Analytic Representation of the Distributional Finite Fourier Transform
Richard D. Carmichael · SIAM Journal on Mathematical Analysis · 1974
We define n-dimensional finite Fourier transforms for functions and distributions which are mappings from $\mathcal{K}(A)$ to $\mathcal{Z}(\bar 2\pi )$ and $\mathcal{K}'(A)$ to $\mathcal{Z}'(\bar 2\pi )$, respectively, where A is an arbitrary n-tuple of positive real numbers. Representation theorems are obtained for the distributional finite Fourier transform in which we relate this transform to entire analytic functions in $\mathbb{C}^n $. A finite convolution is defined, and we construct from it the finite regularization of an element of . $\mathcal{K}'(A)$ which is used to give another representation of the distributional finite Fourier transform. We show that the Fourier transform mapping $\mathcal{K}'$ to $\mathcal{Z}'$ can be obtained as the limit of a sequence of distributional finite Fourier transforms. Further, we give necessary and sufficient conditions for the distributional finite Fourier transform to be represented as the boundary value of a function which is analytic in the tubular radial domain $T^C = \mathbb{R}^n + iC$, C being an open connected cone ; and we use these results to obtain the analytic decomposition of the distributional finite Fourier transform.