The Hilbert transform of almost periodic functions and distributions (Q2754349)

The author finds a space \(B\) of almost periodic functions, closed with respect to the operations of differentiation and the Hilbert transform. The elements of its dual \(B'\) are called semi-almost periodic distributions. The Hilbert transform is defined on \(B'\) and an inversion formula is established. This result is used to find the harmonic function \(u(x,y)\), \(x\in\mathbb{R}\), \(y> 0\), which for fixed \(y\) is an almost periodic function of \(x\) and satisfies \(\lim_{y\to 0+} u(x,y)= f(x)\), where \(f\) is a Schwartz almost periodic distribution, the limit is interpreted in the weak distributional sense and \(u(x,y)= 0(1)\), \(y\to\infty\), uniformly in \(x\in\mathbb{R}\).NEWLINENEWLINEFor the entire collection see [Zbl 0971.00008].