{"id":47,"date":"2023-05-20T22:53:16","date_gmt":"2023-05-20T21:53:16","guid":{"rendered":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/?p=47"},"modified":"2023-05-27T23:25:31","modified_gmt":"2023-05-27T22:25:31","slug":"taylors-theorem","status":"publish","type":"post","link":"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/20\/taylors-theorem\/","title":{"rendered":"Taylor&#8217;s Theorem"},"content":{"rendered":"<h3><em>1. Weierstrass Approximation Theorem<\/em><\/h3>\n<p>We want to show that The set of polynomials\u00a0is <strong>dense<\/strong> in the space of continuous functions. We shall firstly define the Bernstein Polynomials as<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"71\" class=\"alignnone size-full wp-image-92\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation1-1.png\" alt=\"\" \/><\/p>\n<p>where <em>f<\/em> is a continuous function with domain [0,1] and we can thus see that it is uniformly continuous.This means that in \u03b4-\u03b5 language :<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"50\" class=\"alignnone size-full wp-image-93\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation2-1.png\" alt=\"\" \/><\/p>\n<p>here we manually set an interval in the domain. However, we are interested in the general relationship between two outputs without limitation ( other than the distance between two input must be in the range [0,1] ). So we want to now what happens when the |<em>x -\u03be<\/em>| is larger than <em>\u03b4<\/em>.<\/p>\n<p>We recall the definition of norms in linear algebra. For finite vector space, the infinite norm is defined as:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"53\" class=\"alignnone size-full wp-image-94\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation3-1.png\" alt=\"\" \/><\/p>\n<p>It is easy to understand that this gives the largest entry (in terms of magnitude) of a vector. <strong><em>Note that this is true even if there exists more than one maximum entry.<\/em> <\/strong>We expand this property into function space and define <em>M<\/em> to be the infinite norm of <em>f(x)<\/em>. Then for |<em>x -\u03be<\/em>| is larger than <em>\u03b4\u00a0<\/em>we write:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"46\" class=\"alignnone size-full wp-image-95\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation4-1.png\" alt=\"\" \/><\/p>\n<p>Then combine (3) and (4) we obtain<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"56\" class=\"alignnone size-full wp-image-96\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation5-1.png\" alt=\"\" \/><\/p>\n<p>We now want to use this relation to show that Bernstein polynomials can be used to approximate <em>f(x)<\/em>. We notice that the Bernstein polynomial of <em>f(\u03be)<\/em> is just <em>f(\u03be)<\/em>, where binomial expansion has been used to obtain this result<em>. <\/em>Then, using the fact that Bernstein polynomial is linear for\u00a0<em>f(x),<\/em><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1206\" height=\"190\" class=\"alignnone size-full wp-image-98\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation67.png\" alt=\"\" \/>In the second line above we putted the first term of the function into the expression of the polynomial and it came out will some calculation. Obviously, we need to set <em>x=\u03be <\/em>to proceed and this yields<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" width=\"1375\" height=\"53\" class=\"alignnone size-full wp-image-97\" src=\"http:\/\/blogs.imperial.ac.uk\/cocteaupedia\/files\/2023\/05\/equation8.png\" alt=\"\" \/><\/p>\n<p>What does this means? Remember that we have the freedom to make <em>n<\/em> as large as we want. This means that<em>\u00a0<\/em>we can make the difference between Bernstein polynomials and our function as arbitrarily small as we want by approaching <em>n<\/em> to infinity. \u00a0i.e., Bernstein \u00a0polynomials converges to our <em>f(x)<\/em>.<\/p>\n<p>[Literature: Matt Young, MATH 328 Notes, Queen&#8217;s University at Kingston, 2006]<\/p>\n<h3><em>2. Taylor&#8217;s Theorem<\/em><\/h3>\n<p>Now we are ready to prove Taylor&#8217;s Theorem. Knowing polynomials span the space of continuous function, we now assume that the function is also\u00a0<em><strong>infinitely differentiable<\/strong><\/em>. In this special case, instead of using Bernstein polynomials, we want to use the basis consists of <em>1<\/em>,<em>x<\/em>,&#8230;,<em>x<sup>n<\/sup><\/em>. Matching the value of our series and our function at a particular point yields the ordinary form of single variable Taylor series.<\/p>\n<h3><i>3. Vector fields Taylor Expansion<\/i><\/h3>\n<p>To be continued&#8230;<\/p>\n<p>&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>1. Weierstrass Approximation Theorem We want to show that The set of polynomials\u00a0is dense in the space of continuous functions. We shall firstly define the Bernstein Polynomials as where f is a continuous function with domain [0,1] and we can thus see that it is uniformly continuous.This means that in \u03b4-\u03b5 language : here we [&hellip;]<\/p>\n","protected":false},"author":1741,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[8],"tags":[],"class_list":["post-47","post","type-post","status-publish","format-standard","hentry","category-analysis"],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v27.4 - https:\/\/yoast.com\/product\/yoast-seo-wordpress\/ -->\n<title>Taylor&#039;s Theorem - Cocteaupedia<\/title>\n<meta name=\"description\" content=\"Weierstrass Approximation Theorem, Taylor&#039;s Theorem\" \/>\n<meta name=\"robots\" content=\"index, follow, max-snippet:-1, max-image-preview:large, max-video-preview:-1\" \/>\n<link rel=\"canonical\" href=\"https:\/\/blogs.imperial.ac.uk\/cocteaupedia\/2023\/05\/20\/taylors-theorem\/\" \/>\n<meta property=\"og:locale\" content=\"en_GB\" \/>\n<meta property=\"og:type\" content=\"article\" \/>\n<meta property=\"og:title\" content=\"Taylor&#039;s Theorem - 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