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 : https://doi.org/10.5802/ambp.272
Classification : 42B35,  43A32,  35S30
Mots clés : Riemann-Liouville operator, Fourier transform, Paley-Wiener-Schwartz theorems
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     title = {On the range of the {Fourier} transform  connected with {Riemann-Liouville} operator},
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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://archive.numdam.org/articles/10.5802/ambp.272/

[1] Akhiezer, N. Vorlesungen Über Approximations Theorie, Akademieverlag, Berlin, 1953 | MR 61692 | Zbl 0060.16906

[2] Andersson, L. E. On the determination of a function from spherical averages, SIAM. J. Math Anal, Volume 19 (1988), pp. 214-234 | Article | MR 924556 | Zbl 0638.44004

[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 2172801 | Zbl 1131.44002

[4] Bang, H. H. A property of infinitely differentiable functions, Proc. Amer. Math. Soc, Volume 108 (1990), pp. 73-76 | Article | MR 1024259 | Zbl 0707.26015

[5] Boas, R. P. Entire Functions, Academic Press, New-York, 1954 | MR 68627 | Zbl 0058.30201

[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 | Article | MR 781111 | Zbl 0588.44006

[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 | Article | MR 875320 | Zbl 0619.65132

[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 31009 | Zbl 0061.11602

[12] Lebedev, N.N. Special Functions and Their Applications, Dover publications, Inc., New-York, 1972 | MR 350075 | Zbl 0271.33001

[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 | Article | MR 1365228 | Zbl 0845.43005

[14] Rachdi, L. T.; Trimèche, K. Weyl transforms associated with the spherical mean operator, Anal. Appl., Volume 1 (2003), pp. 141-164 (No. 2) | Article | MR 1976612 | Zbl 1045.47038

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

[16] Schwartz, L. Theorie des Distributions, Hermann, Paris, 1978 | MR 209834 | Zbl 0399.46028

[17] Stein, E. M. Functions of exponential type, Ann. of Math., Volume 65, No 2 (1957), pp. 582-592 | Article | MR 85342 | Zbl 0079.13103

[18] Swartz, CH. Convergence of convolution operators, Studia.Math., Volume 42 (1972), pp. 249-257 | MR 315438 | Zbl 0239.46029

[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 616008 | Zbl 0416.44002

[20] Trimèche, K. Inversion of the Lions translation operator using generalized wavelets, Appl. Comput. Harmonic Anal., Volume 4 (1997), pp. 97-112 | Article | MR 1429682 | Zbl 0872.34059

[21] Tuan, Vu Kim On the range of the Hankel and extended Hankel transforms, J. Math. Anal. Appl., Volume 209 (1997), pp. 460-478 | Article | MR 1474619 | Zbl 0881.44004

[22] Watson, G.N. A treatise on the Theory of Bessel functions, 2nd ed. Cambridge Univ. Press., London/New-York, 1966 | MR 1349110 | Zbl 0174.36202

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