On the range of the Fourier transform connected with Riemann-Liouville operator
Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 355-397.

We characterize the range of some spaces of functions by the Fourier transform associated with the Riemann-Liouville operator α ,α0 and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a Paley-Wiener-Schwartz theorems.

DOI : 10.5802/ambp.272
Classification : 42B35, 43A32, 35S30
Mots clés : Riemann-Liouville operator, Fourier transform, Paley-Wiener-Schwartz theorems
Rachdi, Lakhdar Tannech 1 ; Rouz, Ahlem 1

1 Department of Mathematics Faculty of Sciences of Tunis 2092 El Manar 2 Tunis Tunisia
@article{AMBP_2009__16_2_355_0,
     author = {Rachdi, Lakhdar Tannech and Rouz, Ahlem},
     title = {On the range of the {Fourier} transform  connected with {Riemann-Liouville} operator},
     journal = {Annales math\'ematiques Blaise Pascal},
     pages = {355--397},
     publisher = {Annales math\'ematiques Blaise Pascal},
     volume = {16},
     number = {2},
     year = {2009},
     doi = {10.5802/ambp.272},
     zbl = {1179.42019},
     mrnumber = {2568871},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/ambp.272/}
}
TY  - JOUR
AU  - Rachdi, Lakhdar Tannech
AU  - Rouz, Ahlem
TI  - On the range of the Fourier transform  connected with Riemann-Liouville operator
JO  - Annales mathématiques Blaise Pascal
PY  - 2009
SP  - 355
EP  - 397
VL  - 16
IS  - 2
PB  - Annales mathématiques Blaise Pascal
UR  - http://www.numdam.org/articles/10.5802/ambp.272/
DO  - 10.5802/ambp.272
LA  - en
ID  - AMBP_2009__16_2_355_0
ER  - 
%0 Journal Article
%A Rachdi, Lakhdar Tannech
%A Rouz, Ahlem
%T On the range of the Fourier transform  connected with Riemann-Liouville operator
%J Annales mathématiques Blaise Pascal
%D 2009
%P 355-397
%V 16
%N 2
%I Annales mathématiques Blaise Pascal
%U http://www.numdam.org/articles/10.5802/ambp.272/
%R 10.5802/ambp.272
%G en
%F AMBP_2009__16_2_355_0
Rachdi, Lakhdar Tannech; Rouz, Ahlem. On the range of the Fourier transform  connected with Riemann-Liouville operator. Annales mathématiques Blaise Pascal, Tome 16 (2009) no. 2, pp. 355-397. doi : 10.5802/ambp.272. http://www.numdam.org/articles/10.5802/ambp.272/

[1] Akhiezer, N. Vorlesungen Über Approximations Theorie, Akademieverlag, Berlin, 1953 | MR | Zbl

[2] Andersson, L. E. On the determination of a function from spherical averages, SIAM. J. Math Anal, Volume 19 (1988), pp. 214-234 | DOI | MR | Zbl

[3] Baccar, C.; Hamadi, N. B.; Rachdi, L. -T. Inversion formulas for the Riemann-Liouville transform and its dual associated with singular partial differential operators, Internat. J. Math. Math. Sci. 2006, Article ID 86238 (2006), pp. 1-26 | MR | Zbl

[4] Bang, H. H. A property of infinitely differentiable functions, Proc. Amer. Math. Soc, Volume 108 (1990), pp. 73-76 | DOI | MR | Zbl

[5] Boas, R. P. Entire Functions, Academic Press, New-York, 1954 | MR | Zbl

[6] Erdely, A.; all Higher Transcendental Functions, I, Mc Graw-Hill Book Compagny, New-York, 1953

[7] Erdely, A.; all Tables of Integral Transforms, II, Mc Graw-Hill Book Compagny, New-York, 1954

[8] Fawcett, J. A. Inversion of N-dimensional spherical means, SIAM. J. Appl. Math., Volume 45 (1985), pp. 336-341 | DOI | MR | Zbl

[9] Helesten, H.; Anderson, L. E. An inverse method for the processing of synthetic aperture radar data, Inv. Prob., Volume 3 (1987), pp. 111-124 | DOI | MR | Zbl

[10] Herberthson, M. A numerical Implementation of An Inverse Formula for CARABAS Raw Data, National Defense Research Institute, Internal Report D 30430-3.2, Linköping, Sweden, 1986

[11] Kolmogoroff, A. N. On Inequalities Between Upper Bounds of the Successive Derivatives of an Arbitrary Function on an Infinite Interval, 4, Amer. Math. Soc. Translation, 1949 | MR | Zbl

[12] Lebedev, N.N. Special Functions and Their Applications, Dover publications, Inc., New-York, 1972 | MR | Zbl

[13] Nessibi, M. M.; Rachdi, L. -T.; Trimèche, K. Ranges and inversion formulas for spherical mean operator and its dual, J. Math. Anal. Appl., Volume 196 (1995), pp. 861-884 | DOI | MR | Zbl

[14] Rachdi, L. T.; Trimèche, K. Weyl transforms associated with the spherical mean operator, Anal. Appl., Volume 1 (2003), pp. 141-164 (No. 2) | DOI | MR | Zbl

[15] Schwartz, L. Theory of Distributions, I, Hermann, Paris, 1957

[16] Schwartz, L. Theorie des Distributions, Hermann, Paris, 1978 | MR | Zbl

[17] Stein, E. M. Functions of exponential type, Ann. of Math., Volume 65, No 2 (1957), pp. 582-592 | DOI | MR | Zbl

[18] Swartz, CH. Convergence of convolution operators, Studia.Math., Volume 42 (1972), pp. 249-257 | MR | Zbl

[19] Trimèche, K. Transformation intégrale de Weyl et théorème de Paley-Wiener associés à un opérateur différentiel singulier sur (0,+), J. Math. Pures Appl., Volume 60 (1981), pp. 51-98 | MR | Zbl

[20] Trimèche, K. Inversion of the Lions translation operator using generalized wavelets, Appl. Comput. Harmonic Anal., Volume 4 (1997), pp. 97-112 | DOI | MR | Zbl

[21] Tuan, Vu Kim On the range of the Hankel and extended Hankel transforms, J. Math. Anal. Appl., Volume 209 (1997), pp. 460-478 | DOI | MR | Zbl

[22] Watson, G.N. A treatise on the Theory of Bessel functions, 2nd ed. Cambridge Univ. Press., London/New-York, 1966 | MR | Zbl

Cité par Sources :