On dualizing a multivariable Poisson summation formula (Q1375132)

The authors have shown in an earlier paper [Proc. Natl. Acad. Sci. USA 88, No. 16, 7348-7350 (1991; Zbl 0771.42001)] that dualizing a form of the Poisson summation formula yields a pair of linear transformations which map a function \(\phi\) of one variable into a function and its cosine transform in a generalized sense. In this note, the authors consider \(\phi\) such that \(\phi(x)\) and \(\phi'(x)\) are absolutely continuous and \[ \int^\infty_{-\infty}\Biggl\{\Biggl[1+ \Biggl({1\over x}\Biggr)\Biggr][|\phi(x)|+ | x| |\phi'(x)|]+ |\phi''(x)|\Biggr\}dx< \infty. \] It is shown that the transform relations hold in the classical sense in this case. A generalization to any number of dimensions is also presented.