naa a22 4500 510811590 CHVBK 20180411083445.0 cr unu---uuuuu 180411e20130401xx s 000 0 eng 10.1134/S0081543813010185 doi (NATIONALLICENCE)springer-10.1134/S0081543813010185 Traces of the discrete Hilbert transform with quadratic phase [Elektronische Daten] [K. Oskolkov, M. Chakhkiev] 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 sup x 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., sup t 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 Schrödinger equation), and in quantum optics (Talbot's phenomenon). Pleiades Publishing, Ltd., 2013 Oskolkov K. Department of Mathematics, University of South Carolina, 29208, Columbia, SC, USA aut Chakhkiev M. Russian State Social University, ul. Stromynka 18, 107014, Moscow, Russia aut Proceedings of the Steklov Institute of Mathematics SP MAIK Nauka/Interperiodica 280/1(2013-04-01), 248-262 0081-5438 280:1<248 2013 280 11501 https://doi.org/10.1134/S0081543813010185 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1134/S0081543813010185 text/html Onlinezugriff via DOI NATIONALLICENCE 700 1- Oskolkov K. Department of Mathematics, University of South Carolina, 29208, Columbia, SC, USA aut NATIONALLICENCE 700 1- Chakhkiev M. Russian State Social University, ul. Stromynka 18, 107014, Moscow, Russia aut NATIONALLICENCE 773 0- Proceedings of the Steklov Institute of Mathematics SP MAIK Nauka/Interperiodica 280/1(2013-04-01), 248-262 0081-5438 280:1<248 2013 280 11501 Metadata rights reserved Springer special CC-BY-NC licence nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-springer