ICM 1998, Berlin, Aug. 18–27

Abstracts of Short Communications and Posters

Section:

8. Analysis

1991 MS Classification: 42, 43, 44 Pandey, Jagdish N., Carleton University, Ottawa, Canada The Hilbert Transform of Almost Periodic Functions and Distributions Let B_{ap}^{ac} be the space of infinitely differentiable almost periodic functions defined on R closed with the operations of differentiation and Hilbert transformation, and metrized by the scalar product (f, g) = M\{f(t)\overline{g(t)}\}. It is shown that an arbitrary f ∈ (B_{ap}^{ac})′ is a finite linear combination of finite order distributional derivatives of elements of the completion of B_{ap}^{ac} and that the Schwartz space of almost periodic distributions is contained in (B_{ap}^{ac})′. The Hilbert transform Hf of f ∈ (B_{ap}^{ac})′ is defined by the formula ⟨Hf, ϕ⟩ = ⟨f, −Hϕ⟩∀ϕ ∈ B_{ap}^{ac} leading to the inversion formula H^{2}f = −f. We use our result to find an almost periodic (in x) harmonic function U(x, y), x ∈ R, y > 0 satisfying lim_{y→0+} U(x, y) = f, U(x, y) = o(1), y → ∞ uniformly ∀x ∈ R. Many other related results are proved.