Sigui \\(f \\in L^2(\\mathbb{R})\\). Llavors, per la teoria de la transformada de Fourier, \\(\\hat{f} \\in L^2(\\mathbb{R})\\), i per tant \u00e9s localment integrable, \u00e9s a dir, \\(\\hat{f} \\in L^1_{\\text{loc}}(\\mathbb{R})\\).<\/p>\n
Aix\u00f2 ens permet definir, per a qualsevol \\(R > 0\\), la seg\u00fcent funci\u00f3:<\/p>\n
$$
\nS_R f(x) = \\int_{-R}^{R} \\hat{f}(\\xi)\\, e^{2\\pi i x \\xi}\\, d\\xi
\n$$<\/p>\n
Aquesta expressi\u00f3 representa les sumes parcials de Fourier<\/strong>\u00a0de \\(f\\), i constitueix una extensi\u00f3 natural del concepte cl\u00e0ssic de sumes parcials en les s\u00e8ries de Fourier. Ens preguntem ara si aquestes sumes convergeixen cap a \\(f\\) en la norma de \\(L^p(\\mathbb{R})\\), \\(1<p<\\infty\\) i que podem dir quan \\(p=1\\).<\/p>\n Observeu que aquest tipus de converg\u00e8ncia no implica necess\u00e0riament la converg\u00e8ncia puntual gaireb\u00e9 per tot, ni la converg\u00e8ncia uniforme. Tot i aix\u00ed, \u00e9s fonamental per a l\u2019an\u00e0lisi funcional, la teoria de senyals i altres aplicacions.<\/p>\n El ingredients fonamentals per obtenir la resposta a les preguntes proposades s\u00f3n la transforma de Hilbert i els teoremes de Kolmogorov\u00a0 i de F. Riesz, les quals caldr\u00e0 estudiar-los en detall.<\/p>\n","protected":false},"excerpt":{"rendered":" Sigui \\(f \\in L^2(\\mathbb{R})\\). Llavors, per la teoria de la transformada de Fourier, \\(\\hat{f} \\in L^2(\\mathbb{R})\\), i per tant \u00e9s localment integrable, \u00e9s a dir, \\(\\hat{f} \\in L^1_{\\text{loc}}(\\mathbb{R})\\). Aix\u00f2 ens permet definir, per a qualsevol \\(R > 0\\), la seg\u00fcent funci\u00f3: $$ S_R f(x) = \\int_{-R}^{R} \\hat{f}(\\xi)\\, e^{2\\pi i x \\xi}\\, d\\xi $$ Aquesta expressi\u00f3 […]<\/p>\n","protected":false},"author":71,"featured_media":0,"comment_status":"closed","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[29],"tags":[37],"class_list":["post-211","post","type-post","status-publish","format-standard","hentry","category-analisi-matematica","tag-joaquim-martin"],"_links":{"self":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/211","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\/71"}],"replies":[{"embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/comments?post=211"}],"version-history":[{"count":4,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/211\/revisions"}],"predecessor-version":[{"id":220,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/posts\/211\/revisions\/220"}],"wp:attachment":[{"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/media?parent=211"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/categories?post=211"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/mat.uab.cat\/web\/tfg\/wp-json\/wp\/v2\/tags?post=211"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}