Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms

Anderson, B.; Ash, J. M.; Jones, R. L.; Rider, D. G.; Saffari, B.

doi:10.5802/aif.2298

Exponential sums with coefficients

0

1

and concentrated

L^{p}

norms
[Sommes d’exponentielles à coefficients

0

1

et concentration de normes

L^{p}

]

Anderson, B.¹ ; Ash, J. M.² ; Jones, R. L.³ ; Rider, D. G.⁴ ; Saffari, B.⁵

¹ 130 Channing Ln Chapel Hill, NC 27516 (USA)
² DePaul University Department of Mathematical Sciences Chicago, IL 60614 (USA)
³ Conserve School 5400 N. Black Oak Lake Drive Land O’Lakes, WI 54540 (USA)
⁴ University of Wisconsin Department of Mathematics 480 Lincoln Drive Madison, WI 53706-1313 (USA)
⁵ Université de Paris XI (Orsay) Département de Mathématiques Université de Paris XI (Orsay) 91405 Orsay Cedex (France)

Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1377-1404.

Résumé
Abstract

Une somme d’exponentielles de la forme $f (x) = exp (2 π i N_{1} x) + exp (2 π i N_{2} x) + \cdot \cdot \cdot + exp (2 π i N_{m} x)$ , où les $N_{k}$ sont des entiers distincts, est appelée un polynôme trigonométrique idempotent (car $f * f = f$ ) ou, simplement, un idempotent. Nous prouvons que pour tout réel $p > 1$ , et tout $E \subset 𝕋 = ℝ / ℤ$ avec $| E | > 0,$ il existe des idempotents concentrés sur $E$ au sens de la norme $L^{p}$ . Plus précisément, pour tout $p > 1,$ nous calculons explicitement une constante $C_{p} > 0$ telle que pour tout $E$ avec $| E | > 0$ , et tout réel $ϵ > 0$ , on puisse construire un idempotent $f$ tel que le quotient ${(\int_{E} {| f |}^{p} / \int_{𝕋} {| f |}^{p})}^{1 / p}$ soit supérieur à $C_{p} - ϵ$ . 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 $L^{p}$ a lieu ou non pour $p = 1$ .

A sum of exponentials of the form $f (x) = exp (2 π i N_{1} x) + exp (2 π i N_{2} x) + \dots + exp (2 π i N_{m} x)$ , where the $N_{k}$ are distinct integers is called an idempotent trigonometric polynomial (because the convolution of $f$ with itself is $f$ ) or, simply, an idempotent. We show that for every $p > 1,$ and every set $E$ of the torus $𝕋 = ℝ / ℤ$ with $| E | > 0,$ there are idempotents concentrated on $E$ in the $L^{p}$ sense. More precisely, for each $p > 1,$ there is an explicitly calculated constant $C_{p} > 0$ so that for each $E$ with $| E | > 0$ and $ϵ > 0$ one can find an idempotent $f$ such that the ratio ${(\int_{E} {| f |}^{p} / \int_{𝕋} {| f |}^{p})}^{1 / p}$ is greater than $C_{p} - ϵ$ . 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 $L^{p}$ concentration phenomenon fails to occur when $p = 1 .$

MR Zbl | 1 citation dans Numdam

DOI : 10.5802/aif.2298

Classification : 42A05, 42A10, 42A32
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.

Affiliations des auteurs :

Anderson, B. ¹ ; Ash, J. M. ² ; Jones, R. L. ³ ; Rider, D. G. ⁴ ; Saffari, B. ⁵

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     title = {Exponential sums with coefficients $0$ or $1$ and concentrated $L^{p}$ norms},
     journal = {Annales de l'Institut Fourier},
     pages = {1377--1404},
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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, Tome 57 (2007) no. 5, pp. 1377-1404. doi : 10.5802/aif.2298. https://www.numdam.org/articles/10.5802/aif.2298/

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