Exercise 46, § 1

For every integer $k\geqslant 0$, let $D_k\subset [0,1]$ be the set of numbers $j/2^k$ for integer $j$ such that $0\leqslant j\leqslant 2^k$. Let D be the union of the sets $D_k$ for $k\geqslant 0$. For every $k\geqslant 1$, and every $t\in D_{k+1}-D_k$, let us denote by $t_+$ (resp. $t_-$) the smallest element of $D_{k+1}$ such that $t < t_+$ (resp. the greatest element of $D_{k+1}$ such that $t_-< t$); $t_+$ and $t_-$ belong to $D_k$.

a) Let $\sigma : D\rightarrow [0,1]$ be a strictly increasing function such that $\sigma (0) = 0$ and $\sigma (1) = 1$, whose image is dense in $[0,1]$. Then there exists a unique strictly increasing homeomorphism of $[0,1]$ onto itself which extends $\sigma$.

Let $f: [0,1]\rightarrow \mathbf{R}$ be a continuous function. A level $k$ sketch of $f$ is a strictly increasing mapping $\sigma : D_k\rightarrow [0,1]$ such that $\sigma (0) = 0$, $\sigma (1) = 1$ and such that for every integer $j$ such that $0\leqslant j <2^k$, one of the following conditions holds:

(i) $f(\sigma (j/2^k))\not=f(\sigma ((j+ 1)/2^k))$;

(ii) there exists an open interval I contained in $]\sigma (j/2^k), \sigma (j+ 1)/2^k[$ such that, for every $t\in I$, we have

We denote by $t_{\sigma}$ the smallest strictly positive element of $D_k$ such that

inf $|\sigma (t_{\sigma})-\sigma (t')|=$ sup inf $|\sigma (t)-\sigma (t')|$.

$t'<t_{\sigma}t\in D_kt'<t$

A level $k+ 1$ sketch $\tau$ is said to be a good extension of a level $k$ sketch $\sigma$ if $\tau$ extends $\sigma$ and if, for every $t\in D_{k+1}-D_k$, we have

1

2

if $t_-\not=t_{\sigma}$.

b) There exists an increasing homeomorphism $\sigma$ of $[0,1]$ onto itself such that

(i) For every integer $k\geqslant 1$, the restriction $\sigma_k$ of $\sigma$ to $D_k$ is a level $k$ sketch;

(ii) For every integer $k\geqslant 1$, the mapping $\sigma_{k+1}$ is a good extension of $\sigma_k$.

c) Suppose that $f(0) =f(1)$. Let $\sigma$ denote the increasing homeomorphism of $[0,1]$ defined by a mapping $\sigma : D\rightarrow [0,1]$ satisfying the conditions of the preceding question. There exists a constant $C\geqslant 0$ such that the continuous function $f\circ \sigma$, identified with a continuous function on $\mathbf{T}$, satisfies

for every $n\in \mathbf{Z}-\{0\}$ (“Sahakyan's Theorem”; use Exercise 2 of II, p. 262 and the preceding exercise.) d) $\P$ There exists an increasing homeomorphism $\sigma$ of $[0,1]$ onto itself such that the Fourier series of $f\circ \sigma$ converges symmetrically in $\mathscr{C}(\mathbf{T})$ (“Bohr-Pál Theorem”).

e) $\P \P$ Is Sahakyan's Theorem valid for continuous complex-valued functions?

f) $\P \P$ Does there exist a function $f:\mathbf{R}/\mathbf{Z}\rightarrow \mathbf{C}$ whose image has nonempty interior and which satisfies the condition sup$_{h\in\mathbf{Z}-\{0\}}|h||\widehat{f}(h)|<+\infty$?