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On diffraction Fresnel transforms for Boehmians. (English) Zbl 1243.46032
Summary: The theory of the diffraction Fresnel transform is extended to certain spaces of Schwartz distributions. In the context of Boehmian spaces, the diffraction Fresnel transform is obtained as a continuous function. Convergence with respect to $\delta$ and $\Delta$ is also defined.

MSC:
46F12Integral transforms in distribution spaces
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References:
[1] V. Namias, “The fractional order fourier transform and its application to quantum mechanics,” IMA Journal of Applied Mathematics, vol. 25, no. 3, pp. 241-265, 1980. · Zbl 0434.42014 · doi:10.1093/imamat/25.3.241
[2] D. Mendlovic and H. M. Ozaktas, “Fractional Fourier transforms and their optical implementation: I,” Journal of the Optical Society of America A, vol. 10, no. 9, pp. 1875-1881, 1993. · doi:10.1364/JOSAA.10.001875
[3] H. Ozaktas and D. Mendlovic, “Fractional Fourier transforms and their optical implementation. II,” Journal of the Optical Society of America A, vol. 10, no. 12, pp. 2522-2531, 1993. · doi:10.1364/JOSAA.10.002522
[4] L. M. Bernardo and O. D. D. Soares, “Fractional Fourier transforms and optical systems,” Optics Communications, vol. 110, no. 5-6, pp. 517-522, 1994. · doi:10.1016/0030-4018(94)90242-9
[5] A. W. Lohmann, “Image rotation, Wigner rotation, and the fractional Fourier transform,” Journal of the Optical Society of America A, vol. 10, no. 10, pp. 2181-2186, 1993. · doi:10.1364/JOSAA.10.002181
[6] A. Kılı\ccman, “On the fresnel sine integral and the convolution,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 37, pp. 2327-2333, 2003. · Zbl 1026.33002 · doi:10.1155/S0161171203211510
[7] A. Kılı\ccman and B. Fisher, “On the fresnel integrals and the convolution,” International Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 41, pp. 2635-2643, 2003. · Zbl 1043.33001 · doi:10.1155/S0161171203211522 · eudml:50875
[8] L. Mertz, Transformations in Optics, Wiley, New York, NY, USA, 1965.
[9] H. Y. Fan and H. L. Lu, “Wave-function transformations by general SU(1, 1) single-mode squeezing and analogy to fresnel transformations in wave optics,” Optics Communications, vol. 258, no. 1, pp. 51-58, 2006. · doi:10.1016/j.optcom.2005.07.044
[10] S. K. Q. Al-Omari, D. Loonker, P. K. Banerji, and S. L. Kalla, “Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces,” Integral Transforms and Special Functions, vol. 19, no. 6, pp. 453-462, 2008. · Zbl 1215.42007 · doi:10.1080/10652460801936721
[11] R. S. Pathak, Integral Transforms of Generalized Functions and Their Applications, Gordon and Breach Science Publishers, Amsterdam, The Netherlands, 1997.
[12] A. H. Zemanian, Generalized Integral Transformations, Dover Publications, New York, NY, USA, 2nd edition, 1987. · Zbl 0643.46029
[13] S. K. Q. Al-Omari, “The generalized stieltjes and Fourier transforms of certain spaces of generalized functions,” Jordan Journal of Mathematics and Statistics, vol. 2, no. 2, pp. 55-66, 2009. · Zbl 1279.40004
[14] S. K. Q. Al-Omari, “On the distributional Mellin transformation and its extension to Boehmian spaces,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 17, pp. 801-810, 2011. · Zbl 1245.46032
[15] S. K. Q. Al-Omari, “A Mellin transform for a space of lebesgue integrable Boehmians,” International Journal of Contemporary Mathematical Sciences, vol. 6, no. 32, pp. 1597-1606, 2011. · Zbl 1253.46047
[16] T. K. Boehme, “The support of Mikusinski operators,” Transactions of the American Mathematical Society, vol. 176, pp. 319-334, 1973. · Zbl 0268.44005 · doi:10.2307/1996211
[17] P. Mikusiński, “Fourier transform for integrable Boehmians,” Rocky Mountain Journal of Mathematics, vol. 17, no. 3, pp. 577-582, 1987. · Zbl 0629.44005 · doi:10.1216/RMJ-1987-17-3-577
[18] P. Mikusiński, “Convergence of Boehmians,” Japanese Journal of Mathematics, vol. 9, no. 1, pp. 159-179, 1983. · Zbl 0524.44005
[19] R. Roopkumar, “Mellin transform for Boehmians,” Bulletin of the Institute of Mathematics. Academia Sinica, vol. 4, no. 1, pp. 75-96, 2009. · Zbl 1182.46030