### Concerning one Paley-Wiener theorem

00:00 05-01-2011
&amp;lt;p style="TEXT-ALIGN: left; MARGIN: 0cm 0cm 0pt; unicode-bidi: embed; DIRECTION: ltr" class="MsoNormal"&amp;gt;&amp;lt;font face="Arial, Helvetica, sans-serif"&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black; FONT-SIZE: 14px"&amp;gt;In a joint work with S. Tikhonov, we prove weighted analogues of the Paley-Wiener theorem on integrability of the Hilbert transform of an integrable odd function which is monotone on \$mathbb{R}_+\$. This extends Hardy-Littlewood&amp;amp;#39;s and Flett&amp;amp;#39;s results to the case \$p=1\$ under the assumption of (general) monotonicity for an even/odd function.&amp;lt;br /&amp;gt;Further, relations between the Fourier transform and Hilbert transform will be discussed.&amp;lt;/span&amp;gt;&amp;lt;span style="FONT-FAMILY: &amp;amp;#39;Arial&amp;amp;#39;, &amp;amp;#39;sans-serif&amp;amp;#39;; COLOR: black"&amp;gt;&amp;lt;font size="2"&amp;gt;&amp;amp;nbsp;&amp;lt;/font&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;span style="COLOR: black"&amp;gt;&amp;lt;/span&amp;gt;&amp;lt;/font&amp;gt;&amp;lt;/p&amp;gt;