{"id":259,"date":"2025-07-03T11:26:43","date_gmt":"2025-07-03T10:26:43","guid":{"rendered":"https:\/\/mat.uab.cat\/web\/tfg\/?p=259"},"modified":"2025-07-03T17:23:05","modified_gmt":"2025-07-03T16:23:05","slug":"grau-topologic-i-series-de-fourier","status":"publish","type":"post","link":"https:\/\/mat.uab.cat\/web\/tfg\/grau-topologic-i-series-de-fourier\/","title":{"rendered":"Grau topol\u00f2gic i s\u00e8ries de Fourier"},"content":{"rendered":"\n<p style=\"text-align: justify\">Sigui \\(\\mathbb{S}^1\\) el cercle unitat. Donada una funci\u00f3 \\(f:\\mathbb{S}^1\\to \\mathbb{S}^1\\) suficientment regular (per exemple, cont\u00ednua), es pot definir el seu grau topol\u00f2gic \\(\\textrm{deg}\\,f\\) que, informalment, \u00e9s el nombre total de vegades que la imatge \\(f(\\mathbb{S}^1)\\) completa un gir al cercle en sentit antihorari (per tant, \\(\\textrm{deg}\\,f\\) \u00e9s un enter). <a href=\"#my-video\">Exemple gr\u00e0fic<\/a>.<\/p>\n<p style=\"text-align: justify\">Sorprenentment, el grau topol\u00f2gic d&#8217;una funci\u00f3 \\(f\\) prou regular (com ara cont\u00ednuament derivable) es pot calcular a partir dels coeficients de Fourier \\(\\{ \\hat{f}(n)\\}\\) de \\(f\\) amb la identitat<br \/>$$<br \/>\\textrm{deg}\\,f = \\sum_{n\\in\\mathbb{Z}} n |\\hat{f}(n)|^2,<br \/>$$<br \/>on la s\u00e8rie \u00e9s absolutament convergent, ja que \\(f\\in C^1(\\mathbb{S}^1)\\).<\/p>\n<p style=\"text-align: justify\">A partir d&#8217;aquesta identitat tan interessant s&#8217;obre un ventall molt ric de possibles l\u00ednies d\u2019estudi [1, 3], algunes de les quals poden resultar especialment adequades per a projectes de TFG:<\/p>\n\n\n<ul class=\"wp-block-list\">\n<li><p style=\"text-align: justify\">Es pot definir el concepte de grau topol\u00f2gic per a funcions no cont\u00ednues a partir de la identiat anterior? Per exemple, en determinats espais de Sobolev o en l&#8217;espai de les funcions \\(VMO\\)? Una proposta de TFG \u00e9s l&#8217;estudi de l&#8217;extensi\u00f3 del concepte de grau topol\u00f2gic a diversos espais de funcions [3, 6, 7]. <\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">En els casos en els quals \\(f\\) \u00e9s cont\u00ednua per\u00f2 \\(\\displaystyle \\sum_{n\\in\\mathbb{Z}} |n| |\\hat{f}(n)|^2=\\infty\\), es pot caracteritzar \\(\\textrm{deg}\\,f\\) a partir d&#8217;algun m\u00e8tode de sumaci\u00f3 alternatiu de la s\u00e8rie \\(\\displaystyle \\sum_{n\\in\\mathbb{Z}} n |\\hat{f}(n)|^2\\)? Per exemple, per a funcions \\(\\alpha-\\)H\u00f6lder cont\u00ednues amb \\(\\alpha&gt;\\frac{1}{3}\\) es t\u00e9 que $$ \\textrm{deg}\\,f = \\lim_{t\\to 0}\\sum_{n\\in\\mathbb{Z}} |\\hat{f}(n)|^2 \\frac{\\sin nt}{t}. $$ Una proposta de TFG \u00e9s l&#8217;estudi dels diversos m\u00e8todes de sumaci\u00f3 que permeten recuperar el grau topol\u00f2gic d&#8217;una funci\u00f3 en espais de funcions menys regulars [3, 4, 5, 6] (aquesta proposta est\u00e0 molt relacionada amb l&#8217;anterior). <\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">\u00c9s la identitat \\(\\textrm{deg}\\,f = \\displaystyle\\sum_{n\\in\\mathbb{Z}} n |\\hat{f}(n)|^2\\) certa per a totes les funcions cont\u00ednues? La resposta \u00e9s negativa. Una proposta de TFG \u00e9s l&#8217;estudi del contraexemple donat en l&#8217;article de recerca [2] (<strong>dificultat molt alta<\/strong>). <\/p><\/li>\n<\/ul>\n\n\n\n<h3>Bibliografia (orientativa)<\/h3>\n\n\n\n<ol class=\"wp-block-list\">\n<li><p style=\"text-align: justify\">H. Brezis, <em>New questions related to the topological degree. <\/em>The unity of mathematics. In honor of the ninetieth birthday of I. M. Gelfand, pp. 137-171. Birkh\u00e4user, Boston, MA, 2006.<\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">J. Bourgain and G. Kozma,<em> One cannot hear the winding number<\/em>, J. Eur. Math. Soc. <strong>9<\/strong> (4) (2007), 637-658.<\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">H. Brezis and P. Mironescu, <em>Sobolev maps to the circle. From the perspective of analysis, geometry, and topology<\/em>. Birkh\u00e4user, New York, 2021.<\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">J.-P. Kahane, <em>Winding numbers and Fourier series<\/em>, Proc. Steklov Inst. Math. <strong>273<\/strong> (2011), 191-195.<\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">J.-P. Kahane, <em>Winding numbers and summation processes<\/em>, Complex Var. Elliptic Equ. <strong>55<\/strong> (8-10) (2010), 911-922.<\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">J. Korevaar, <em>On a question of Br\u00e9zis and Nirenberg concerning the degree of circle maps<\/em>, Sel. Math., New Ser. <strong>5<\/strong> (1999), 107-122.<\/p><\/li>\n\n\n\n<li><p style=\"text-align: justify\">L. Nirenberg, <em>Degree theory beyond continuous maps<\/em>, CWI Q. <strong>9<\/strong> (1-2) (1996), 113-120.<\/p><\/li>\n<\/ol>\n\n\n\n<figure id=\"my-video\" class=\"wp-block-video\"><video controls src=\"http:\/\/mat.uab.cat\/web\/tfg\/wp-content\/uploads\/sites\/42\/2025\/07\/winding_number_ex1-1.mp4\"><\/video><figcaption class=\"wp-element-caption\"> Demostraci\u00f3 gr\u00e0fica de \\(\\textrm{deg}\\, f=3\\) per a la funci\u00f3 \\(f(e^{it}) = e^{3it}\\).<\/figcaption><\/figure>\n\n\n\n<figure class=\"wp-block-video\"><video controls src=\"https:\/\/mat.uab.cat\/web\/tfg\/wp-content\/uploads\/sites\/42\/2025\/07\/winding_number_ex2.mp4\"><\/video><figcaption class=\"wp-element-caption\"> Demostraci\u00f3 gr\u00e0fica de \\(\\textrm{deg}\\, f=1\\) per a la funci\u00f3 \\(f(e^{it}) = e^{i(t+\\sin(3t))}\\).<\/figcaption><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Sigui \\(\\mathbb{S}^1\\) el cercle unitat. Donada una funci\u00f3 \\(f:\\mathbb{S}^1\\to \\mathbb{S}^1\\) suficientment regular (per exemple, cont\u00ednua), es pot definir el seu grau topol\u00f2gic \\(\\textrm{deg}\\,f\\) que, informalment, \u00e9s el nombre total de vegades que la imatge \\(f(\\mathbb{S}^1)\\) completa un gir al cercle en sentit antihorari (per tant, \\(\\textrm{deg}\\,f\\) \u00e9s un enter). Exemple gr\u00e0fic. Sorprenentment, el grau topol\u00f2gic [&hellip;]<\/p>\n","protected":false},"author":73,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29,31],"tags":[38],"class_list":["post-259","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","category-geometria-i-topologia","tag-alberto-debernardi-pinos"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/259","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/users\/73"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=259"}],"version-history":[{"count":25,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/259\/revisions"}],"predecessor-version":[{"id":296,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/259\/revisions\/296"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=259"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=259"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=259"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}