Divergence and Almost Everywhere Convergence of Fourier Series

Use $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ with Haar measure $m(\mathbb{T})=1$ and the negative-sign Fourier coefficients and symmetric partial sums of Period-one Fourier coefficients, partial sums, and convolution on the torus. For $f\in L^1(\mathbb{T})$ define the Carleson maximal partial-sum operator by

$Cf(x):={\sup }_{N\in {\mathbb{Z}}_{\ge 0}}\mathrm{\mid }{S}_{N}f(x)\mathrm{\mid }\in [0,\mathrm{\infty }]\mathrm{.}$

Each coefficient is a finite integral independent of the representative of $f$, and each $S_{N}f$ is a continuous trigonometric polynomial. Thus $C$ depends only on the $L^1$ class, at every point. For every real $a$, the set $\{Cf>a\}={\bigcup }_{N\ge 0}\{\mathrm{\mid }{S}_{N}f\mathrm{\mid }>a\}$ is measurable.

Linearity of finite Fourier sums gives $C(f+g)\le Cf+Cg$ and $C(af)=\mathrm{\mid }a\mathrm{\mid }Cf$ for nonzero scalars $a$. Also $C0=0$; with $0\cdot \mathrm{\infty }:=0$ the homogeneity identity holds for $a=0$ as well. This is sublinearity with extended nonnegative values. Including $N=0$ retains the constant Fourier coefficient.

On $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ with Haar mass one, for each integer $N\ge 0$ and each prescribed $x_0\in \mathbb{T}$, let $T_{N,x_0}f=S_{N}f(x_0)$. Over either $\mathbb{R}$ or $\mathbb{C}$, on the continuous periodic functions with supremum norm,

$\mathrm{\parallel }{T}_{N,x_0}\mathrm{\parallel }=\mathrm{\parallel }{S}_N:C(\mathbb{T})\to C(\mathbb{T})\mathrm{\parallel }={\int }_{\mathbb{T}}\mathrm{\mid }{D}_N(t)\mathrm{\mid }dm(t)\mathrm{.}$

For $N\ge 1$ these norms are at least $\frac{1}{3\pi }\log (N+1)$, and hence are unbounded as $N\to \mathrm{\infty }$.

The harmonic lower estimate is included here to support the functional and operator norm assertion. It reuses the classical Lebesgue-constant calculation; it is not a separate growth theorem.

Continuity of a one-periodic real function guarantees convergence of its Fourier series at a prescribed point.

More precisely, assume DC. For every $x_0\in \mathbb{T}$ there exists $f\in C(\mathbb{T},\mathbb{R})$ such that

${\sup }_{N\ge 0}\mathrm{\mid }{S}_{N}f(x_0)\mathrm{\mid }=\mathrm{\infty }\mathrm{.}$

Assume countable choice and fix $1\le p<\mathrm{\infty }$. On the period-one torus with normalized Haar measure, suppose a finite constant $A\ge 0$ satisfies

$m\{Cg>\lambda \}\le A{\lambda }^{-p}\mathrm{\parallel }g{\mathrm{\parallel }}_p^p(g\in L^p(\mathbb{T}),\lambda >0)\mathrm{.}$

Then for every $f\in L^p(\mathbb{T})$, $S_{N}f(x)\to f(x)$ almost everywhere. The conclusion holds for any measurable representative of $f$.

For every integer $M\ge 1$ there exist a nonnegative trigonometric polynomial $g_M$ on $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ and a measurable set $A_M\subseteq \mathbb{T}$ such that, for normalized Haar measure,

$\mathrm{\parallel }{g}_M{\mathrm{\parallel }}_1=1,m(A_M)>1-2^{-M},{\inf }_{x\in A_M}{\sup }_{N\ge 1}\mathrm{\mid }{S}_N{g}_M(x)\mathrm{\mid }>2^M\mathrm{.}$

The supremum is of the same partial sums as Carleson maximal partial-sum operator. This is the integer version of Grafakos, Lemma 4.2.4, including nonnegativity from its construction. The proof is not supplied here. “Block” does not mean disjoint frequency support.

Grafakos's Lemma 4.2.2 aligns phases: if $1,x_1,\dots ,x_n$ are linearly independent over $\mathbb{Q}$, then for any unimodular $z_j$ and $\epsilon >0$ some integer $L$ satisfies $\mathrm{\mid }{e}^{2\pi iL{x}_j}-z_j\mathrm{\mid }<\epsilon$ for all $j$. Its Fourier-averaging argument is not established locally.

Lemma 4.2.3 constructs atomic probability measures ${\mu }_n$ with ${\sup }_{L\ge 1}\mathrm{\mid }{D}_L\ast {\mu }_n\mathrm{\mid }\ge c\log n$ almost everywhere, for an absolute $c>0$. Rational independence supplies simultaneous alignment of the kernel terms. Lemma 4.2.4 then selects a finite maximal truncation and smooths ${\mu }_n$ with a Fejer kernel. Positivity and mass one survive this smoothing, while the finitely many partial sums remain close enough to preserve the required height. These are descriptions of unproved source machinery, not local facts available as dependencies.

There exists $f\in L^1(\mathbb{T})$, where $\mathbb{T}=\mathbb{R}\mathrm{/}\mathbb{Z}$ has Haar mass one, such that

${\sup }_{N\ge 0}\mathrm{\mid }{S}_{N}f(x)\mathrm{\mid }=\mathrm{\infty }\text{for almost every }x\in \mathbb{T}\mathrm{.}$

Equivalently, the function in Carleson maximal partial-sum operator is infinite almost everywhere for this $f$. Every individual partial sum is finite. The Fourier series therefore diverges almost everywhere.

This records Grafakos, Theorem 4.2.1 and the explicit conclusion (4.2.13). No local proof is supplied, and the claim is not strengthened to divergence at every point.

The source forms a summable weighted series of the polynomials in Kolmogorov polynomial blocks — recorded construction lemma ‡. It chooses weights and polynomial degrees recursively. The large contribution of the current polynomial must exceed both the contribution of earlier polynomials and the tail; those two errors require separate estimates. A full-measure limsup of the good sets supplies arbitrarily large partial sums of the final integrable function. The polynomial construction and this summation argument remain external, so the linked record is a bibliographic mention rather than a logical prerequisite.

On the period-one torus with normalized Haar measure, for every $1<p<\mathrm{\infty }$ there is a finite constant $C_p$ such that

$\mathrm{\parallel }Cf{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }f{\mathrm{\parallel }}_p(f\in L^p(\mathbb{T})),$

where $C$ is Carleson maximal partial-sum operator. Consequently,

$S_{N}f(x)⟶f(x)\text{for almost every }x\in \mathbb{T}\mathrm{.}$

Carleson's original result concerns $p=2$; Hunt obtained the full open range. This is recorded literature, with no local proof of the maximal estimate. Laugesen's Theorem 8.7 and its omitted-proof discussion give the torus convergence statement and the required strong maximal estimate; the source's period $2\pi$ is rescaled by $t=2\pi x$ with normalized measure.

The norm here is $\mathrm{\parallel }{\sup }_N\mathrm{\mid }{S}_{N}f\mathrm{\mid }{\mathrm{\parallel }}_p$. Uniformly bounding the separate numbers $\mathrm{\parallel }{S}_{N}f{\mathrm{\parallel }}_p$ does not supply this estimate. The endpoint $p=1$ is excluded.

Assume countable choice and let $1<p<\mathrm{\infty }$. Suppose explicitly that a finite $C_p\ge 0$ satisfies $\mathrm{\parallel }Cg{\mathrm{\parallel }}_p\le C_p\mathrm{\parallel }g{\mathrm{\parallel }}_p$ for every $g\in L^p(\mathbb{T})$, with symmetric sums and normalized Haar measure. Then every $f\in L^p(\mathbb{T})$ satisfies $S_{N}f\to f$ almost everywhere.

Carleson–Hunt maximal bound and almost-everywhere convergence — recorded theorem ‡ records that the strong estimate is true. This proof establishes the implication from that estimate as an explicit hypothesis; it does not prove or discharge the estimate itself.

The Lacey–Thiele route to Carleson–Hunt maximal bound and almost-everywhere convergence — recorded theorem ‡ uses a decomposition in time and frequency. Lacey's survey works with a real-line model; it is not a local transference argument to the torus.

In §3, tiles are organized into trees. Lemma 3.6 reduces residual density and controls the total length of selected tree tops by inverse density. Lemma 3.9 reduces residual size with an inverse-square size bound on that total length. Lemma 3.11 bounds a tree contribution by its top length times its size and density. Matching density with squared size balances these estimates, and (3.13)–(3.16) leave scale contributions bounded by multiples of $\min \{2^{-n},2^n\}$, summable over $n\in \mathbb{Z}$.

The $L^p$ extension in §7 uses distributional estimates and interpolation. For the large testing-set case, its opening removes an exceptional set defined by a maximal function and separately treats tiles inside and outside that set. This is a sourced roadmap of one method. The density, size, tree and exceptional-set estimates have not been proved on this page.

Assume countable choice. The assertion in Kolmogorov almost-everywhere divergence — recorded theorem ‡ prevents extending Carleson–Hunt maximal bound and almost-everywhere convergence — recorded theorem ‡ to all of $L^1(\mathbb{T})$. This comparison relies on the recorded Kolmogorov existence theorem, whose proof is not supplied here.

Indeed, a weak $(1,1)$ estimate $m\{Cg>\lambda \}\le A\mathrm{\parallel }g{\mathrm{\parallel }}_1\mathrm{/}\lambda$ for all $g\in L^1$ and $\lambda >0$ would imply almost-everywhere convergence for every such $g$ by A weak maximal bound implies almost-everywhere Fourier convergence. That implication is incompatible with the recorded witness. A strong $(1,1)$ estimate would imply this weak estimate by Chebyshev-Markov inequality for the integral applied to $Cg$.

This is an interpretation of external literature, not an independently proved weak-endpoint theorem. Failure at one prescribed point for a continuous function is a different phenomenon from failure on a full-measure set for an integrable function.