Local and global properties of functions and their Fourier transforms
It is known that if an integrable function on the circle group \(\mathbb{T}\) with nonnegative Fourier coefficients is square-integrable in some neighbourhood of 0, then it is square-integrable on \(\mathbb{T}\) (by N. Wiener). However, an analogous result does not hold for the reals \(\mathbb{R}\) (by Kawazoe, Onoe and Tachizawa). The author investigates integrable functions with nonnegative Fourier transforms through the amalgam \((L^p,\ell^q)\), and he gives another proof for the latter. One of his results is as follows. Theorem: The following conditions are equivalent for any function in \(L^1(\mathbb{R})\) with a nonnegative transform. (a) The function is square-integrable in some neighbourhood of 0. (b) The function belongs to \((L^2,\ell^\infty)\). (c) The transform of the function belongs to \((L^1,\ell^2)\).
integrable functions with nonnegative Fourier transforms
scientific article; zbMATH DE number 1036424