logo AFST
Lower bounds for the Dyadic Hilbert transform
Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 1, pp. 265-284.

Dans cet article, nous établissons des bornes pour la transformée de Hilbert dyadique (Haar shift) de la forme Шf L 2 (K) C(I,K)f L 2 (I) I et K sont des intervalles dyadiques et f est à support dans I. Si IK de telles bornes existent sans condition supplémentaire sur f alors que dans les cas KI et KI= une telle borne n’existe que si on impose une condition sur la dérivée de f. Dans le dernier cas nous établissons une borne de la forme Шf L 2 (K) C(I,K)|f I |f I est la moyenne de f sur I. Ce travail permet ainsi une meilleure compréhension du problème similaire pour la transformée de Hilbert sur .

In this paper, we seek lower bounds of the dyadic Hilbert transform (Haar shift) of the form Шf L 2 (K) C(I,K)f L 2 (I) where I and K are two dyadic intervals and f supported in I. If IK, such bounds exist while in the other cases KI and KI= such bounds are only available under additional constraints on the derivative of f. In the later case, we establish a bound of the form Шf L 2 (K) C(I,K)|f I | where f I is the mean of f over I. This sheds new light on the similar problem for the usual Hilbert transform.

Reçu le :
Accepté le :
Publié le :
DOI : https://doi.org/10.5802/afst.1569
Classification : 42B20
Mots clés : Dyadic Hilbert transform, Haar Shift, BMO
@article{AFST_2018_6_27_1_265_0,
     author = {Philippe Jaming and Elodie Pozzi and Brett D. Wick},
     title = {Lower bounds for the {Dyadic} {Hilbert} transform},
     journal = {Annales de la Facult\'e des sciences de Toulouse : Math\'ematiques},
     pages = {265--284},
     publisher = {Universit\'e Paul Sabatier, Toulouse},
     volume = {Ser. 6, 27},
     number = {1},
     year = {2018},
     doi = {10.5802/afst.1569},
     language = {en},
     url = {https://afst.centre-mersenne.org/articles/10.5802/afst.1569/}
}
TY  - JOUR
AU  - Philippe Jaming
AU  - Elodie Pozzi
AU  - Brett D. Wick
TI  - Lower bounds for the Dyadic Hilbert transform
JO  - Annales de la Faculté des sciences de Toulouse : Mathématiques
PY  - 2018
DA  - 2018///
SP  - 265
EP  - 284
VL  - Ser. 6, 27
IS  - 1
PB  - Université Paul Sabatier, Toulouse
UR  - https://afst.centre-mersenne.org/articles/10.5802/afst.1569/
UR  - https://doi.org/10.5802/afst.1569
DO  - 10.5802/afst.1569
LA  - en
ID  - AFST_2018_6_27_1_265_0
ER  - 
Philippe Jaming; Elodie Pozzi; Brett D. Wick. Lower bounds for the Dyadic Hilbert transform. Annales de la Faculté des sciences de Toulouse : Mathématiques, Série 6, Tome 27 (2018) no. 1, pp. 265-284. doi : 10.5802/afst.1569. https://afst.centre-mersenne.org/articles/10.5802/afst.1569/

[1] Reema Al-Aifari; Michel Defrise; Alexander Katsevich Asymptotic analysis of the SVD of the truncated Hilbert transform with overlap, SIAM J. Math. Anal., Volume 47 (2015) no. 1, pp. 797-824 | Article

[2] Reema Al-Aifari; Alexander Katsevich Spectral analysis of the truncated Hilbert transform with overlap, SIAM J. Math. Anal., Volume 46 (2014) no. 1, pp. 192-213 | Article

[3] Reema Al-Aifari; Lillian B. Pierce; Stefan Steinerberger Lower bounds for the truncated Hilbert transform, Rev. Mat. Iberoam., Volume 32 (2016) no. 1, pp. 23-56 | Article

[4] Hedy Attouch; Giuseppe Buttazzo; Gérard Michaille Variational analysis in Sobolev and BV spaces: applications to PDEs and optimization, Series on Optimization, Volume 6, SIAM, 2006, xii+634 pages

[5] Matias Courdurier; Frédéric Noo; Michel Defrise; Hiroyuki Kudo Solving the interior problem of computed tomography using a priori knowledge, Inverse Probl., Volume 24 (2008) no. 6 (Article ID 065001, 27 p.) | Article

[6] Bernard Dacorogna Direct Methods in the Calculus of Variations, Applied Mathematical Sciences, Volume 78, Springer, 2008, xii+619 pages

[7] Michel Defrise; Frédéric Noo; Rolf Clackdoyle; Hiroyuki Kudo Truncated Hilbert transform and image reconstruction from limited tomographic data, Inverse Probl., Volume 22 (2006) no. 3, pp. 1037-1053 | Article

[8] Tuomas Hytönen On Petermichl’s dyadic shift and the Hilbert transform, C. R. Math., Acad. Sci. Paris, Volume 346 (2008) no. 21–22, pp. 1133-1136 | Article

[9] Tuomas Hytönen The sharp weighted bound for general Calderón-Zygmund operators, Ann. Math., Volume 175 (2012) no. 3, pp. 1473-1506 | Article

[10] Alexander Katsevich Singular value decomposition for the truncated Hilbert transform, Inverse Probl., Volume 26 (201) no. 11 (Article ID 115011, 12 p.)

[11] Alexander Katsevich Singular value decomposition for the truncated Hilbert transform. II., Inverse Probl., Volume 27 (2011) no. 7 (Article ID 075006, 7 p.) | Article

[12] E. Katsevich; Alexander Katsevich; Ge Wang Stability of the interior problem with polynomial attenuation in the region of interest, Inverse Probl., Volume 28 (2012) no. 6 (Article ID 065022, 28 p.) | Article

[13] Hiroyuki Kudo; Matias Courdurier; Frédéric Noo; Michel Defrise Tiny a priori knowledge solves the interior problem in computed tomography, Phys. Med. Biol., Volume 53 (2008) no. 9, pp. 2207-2231 | Article

[14] Frank Natterer The mathematics of computerized tomography, Classics in Applied Mathematics, Volume 32, SIAM, 2007, xvii+222 pages

[15] Fedor Nazarov; Alexander Reznikov; Vasily Vasyunin; Alexander Volberg A Bellman function counterexample to the A 1 conjecture: the blow-up of the weak norm estimates of weighted singular operators (2015) (https://arxiv.org/abs/1506.04710)

[16] Fedor Nazarov; Alexander Volberg The Bellman function, the two-weight Hilbert transform, and embeddings of the model spaces K θ , J. Anal. Math., Volume 87 (2002), pp. 385-414 | Article

[17] Stefanie Petermichl Dyadic shifts and a logarithmic estimate for Hankel operators with matrix symbol, C. R. Math., Acad. Sci. Paris, Volume 330 (2000) no. 6, pp. 455-460 | Article

[18] Stefanie Petermichl The sharp bound for the Hilbert transform on weighted Lebesgue spaces in terms of the classical A p characteristic, Am. J. Math., Volume 129 (2007) no. 5, pp. 1355-1375 | Article

[19] David Slepian Some comments on Fourier analysis, uncertainty and modeling, SIAM Rev., Volume 25 (1983), pp. 379-393 | Article

[20] Francesco G. Tricomi Integral Equations, Dover Books on Mathematics, Dover publications, 1985, iv+254 pages

[21] Yangbo Ye; Hengyong Yu; Ge Wang Exact interior reconstruction with cone-beam CT, International Journal of Biomedical Imaging, Volume 2007 (2007) (Article ID 10693, 5 p.) | Article

Cité par Sources :