A sum of exponentials of the form , where the are distinct integers is called an idempotent trigonometric polynomial (because the convolution of with itself is ) or, simply, an idempotent. We show that for every and every set of the torus with there are idempotents concentrated on in the sense. More precisely, for each there is an explicitly calculated constant so that for each with and one can find an idempotent such that the ratio is greater than . This is in fact a lower bound result and, though not optimal, it is close to the best that our method gives. We also give both heuristic and computational evidence for the still open problem of whether the concentration phenomenon fails to occur when
Une somme d’exponentielles de la forme , où les sont des entiers distincts, est appelée un polynôme trigonométrique idempotent (car ) ou, simplement, un idempotent. Nous prouvons que pour tout réel , et tout avec il existe des idempotents concentrés sur au sens de la norme . Plus précisément, pour tout nous calculons explicitement une constante telle que pour tout avec , et tout réel , on puisse construire un idempotent tel que le quotient soit supérieur à . Ceci est en fait un théorème de minoration qui, bien que non optimal, est proche du meilleur résultat que notre méthode puisse fournir. Nous présentons également des considérations heuristiques et aussi numériques concernant le problème (toujours ouvert) de savoir si le phénomène de concentration a lieu ou non pour .
Keywords: Idempotents, idempotent trigonometric polynomials, $L^{p}$ norms, Dirichlet kernel, concentrating norms, sums of exponentials, $L^{1}$ concentration conjecture, weak restricted operators.
Mot clés : idempotents, polynômes trigonométriques idempotents, normes $L^{p}$, noyau de Dirichlet, concentration de normes, sommes d’exponentielles, conjecture de concentration en norme $L^{1}$, opérateurs faiblement restreints.
Anderson, B. 1; Ash, J. M. 2; Jones, R. L. 3; Rider, D. G. 4; Saffari, B. 5
@article{AIF_2007__57_5_1377_0, author = {Anderson, B. and Ash, J.~M. and Jones, R.~L. and Rider, D. G. and Saffari, B.}, title = {Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms}, journal = {Annales de l'Institut Fourier}, pages = {1377--1404}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {57}, number = {5}, year = {2007}, doi = {10.5802/aif.2298}, mrnumber = {2364133}, zbl = {1133.42004}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.2298/} }
TY - JOUR AU - Anderson, B. AU - Ash, J. M. AU - Jones, R. L. AU - Rider, D. G. AU - Saffari, B. TI - Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms JO - Annales de l'Institut Fourier PY - 2007 SP - 1377 EP - 1404 VL - 57 IS - 5 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.2298/ DO - 10.5802/aif.2298 LA - en ID - AIF_2007__57_5_1377_0 ER -
%0 Journal Article %A Anderson, B. %A Ash, J. M. %A Jones, R. L. %A Rider, D. G. %A Saffari, B. %T Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms %J Annales de l'Institut Fourier %D 2007 %P 1377-1404 %V 57 %N 5 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.2298/ %R 10.5802/aif.2298 %G en %F AIF_2007__57_5_1377_0
Anderson, B.; Ash, J. M.; Jones, R. L.; Rider, D. G.; Saffari, B. Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms. Annales de l'Institut Fourier, Volume 57 (2007) no. 5, pp. 1377-1404. doi : 10.5802/aif.2298. https://aif.centre-mersenne.org/articles/10.5802/aif.2298/
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