Traces of the discrete Hilbert transform with quadratic phase

Abstract

For the function \(H:\mathbb{R}^2 \mapsto \mathbb{C}\), \(H: = (p.v.)\sum\nolimits_{n \in \mathbb{Z}\backslash \{ 0\} } {\tfrac{{\exp \left\{ {\pi i\left( {tn^2 + 2xn} \right)} \right\}}} {{2\pi in}}}\) of two real variables (t, x) ∈ ℝ2, we study the uniform moduli of continuity and the variations of the restrictions H|t and H|x onto the lines parallel to the coordinate axes x = 0 and t = 0. Smoothness of such restrictions is primarily determined by the Diophantine approximation of the fixed parameter. Generalized (weak) variations are also studied, and it is shown in particular that supx w4[H|x] < ∞ where w4 denotes the weak quartic variation. Previously it was known that uniformly in the parameter t ∈ ℝ, the restriction H|t is a function of bounded weak quadratic variation in the variable x, i.e., supt w2[H|t] < ∞. The function H has multiple applications: in the study of the spectra of uniform convergence (P.L. Ul’yanov’s problem), in the incomplete Gaussian sums (where it plays the role of the generating function), in the partial differential equations of mathematical physics (in the Cauchy problem for the Schrodinger equation), and in quantum optics (Talbot’s phenomenon).