{"id":217787,"date":"2025-07-13T17:05:44","date_gmt":"2025-07-13T22:05:44","guid":{"rendered":"https:\/\/lifeboat.com\/blog\/2025\/07\/replicating-kolmogorovs-counterexample-for-fourier-series-in-context-of-fourier-transforms"},"modified":"2025-07-13T17:05:44","modified_gmt":"2025-07-13T22:05:44","slug":"replicating-kolmogorovs-counterexample-for-fourier-series-in-context-of-fourier-transforms","status":"publish","type":"post","link":"https:\/\/lifeboat.com\/blog\/2025\/07\/replicating-kolmogorovs-counterexample-for-fourier-series-in-context-of-fourier-transforms","title":{"rendered":"Replicating Kolmogorov\u2019s Counterexample for Fourier Series in Context of Fourier Transforms"},"content":{"rendered":"<p>It is a famous <i>result<\/i> of <a href=\"http:\/\/matwbn.icm.edu.pl\/ksiazki\/fm\/fm4\/fm4127.pdf\" rel=\"noreferrer\">Kolmogorov<\/a> that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More specifically, he exhibited an $f\\in L^{1}(\\mathbb{T})$ such that.<\/p>\n<p>\\begin{align*} \\sup_{N\\geq 1}\\left|S_{N}f(x)ight|=\\sup_{N\\geq 1}\\left|(f\\ast D_{N})(x)ight|=\\infty, \\qquad\\forall \\text{ a.e. } x\\in\\mathbb{T}, \\end{align*} where $S_{N}$ is the $N^{th}$ partial sum and $D_{N}$ is the $N^{th}$ Dirichlet kernel given below. \\begin{align*} S_{N}f(x):=\\sum_{\\left|night|\\leq N}\\widehat{f}(n)e^{2\\pi inx}, \\quad D_{N}(x):=\\dfrac{\\sin 2\\pi(N+\\frac{1}{2})x}{\\sin \\pi x} \\end{align*} and we identify $\\mathbb{T}$ with the unit interval $[0,1]$.<\/p>\n<p>I have read, for example pg. 118 in [Pinsky], that Kolmogorov\u2019s counterexample can be replicated in the context of the Fourier transform on the real line $\\mathbb{R}$, showing that $L^{1}$ pointwise Fourier inversion can fail quite horribly. If my understanding is correct, then the following claim is true:<\/p>\n","protected":false},"excerpt":{"rendered":"<p>It is a famous result of Kolmogorov that there exists a (Lebesgue) integrable function on the torus such that the partial sums of Fourier series of $f$ diverge almost everywhere (a.e.). More specifically, he exhibited an $f\\in L^{1}(\\mathbb{T})$ such that. \\begin{align*} \\sup_{N\\geq 1}\\left|S_{N}f(x)ight|=\\sup_{N\\geq 1}\\left|(f\\ast D_{N})(x)ight|=\\infty, \\qquad\\forall \\text{ a.e. } x\\in\\mathbb{T}, \\end{align*} where $S_{N}$ is the [\u2026]<\/p>\n","protected":false},"author":709,"featured_media":0,"comment_status":"open","ping_status":"closed","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[20],"tags":[],"class_list":["post-217787","post","type-post","status-publish","format-standard","hentry","category-futurism"],"_links":{"self":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/217787","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/users\/709"}],"replies":[{"embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/comments?post=217787"}],"version-history":[{"count":0,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/posts\/217787\/revisions"}],"wp:attachment":[{"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/media?parent=217787"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/categories?post=217787"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/lifeboat.com\/blog\/wp-json\/wp\/v2\/tags?post=217787"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}