sigma \}\) and \[ f(z)=\int_{{\mathbb{R}}^ n}e^{i<z,t>}g(t)dt. \] A number of corollaries are obtained. They generalize results of previous articles of the authors devoted to the same problems. There are also conditions on some special class of pairs (u,v) which are sufficient for \(F^ p_ q\).
scientific article; zbMATH DE number 4190129