Comptes Rendus
no. 21-22
Harmonic Analysis
Extended solution of Boas' conjecture on Fourier transforms
[Solutions prolongées de la conjecture de Boas sur les transformées de Fourier]

Présenté par : Jean-Pierre Kahane

Elijah Liflyand1 ; Sergey Tikhonov2, 3
1 Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
2 Scuola Normale Superiore, 56126 Pisa, Italy
3 ICREA, Passeig Lluís Companys, 23, 08010 Barcelona, Spain
Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1137-1142.

On étudie des inégalités Lp→Lq à poids pour des transformées de Fourier, en particulier on formule une conjecture de Boas traduisant une intégrabilité pour des fonctions dans le cas où le poids est une puissance lorsque l'une des fonctions est monotone et p=q. Nous donnons des versions unidimensionnelles et multidimensionnelles (dans le cas de fonctions radiales) pour p⩾q ou p⩽q et pour une classe définie de fonctions généralement monotones.

Weighted Lp→Lq Fourier inequalities are studied. We prove Boas' conjecture on integrability with power weights of the Fourier transform. One-dimensional as well as multidimensional versions (for radial functions) are obtained for general monotone functions.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2008.07.029
Elijah Liflyand 1 ; Sergey Tikhonov 2, 3

1 Department of Mathematics, Bar-Ilan University, 52900 Ramat-Gan, Israel
2 Scuola Normale Superiore, 56126 Pisa, Italy
3 ICREA, Passeig Lluís Companys, 23, 08010 Barcelona, Spain 
@article{CRMATH_2008__346_21-22_1137_0,
     author = {Elijah Liflyand and Sergey Tikhonov},
     title = {Extended solution of {Boas'} conjecture on {Fourier} transforms},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1137--1142},
     publisher = {Elsevier},
     volume = {346},
     number = {21-22},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.029},
     language = {en},
}
TY  - JOUR
AU  - Elijah Liflyand
AU  - Sergey Tikhonov
TI  - Extended solution of Boas' conjecture on Fourier transforms
JO  - Comptes Rendus. Mathématique
PY  - 2008
SP  - 1137
EP  - 1142
VL  - 346
IS  - 21-22
PB  - Elsevier
DO  - 10.1016/j.crma.2008.07.029
LA  - en
ID  - CRMATH_2008__346_21-22_1137_0
ER  - 
%0 Journal Article
%A Elijah Liflyand
%A Sergey Tikhonov
%T Extended solution of Boas' conjecture on Fourier transforms
%J Comptes Rendus. Mathématique
%D 2008
%P 1137-1142
%V 346
%N 21-22
%I Elsevier
%R 10.1016/j.crma.2008.07.029
%G en
%F CRMATH_2008__346_21-22_1137_0
Elijah Liflyand; Sergey Tikhonov. Extended solution of Boas' conjecture on Fourier transforms. Comptes Rendus. Mathématique, Volume 346 (2008) no. 21-22, pp. 1137-1142. doi : 10.1016/j.crma.2008.07.029. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.029/

[1] R. Askey; S. Wainger Integrability theorems for Fourier series, Duke Math. J., Volume 33 (1966), pp. 223-228

[2] J.J. Benedetto; H.P. Heinig Weighted Fourier inequalities: new proofs and generalizations, J. Fourier Anal. Appl., Volume 9 (2003), pp. 1-37

[3] R.P. Boas The integrability class of the sine transform of a monotonic function, Studia Math., Volume XLIV (1972), pp. 365-369

[4] J.S. Bradley Hardy inequalities with mixed norms, Canad. Math. Bull., Volume 21 (1978), pp. 405-408

[5] M.I. Dyachenko; S. Tikhonov Convergence of trigonometric series with general monotone coefficients, C. R. Math. Acad. Sci. Paris, Ser. I, Volume 345 (2007) no. 3, pp. 123-126

[6] E. Liflyand, S. Tikhonov, The Fourier transforms of general monotone functions, in: Analysis and Mathematical Physics, Birkhäuser, 2009

[7] C. Sadosky; R.L. Wheeden Some weighted norm inequalities for the Fourier transform of functions with vanishing moments, Trans. Amer. Math. Soc., Volume 300 (1987), pp. 521-533

[8] Y. Sagher Integrability conditions for the Fourier transform, J. Math. Anal. Appl., Volume 54 (1976), pp. 151-156

[9] S.G. Samko; A.A. Kilbas; O.I. Marichev Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach, New York, NY, 1993

[10] S. Tikhonov Trigonometric series with general monotone coefficients, J. Math. Anal. Appl., Volume 326 (2007), pp. 721-735

[11] S. Tikhonov Best approximation and moduli of smoothness: computation and equivalence theorems, J. Approx. Theory, Volume 153 (2008) no. 1, pp. 19-39

Cité par Sources :

Commentaires - Politique

Ces articles pourraient vous intéresser

Weighted Paley–Wiener theorem on the Hilbert transform

Elijah Liflyand; Sergey Tikhonov

C. R. Math (2010)

Convergence of trigonometric series with general monotone coefficients

Mikhail Dyachenko; Sergey Tikhonov

C. R. Math (2007)