Exponential sums with coefficients 0 or 1 and concentrated L p norms
[Sommes d’exponentielles à coefficients 0 ou 1 et concentration de normes L p ]
Annales de l'Institut Fourier, Tome 57 (2007) no. 5, pp. 1377-1404.

Une somme d’exponentielles de la forme f(x)=exp2πiN 1 x+exp2πiN 2 x+···+exp2πiN 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 E𝕋=/ 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 E |f| p / 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)=exp2πiN 1 x+exp2πiN 2 x++exp2πiN 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=/ 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 E |f| p / 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.

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.
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)
@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},
     zbl = {1133.42004},
     mrnumber = {2364133},
     language = {en},
     url = {http://archive.numdam.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  - http://archive.numdam.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 http://archive.numdam.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, Tome 57 (2007) no. 5, pp. 1377-1404. doi : 10.5802/aif.2298. http://archive.numdam.org/articles/10.5802/aif.2298/

[1] Anderson, Bruce; Ash, J. Marshall; Jones, Roger L.; Rider, Daniel G.; Saffari, Bahman L p norm local estimates for exponential sums, C. R. Acad. Sci. Paris Sér. I Math., Volume 330 (2000) no. 9, pp. 765-769 | DOI | MR | Zbl

[2] Ash, J. Marshall Weak restricted and very restricted operators on L 2 , Trans. Amer. Math. Soc., Volume 281 (1984) no. 2, pp. 675-689 | MR | Zbl

[3] Ash, J. Marshall Triangular Dirichlet kernels and growth of L p Lebesgue constants (2004) (preprint, http://condor.depaul.edu/~mash/YudinLp.pdf)

[4] Ash, J. Marshall; Jones, Roger L.; Saffari, Bahman Inégalités sur des sommes d’exponentielles, C. R. Acad. Sci. Paris Sér. I Math., Volume 296 (1983) no. 22, pp. 899-902 | Zbl

[5] Cowling, Michael Some applications of Grothendieck’s theory of topological tensor products in harmonic analysis, Math. Ann., Volume 232 (1978) no. 3, pp. 273-285 | DOI | Zbl

[6] Déchamps, Myriam; Queffélec, Hervé; Piquard, Françoise Estimations locales de sommes d’exponentielles, Harmonic analysis: study group on translation-invariant Banach spaces (Publ. Math. Orsay), Volume 84, Univ. Paris XI, Orsay, 1984, pp. Exp. No. 1, 16 | Zbl

[7] Dechamps-Gondim, Myriam; Piquard-Lust, Françoise; Queffélec, Hervé Estimations locales de sommes d’exponentielles, C. R. Acad. Sci. Paris Sér. I Math., Volume 297 (1983) no. 3, pp. 153-156 | Zbl

[8] Harman, Glyn Metric number theory, London Mathematical Society Monographs. New Series, 18, The Clarendon Press Oxford University Press, New York, 1998 | MR | Zbl

[9] Kahane, Jean-Pierre, 1982 (personal communication)

[10] Kahane, Jean-Pierre Some random series of functions, Cambridge Studies in Advanced Mathematics, 5, Cambridge University Press, Cambridge, 1985 | MR | Zbl

[11] Körner, T. W. Fourier analysis, Cambridge University Press, Cambridge, 1989 | MR | Zbl

[12] Montgomery, H. L., November 1980 and October 1982 (personal communications)

[13] Nazarov, F. L.; Podkorytov, A. N. The behavior of the Lebesgue constants of two-dimensional Fourier sums over polygons, St. Petersburg Math. J., Volume 7 (1996), pp. 663-680 | MR | Zbl

[14] Pichorides, S. K., 1980 (personal communication)

[15] Yudin, A. A.; Yudin, V. A. Polygonal Dirichlet kernels and growth of Lebesgue constants, Mat. Zametki, Volume 37 (1985) no. 2, p. 220-236, 301 English translation: Math. Notes, 37 (1985), no. 1-2, 124–135 | MR | Zbl

[16] Zygmund, A. Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959 | MR | Zbl

Cité par Sources :