Divergence and Almost Everywhere Convergence of Fourier Series

Assume the Axiom of Countable Choice for the torus $L^1$ integrals. 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 $f$.

Assume the Axiom of Choice and let $1<p<\mathrm{\infty }$. Every complex $f\in L^p(\mathbb{T})$ satisfies $S_{N}f\to f$ almost everywhere, with symmetric partial sums and normalized Haar measure on the period-one torus. The conclusion holds for every measurable representative of $f$.

Assume the Axiom of Choice (The Axiom of Choice) for the Fourier and measure interfaces of the following proved suppliers. The local conclusion is Carleson hunt maximal inequality on the torus, a strong maximal estimate for symmetric partial sums on the normalized period-one torus, at each $1<p<\mathrm{\infty }$.

The real-line tile argument uses Carleson density selection and Carleson size selection to select forests and Carleson single tree estimate to bound a tree contribution. The exceptional-set argument and distribution bootstrap of Hunt exceptional set and distribution estimates give finite tile models uniform restricted weak type at every exponent strictly between one and infinity. Then Carleson restricted weak interpolation gives uniform strong bounds at the desired exponent.

The packet averaging and reconstruction argument in Wave packet model dominates the linearised carleson operator passes those model bounds to the real-line Carleson maximal operator on Schwartz inputs. Finally Carleson real line to torus transfer uses a slowly varying Schwartz window on trigonometric polynomials, periodic averaging, Fejer approximation and monotone convergence to obtain the torus estimate. This supplies the transference argument as well as the time-frequency model estimates. The local theorem assembles these interfaces; the assumption AC is inherited from them.

Assume the Axiom of Choice (The Axiom of Choice). The local theorem Kolmogorov lone fourier series diverges almost everywhere supplies an integrable function with unbounded symmetric Fourier partial sums almost everywhere. Thus a convergence theorem for every $L^1(\mathbb{T})$ function would be false.

In particular there is no finite $A\ge 0$ such that $m\{Cg>\lambda \}\le A\mathrm{\parallel }g{\mathrm{\parallel }}_1\mathrm{/}\lambda$ for all $g\in L^1$ and $\lambda >0$. Such a weak $(1,1)$ estimate would imply almost-everywhere convergence for every such $g$ by A weak maximal bound implies almost-everywhere Fourier convergence at $p=1$. Applying that conclusion to the supplied witness contradicts its almost-everywhere unboundedness, because a convergent complex sequence is bounded and the union of the two exceptional null sets remains null.

A strong $(1,1)$ estimate $\mathrm{\parallel }Cg{\mathrm{\parallel }}_1\le B\mathrm{\parallel }g{\mathrm{\parallel }}_1$ with finite $B$ would imply the forbidden weak estimate by Chebyshev-Markov inequality for the integral applied to $Cg$, so no such strong estimate exists either. All operators here use symmetric partial sums on the normalized period-one torus. AC supplies the witness and implies the countable choice needed by the maximal convergence principle. Failure at one prescribed point for a continuous function is a different assertion from this failure on a full-measure set for an integrable function.