Note on Boehmians for Class of Optical Fresnel Wavelet Transforms
We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results....
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| Vydáno v: | Journal of function spaces Ročník 2012; číslo 2012; s. 1 - 14 |
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| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
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Cairo, Egypt
Hindawi Puplishing Corporation
01.01.2012
Hindawi Publishing Corporation John Wiley & Sons, Inc Wiley |
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| ISSN: | 0972-6802, 2314-8896, 1758-4965, 2314-8888 |
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| Abstract | We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect to δ and Δ convergence are discussed in some details. |
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| AbstractList | We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect to δ and Δ convergence are discussed in some details. We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect to delta and [Delta] convergence are discussed in some details. We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect to δand Δ convergence are discussed in some details. We extend the Fresnel-wavelet transform to the context of generalized functions, namely, Boehmians. At first, we study the Fresnel-wavelet transform in the sense of distributions of compact support. Based on this concept, we introduce two new spaces of Boehmians and proving certain related results. Further, we show that the extended transform establishes a linear and an isomorphic mapping between the Boehmian spaces. Moreover, conditions of continuity of the extended transform and its inverse with respect to δ and Δ convergence are discussed in some details. |
| Author | Kiliçman, Adem al-Omari, S. K. Q. |
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| CitedBy_id | crossref_primary_10_1002_mma_5304 crossref_primary_10_1007_s11253_018_1435_x crossref_primary_10_1007_s11868_014_0102_8 crossref_primary_10_1007_s13370_015_0382_z crossref_primary_10_1155_2013_539240 crossref_primary_10_1155_2014_463901 crossref_primary_10_1155_2013_348701 crossref_primary_10_1155_2014_612391 crossref_primary_10_5269_bspm_v38i1_36468 |
| Cites_doi | 10.1080/10652460600856534 10.1016/j.optcom.2005.07.044 10.1080/10652460801936721 10.1364/JOSAA.10.002181 10.1080/10652469608819095 10.1063/1.528825 10.1080/10652460903335061 10.1364/JOSAA.10.001875 10.1155/2011/712746 10.1016/j.optcom.2009.06.031 10.1155/S0161171203211510 10.1016/0030-4018(94)90242-9 10.1023/A:1013719619356 10.2307/2160805 10.1364/JOSAA.10.002522 10.1155/S0161171203211522 10.1093/imamat/39.2.159 10.1216/RMJ-1987-17-3-577 10.1063/1.1665805 10.1093/imamat/25.3.241 |
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| Contributor | Kiliçman, Adem |
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| Copyright | Copyright © 2012 S. K. Q. Al-Omari and A. Kılıçman. Copyright © 2012 S. K. Q. Al-Omari and A. Kiliçman. S. K. Q. Al-Omari et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. |
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| References_xml | – reference: Goodman J. W. Introductionto Fourier Optics 1972 New York, NY, USA Mc Graw-Hill – volume: 17 start-page: 759 issue: 11 year: 2006 end-page: 768 ident: 1 article-title: Tempered distributional Fourier sine (cosine) transform – volume: 17 start-page: 577 issue: 3 year: 1987 end-page: 582 ident: 8 article-title: Fourier transform for integrable Boehmians – volume: 51(126) start-page: 463 issue: 3 year: 2001 end-page: 471 ident: 18 article-title: A comparison on the commutative neutrix convolution of distributions and the exchange formula – reference: Pathak R. S. Integral Transforms of Generalized Functions and Their Applications 1997 Amsterdam, The Netherlands Gordon and Breach Science xvi+415 1630717 – volume: 30 start-page: 753 issue: 8 year: 1999 end-page: 762 ident: 19 article-title: On the commutative neutrix product of distributions – volume: 176 start-page: 319 year: 1973 end-page: 334 ident: 23 article-title: The support of Mikusiński operators – volume: 12 year: 1971 end-page: 9 ident: 10 article-title: Linear canonical transformations and their unitary representations – volume: 4 start-page: 121562 year: 1965 end-page: 121569 ident: 5 article-title: On the propagation of Gaussian beams of light through Lenslike media including those with a loss or gain variation – volume: 10 start-page: 2181 issue: 10 year: 1993 end-page: 2186 ident: 6 article-title: Image rotation, Wigner rotation, and the fractional Fourier transform – volume: 19 start-page: 453 issue: 5-6 year: 2008 end-page: 462 ident: 22 article-title: Fourier sine (cosine) transform for ultradistributions and their extensions to tempered and ultraBoehmian spaces – volume: 258 start-page: 51 issue: 1 year: 2006 end-page: 58 ident: 3 article-title: Wave-function transformations by general SU(1, 1) single-mode squeezing and analogy to Fresnel transformations in wave optics – volume: 25 start-page: 241 issue: 3 year: 1980 end-page: 265 ident: 26 article-title: The fractional order Fourier transform and its application to quantum mechanics – volume: 21 start-page: 367 issue: 5-6 year: 2010 end-page: 379 ident: 21 article-title: A new integral transform and associated distributions – reference: Mertz L. 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Wavelets, an Analysis Tool 1995 Oxford, UK The Clarendon Press xiv+423 Oxford Mathematical Monographs 1367088 – volume: 282 start-page: 3734 issue: 18 year: 2009 end-page: 3736 ident: 4 article-title: Optical Fresnel transformation and quantum tomography – start-page: 2635 issue: 41 year: 2003 end-page: 2643 ident: 15 article-title: On the Fresnel integrals and the convolution – volume: 2011 year: 2011 end-page: 11 ident: 11 article-title: On diffraction Fresnel transforms for Boehmians – volume: 4 start-page: 77 issue: 1-2 year: 1996 end-page: 82 ident: 28 article-title: A commutative neutrix product of ultradistributions – volume: 39 start-page: 159 issue: 2 year: 1987 end-page: 175 ident: 7 article-title: On Namias's fractional Fourier transforms – volume: 123 start-page: 813 issue: 3 year: 1995 end-page: 817 ident: 25 article-title: Tempered Boehmians and ultradistributions – volume: 10 start-page: 2522 issue: 12 year: 1993 end-page: 2531 ident: 9 article-title: Fractional Fourier transforms and their optical implementation. 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| SubjectTerms | Continuity Convergence Deltas Fourier transforms Function space Inverse Mapping Optics Studies Transforms Wavelet transforms |
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| Title | Note on Boehmians for Class of Optical Fresnel Wavelet Transforms |
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