Generalization of a theorem of Boas to a class of integral transforms
\textit{R. Boas} [Trans. Am. Math. Soc. 40, 287-308 (1936; Zbl 0015.21301)] proved that if \(f\in L^{2}({\mathbb R})\) \(({\mathbb R}=(-\infty,\infty))\), then a necessary and sufficient condition that \(f\) vanishes everywhere on \((-1,1)\) is that \((B(\widetilde{B}f))(\omega)=-\widetilde{f}(\omega)\), where \[ (Bf)(x)={\frac{1}{\pi}}\int^{\infty}_{0} {\frac{f(x+t)-f(x-t)}{t^{2}}}\sin t dt, \quad \widetilde{f}(\omega)= {\frac{1}{2\pi i}}\int^{\infty}_{-\infty}f(x)e^{-\omega x} dx. \] The paper is devoted to obtain an analogue of Boas' result for the integral transform of the form \[ \Phi [F](x)=\int^{\infty}_{a}\varphi (x,\lambda)F(\lambda) d\rho(\lambda), \] where \(a=0\) or \(-\infty\), \(F\in L^{2}(a,\infty)\), \(d\rho (\lambda)\) is an absolutely continuous measure on \((a,\infty)\) and \(\varphi (x,\lambda)\) is a solution of a certain singular Sturm-Liouville problem for the ordinary differential equation of the second kind \[ {\frac{d^{2}\varphi}{dx^{2}}}-q(x)\varphi =-\lambda \varphi \] on the half-axis \({\mathbb R}_{+}=[0,\infty)\). Necessary and sufficient conditions are established for a function \(f=\Phi [F]\) with \(F\in L^{2}({\mathbb R}_{+},d\rho)\) to be vanished in a neighborhood of \(\lambda_{0}\in {\mathbb R}_{+}\). Applications are given to the Fourier sine and cosine transforms and to the Weber transform.
scientific article; zbMATH DE number 1563157
scientific article