https://en.formulasearchengine.com/index.php?action=history&feed=atom&title=Calabi_triangle 2026-05-24T19:44:41Z Revision history for this page on the wiki MediaWiki 1.43.0-wmf.28 https://en.formulasearchengine.com/index.php?title=Calabi_triangle&diff=28217&oldid=prev 2013-05-28T21:48:14Z

<p>Corrected statement of the ratio</p> <p><b>New page</b></p><div>In [[mathematical analysis]], the concept of a &#039;&#039;&#039;mean-periodic function&#039;&#039;&#039; is a generalization introduced by [[Jean Delsarte]], of the concept of a [[periodic function]].[http://www.math.tifr.res.in/~publ/ln/tifr15.pdf]<br /> <br /> Consider a [[complex number|complex]]-valued function &#039;&#039;ƒ&#039;&#039; of a [[real number|real]] variable. The function &#039;&#039;ƒ&#039;&#039; is periodic with period &#039;&#039;a&#039;&#039; precisely if for all real &#039;&#039;x&#039;&#039;, we have &#039;&#039;ƒ&#039;&#039;(&#039;&#039;x&#039;&#039;)&amp;nbsp;&amp;minus;&amp;nbsp;&#039;&#039;ƒ&#039;&#039;(&#039;&#039;x&#039;&#039;&amp;nbsp;&amp;minus;&amp;nbsp;&#039;&#039;a&#039;&#039;)&amp;nbsp;=&amp;nbsp;0. This can be written as<br /> <br /> : &lt;math&gt; \int f(x-y) \, d\mu(y) = 0\qquad\qquad(1) &lt;/math&gt;<br /> <br /> where &lt;math&gt;\mu&lt;/math&gt; is the difference between the [[Dirac delta function|Dirac measures]] at&amp;nbsp;0&amp;nbsp;and&amp;nbsp;&#039;&#039;a&#039;&#039;. A mean-periodic function is a function &#039;&#039;ƒ&#039;&#039; satisfying (1) for some nonzero measure &lt;math&gt;\mu&lt;/math&gt; with compact (hence bounded) support.<br /> <br /> == External links ==<br /> <br /> * [http://www.math.tifr.res.in/~publ/ln/tifr15.pdf Lectures on Mean Periodic Functions, by J. P. Kahane]<br /> <br /> [[Category:Mathematical analysis]]<br /> <br /> {{mathanalysis-stub}}</div>

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