Exercise 51, § 1

We identify $\mathbf{T}$ with the interval $[0,1[$. For $n\in \mathbf{Z}$, let $e_n$ be the unitary character of $\mathbf{T}$ defined by $t\mapsto$ exp(2$i\pi nt$). For every measure $\mu\in \mathscr{M}^1(\mathbf{T})$ and every integer $m\geqslant 1$, one sets

for the $m$-th symmetric partial sum of the Fourier series of $\mu$.

A finite family $(y_1, . . . , y_k)$ of real numbers is said to be $\mathbf{T}$-independent if $(1, y_1, . . . , y_k)$ is linearly independent over $\mathbf{Q}$. This amounts to saying that the equation

in $\mathbf{T}$, with $m_j\in \mathbf{Z}$, has for unique solution the zero family. For $t\in \mathbf{R}$, one denotes by $t-(y_1, . . . , y_k)$ the family $(t-y_j)_j$.

a) Let $n\geqslant 10^6$ be an integer. There exist real numbers $x_1, . . . ,x_n$ in $[0,1]$ such that

and such that the family $E_n= (x_1, . . . , x_n)$ is $\mathbf{T}$-independent.

One sets

$n$

$\mu_n=\frac{1}{n}\sum\varepsilon_{x_j}$.

$j=1$

It is a probability measure on $\mathbf{T}$.

b) Let $t\in \mathbf{R}$. One has

$\frac{1}{n}\sum_{j=1}^n|$ sin($\pi (1t-x_j)$)$|\geqslant \frac{1}{5}$ log $n$.

c) Let $t\in \mathbf{R}$ be such that the family $t-E_n$ is $\mathbf{T}$-independent. There exists $m\in \mathbf{N}^*$ such that

$|S_m(\mu_n)(t)|\geqslant \frac{2}{15}$ log $n$.

(apply the theorem of Kronecker, cf. exercise 34).

d) Let $t\in \mathbf{R}$ be such that the family $t-E_n$ is not $\mathbf{T}$-independent and let $m_1, . . . ,m_n$ be integers, not all zero, such that

in $\mathbf{T}$. If two of the integers $m_j$ are nonzero, there exists $m\geqslant 0$ such that

$|S_m(\mu_n)(t)|\geqslant \frac{1}{10}$ log $n$ (let $j$ be such that $m_j\not= 0$ and $|x_j-t|$ is maximal; note that the family $t-(x_i)_{i\not=j}$ is $\mathbf{T}$-independent and apply the theorem of Kronecker).

e) We retain the assumption of the preceding question, but we suppose that only one of the coefficients $m_j$ is nonzero, and that $t \notin E_n$. There exists $m\geqslant 0$ such that

$|S_m(\mu_n)(t)|\geqslant \frac{1}{10}$ log $n$

(observe that the family $m_j(t-x_i)_{i\not=j}$ is $\mathbf{T}$-independent and apply Kronecker's theorem).

f) We suppose that $t\in E_n$. There exists $m\geqslant 0$ such that

$|S_m(\mu_n)(t)|\geqslant \frac{1}{10}$ log $n$

(the family $(x_i)_{x_i\not=t}$ is $\mathbf{T}$-independent).

g) There exists an integer $M\geqslant 0$ such that, for all $t\in \mathbf{T}$, one has

sup$_{0\leqslant m\leqslant M}|S_m(\mu_n)(t)|\geqslant \frac{1}{20}$ log $n$.

h) There exists a function $f_n\in \mathscr{C}^{\infty}(\mathbf{T})$ and an integer $M\geqslant 0$ such that

Supp($f_n$)$\subset \bigcup_{j=1}^n[\frac{j}{n}-\frac{2}{10^4n},\frac{j}{n}+\frac{2}{10^4n}]$

sup$_{0\leqslant m\leqslant M}|S_m(f_n)(t)|\geqslant \frac{1}{40}$ log $n$, for all $t\in \mathbf{T}$.

i) There exists $M\geqslant 1$ and a finite family $(a_n)_{|n|\leqslant M}$ of complex numbers such that the function

satisfies

$\int_{\mathbf{T}}|g_M(t)|dt\leqslant 2$, inf$_{t\in\mathbf{T}0}$sup$_{\leqslant m\leqslant M}|S_m(g_M)(t)|\geqslant \frac{1}{40}$ log $n$.

j) There exists $\varphi \in L^1(\mathbf{T})$ such that $\widehat{\varphi}(h) = 0$ if $h <0$ and such that the series

diverges for all $t\in \mathbf{T}$ ("Kolmogorov's theorem").

(There exists a sequence $N_k$ of integers and a sequence of functions

such that $\|g_k\|_1\leqslant 2$ and

inf$_{t\in\mathbf{T}0}$sup$_{\leqslant m\leqslant N_k}|S_m(g_k)(t)|\geqslant 2^{2k}$.

Let $\widetilde{g}_k=e_{M_k}g_k$, where $M_k$ is a sufficiently large integer; put $\varphi =\sum_{k\geqslant 1}2^{-k}\widetilde{g}_k.)$