# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Quasi-discrete spectrum and the Wiener-Wintner ergodic theorem for polynomials. (Spectre quasi-discret et théorème ergodique de Wiener-Wintner pour les polynômes.) (French) Zbl 0803.47013

Let $\left({\Omega },\mu \right)$ be a separable probability space, $T:{\Omega }\to {\Omega }$ be measure-preserving. Fix $f\in {L}^{1}$. The ergodic theorem of Wiener-Wintner asserts that for $\mu$-a.e. $\omega$,

$\frac{1}{n}\sum _{k=0}^{n-1}exp\left(ik\alpha \right)·f\left({T}^{k}\omega \right)\phantom{\rule{1.em}{0ex}}\text{converges}\phantom{\rule{4.pt}{0ex}}\text{for}\phantom{\rule{4.pt}{0ex}}\text{all}\phantom{\rule{4.pt}{0ex}}\alpha \in ℝ·$

The author uses the Van der Corput inequality to show the following generalization: For $\mu$-a.e. $\omega$,

$\frac{1}{n}\sum _{k=0}^{n-1}\phi \left(P\left(k\right)\right)·f\left({T}^{k}\omega \right)\phantom{\rule{1.em}{0ex}}\text{converges}$

for all real polynomials $P$ and all continuous periodic functions $\phi$ on $ℝ$. If, in addition, $T$ is weakly mixing and $\int fd\mu =0$, then for $\mu$-a.e. $\omega$, for all $M\in ℕ$ and all continuous periodic $\phi$,

$\underset{P}{sup}\left|\frac{1}{n}\sum _{k=0}^{n-1}\phi \left(P\left(k\right)\right)·f\left({T}^{k}\omega \right)\right|\to 0,$

where the supremum is taken over all real polynomials of degree bounded by $M$.

The proof of the Wiener-Wintner theorem is based on an equivalence which the author generalizes in the following way: Let ${E}_{0}$ be the set of eigenvalues of $T$ (as an operator on ${L}^{2}$) and, for $m\ge 1$, set ${E}_{m}:=\left\{f\in {L}^{2}:|f|\equiv 1,\phantom{\rule{4pt}{0ex}}f\circ T·\overline{f}\in {E}_{m-1}\right\}$. Suppose the linear span of ${\bigcup }_{m}{E}_{m}$ is dense in ${L}^{2}$, $f\in {L}^{1}$ and $M\ge 1$. Then $\int fgd\mu =0$ holds for all $g\in {E}_{M}$ if and only if for $\mu$-a.e. $\omega$, for all real polynomials of degree less than or equal to $M$ and every continuous periodic $\phi$,

$\frac{1}{n}\sum _{k=0}^{n-1}\phi \left(P\left(k\right)\right)·f\left({T}^{k}\omega \right)\to 0·$

.

##### MSC:
 47A35 Ergodic theory of linear operators 28D05 Measure-preserving transformations