1.0Microsoft Researchhttps://www.microsoft.com/en-us/researchJeff Runninghttps://www.microsoft.com/en-us/research/people/jeffrunn/rich600338<blockquote class="wp-embedded-content" data-secret="rjRT53NWZ7"><a href="https://www.microsoft.com/en-us/research/video/fourier-sampling-and-beyond/">Fourier Sampling and Beyond</a></blockquote><iframe sandbox="allow-scripts" security="restricted" src="https://www.microsoft.com/en-us/research/video/fourier-sampling-and-beyond/embed/#?secret=rjRT53NWZ7" width="600" height="338" title="“Fourier Sampling and Beyond” — Microsoft Research" data-secret="rjRT53NWZ7" frameborder="0" marginwidth="0" marginheight="0" scrolling="no" class="wp-embedded-content"></iframe><script type="text/javascript"> /* <![CDATA[ */ /*! This file is auto-generated */ !function(d,l){"use strict";l.querySelector&&d.addEventListener&&"undefined"!=typeof URL&&(d.wp=d.wp||{},d.wp.receiveEmbedMessage||(d.wp.receiveEmbedMessage=function(e){var t=e.data;if((t||t.secret||t.message||t.value)&&!/[^a-zA-Z0-9]/.test(t.secret)){for(var s,r,n,a=l.querySelectorAll('iframe[data-secret="'+t.secret+'"]'),o=l.querySelectorAll('blockquote[data-secret="'+t.secret+'"]'),c=new RegExp("^https?:$","i"),i=0;i<o.length;i++)o[i].style.display="none";for(i=0;i<a.length;i++)s=a[i],e.source===s.contentWindow&&(s.removeAttribute("style"),"height"===t.message?(1e3<(r=parseInt(t.value,10))?r=1e3:~~r<200&&(r=200),s.height=r):"link"===t.message&&(r=new URL(s.getAttribute("src")),n=new URL(t.value),c.test(n.protocol))&&n.host===r.host&&l.activeElement===s&&(d.top.location.href=t.value))}},d.addEventListener("message",d.wp.receiveEmbedMessage,!1),l.addEventListener("DOMContentLoaded",function(){for(var e,t,s=l.querySelectorAll("iframe.wp-embedded-content"),r=0;r<s.length;r++)(t=(e=s[r]).getAttribute("data-secret"))||(t=Math.random().toString(36).substring(2,12),e.src+="#?secret="+t,e.setAttribute("data-secret",t)),e.contentWindow.postMessage({message:"ready",secret:t},"*")},!1)))}(window,document); //# sourceURL=https://www.microsoft.com/en-us/research/wp-includes/js/wp-embed.min.js /* ]]> */ </script> https://www.microsoft.com/en-us/research/wp-content/uploads/2016/02/fourier-sampling-and-beyond-1.jpg320240The Fast Fourier Transform (FFT) is a fundamental algorithm that computes the Discrete Fourier Transform of an n-dimensional signal in O(n log n) time. It is unknown whether the running time can be improved in general. However, in applications such as image, audio, and video compression where the output is “sparse” (i.e., k n coordinates […]