Numerical Applications of Tikhonov Regularization for the Fourier Multiplier Operators

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Tikhonov Regularization for the Fourier Multiplier Operators</title> <meta name="description" content="Numerical Applications of Tikhonov Regularization for the Fourier Multiplier Operators"> <meta name="keywords" content="Fourier multiplier operators, Gauss-Kronrod method of integration, Paley-Wiener space, Tikhonov regularization."> <meta name="viewport" content="width=device-width, initial-scale=1, minimum-scale=1, maximum-scale=1, user-scalable=no"> <meta charset="utf-8"> <meta name="citation_title" content="Numerical Applications of Tikhonov Regularization for the Fourier Multiplier Operators"> <meta name="citation_author" content="Fethi Soltani"> <meta name="citation_author" content="Adel Almarashi"> <meta name="citation_author" content="Idir Mechai"> <meta name="citation_publication_date" content="2016/04/03"> <meta name="citation_journal_title" content="International Journal of Mathematical and Computational Sciences"> <meta name="citation_volume" content="10"> <meta 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<a href="https://publications.waset.org/search?q=Fethi%20Soltani">Fethi Soltani</a>, <a href="https://publications.waset.org/search?q=Adel%20Almarashi"> Adel Almarashi</a>, <a href="https://publications.waset.org/search?q=Idir%20Mechai"> Idir Mechai</a> </p> <p class="card-text"><strong>Abstract:</strong></p> Tikhonov regularization and reproducing kernels are the most popular approaches to solve ill-posed problems in computational mathematics and applications. And the Fourier multiplier operators are an essential tool to extend some known linear transforms in Euclidean Fourier analysis, as: Weierstrass transform, Poisson integral, Hilbert transform, Riesz transforms, Bochner-Riesz mean operators, partial Fourier integral, Riesz potential, Bessel potential, etc. Using the theory of reproducing kernels, we construct a simple and efficient representations for some class of Fourier multiplier operators Tm on the Paley-Wiener space Hh. In addition, we give an error estimate formula for the approximation and obtain some convergence results as the parameters and the independent variables approaches zero. Furthermore, using numerical quadrature integration rules to compute single and multiple integrals, we give numerical examples and we write explicitly the extremal function and the corresponding Fourier multiplier operators. <iframe src="https://publications.waset.org/10004459.pdf" style="width:100%; height:400px;" frameborder="0"></iframe> <p class="card-text"><strong>Keywords:</strong> <a href="https://publications.waset.org/search?q=Fourier%20multiplier%20operators" title="Fourier multiplier operators">Fourier multiplier operators</a>, <a href="https://publications.waset.org/search?q=Gauss-Kronrod%20method%0D%0Aof%20integration" title=" Gauss-Kronrod method of integration"> Gauss-Kronrod method of integration</a>, <a href="https://publications.waset.org/search?q=Paley-Wiener%20space" title=" Paley-Wiener space"> Paley-Wiener space</a>, <a href="https://publications.waset.org/search?q=Tikhonov%20regularization." title=" Tikhonov regularization."> Tikhonov regularization.</a> </p> <p class="card-text"><strong>Digital Object Identifier (DOI):</strong> <a href="https://doi.org/10.5281/zenodo.1124367" target="_blank">doi.org/10.5281/zenodo.1124367</a> </p> <a href="https://publications.waset.org/10004459/numerical-applications-of-tikhonov-regularization-for-the-fourier-multiplier-operators" class="btn btn-primary btn-sm">Procedia</a> <a href="https://publications.waset.org/10004459/apa" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">APA</a> <a href="https://publications.waset.org/10004459/bibtex" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">BibTeX</a> <a href="https://publications.waset.org/10004459/chicago" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Chicago</a> <a href="https://publications.waset.org/10004459/endnote" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">EndNote</a> <a href="https://publications.waset.org/10004459/harvard" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">Harvard</a> <a href="https://publications.waset.org/10004459/json" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">JSON</a> <a href="https://publications.waset.org/10004459/mla" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">MLA</a> <a href="https://publications.waset.org/10004459/ris" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">RIS</a> <a href="https://publications.waset.org/10004459/xml" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">XML</a> <a href="https://publications.waset.org/10004459/iso690" target="_blank" rel="nofollow" class="btn btn-primary btn-sm">ISO 690</a> <a href="https://publications.waset.org/10004459.pdf" target="_blank" class="btn btn-primary btn-sm">PDF</a> <span class="bg-info text-light px-1 py-1 float-right rounded"> Downloads <span class="badge badge-light">1527</span> </span> <p class="card-text"><strong>References:</strong></p> <br>[1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 1948, (68):337–404. <br>[2] T. Matsuura, S. Saitoh and D.D. Trong, Inversion formulas in heat conduction multidimensional spaces, J. Inv. Ill-posed Problems 2005, (13):479–493. <br>[3] T. Matsuura and S. Saitoh, Analytical and numerical inversion formulas in the Gaussian convolution by using the Paley-Wiener spaces, Appl. Anal. 2006, (85):901–915. <br>[4] S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc. 1983,89:74–78. <br>[5] S. Saitoh, The Weierstrass transform and an isometry in the heat equation, Appl. Anal. 1983, (16):1–6. <br>[6] S. Saitoh, Approximate real inversion formulas of the Gaussian convolution, Appl. Anal. 2004, (83):727–733. <br>[7] S. Saitoh, Best approximation, Tikhonov regularization and reproducing kernels, Kodai Math. J. 28 (2005) 359–367. <br>[8] F. Soltani, Littlewood-Paley g-function in the Dunkl analysis on Rd, J. Inequal. Pure Appl. Math. 2005. <br>[9] F. Soltani, Inversion formulas in the Dunkl-type heat conduction on Rd, Appl. Anal. 2005, (84):541–553. <br>[10] F. Soltani, Best approximation formulas for the Dunkl L2-multiplier operators on Rd, Rocky Mountain J. Math. 2012, (42):305–328. <br>[11] F. Soltani, Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator, Acta Math. Sci. 2013, 33B(2):430–442. <br>[12] F. Soltani, Operators and Tikhonov regularization on the Fock space, Int. Trans. Spec. Funct. 2014, 25(4):283–294. <br>[13] F. Soltani and A. Nemri, Analytical and numerical approximation formulas for the Fourier multiplier operators, Complex Anal. Oper. Theory, 2015, 9(1):121–138. <br>[14] F. Soltani and A. Nemri, Analytical and numerical applications for the Fourier multiplier operators on Rn × (0,∞), Appl. Anal. http://dx.doi.org/10.1080/00036811.2014.937432. <br>[15] M. Yamada, T. Matsuura and S. Saitoh, Representations of inverse functions by the integral transform with the sign kernel, Frac. Calc. Appl. Anal. 2007, (2):161–168. </div> </div> </div> </main> <footer> <div id="infolinks" class="pt-3 pb-2"> <div class="container"> <div style="background-color:#f5f5f5;" class="p-3"> <div class="row"> <div class="col-md-2"> <ul class="list-unstyled"> About <li><a href="https://waset.org/page/support">About Us</a></li> <li><a href="https://waset.org/page/support#legal-information">Legal</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/WASET-16th-foundational-anniversary.pdf">WASET celebrates its 16th foundational anniversary</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Account <li><a href="https://waset.org/profile">My Account</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Explore <li><a href="https://waset.org/disciplines">Disciplines</a></li> <li><a href="https://waset.org/conferences">Conferences</a></li> <li><a href="https://waset.org/conference-programs">Conference Program</a></li> <li><a href="https://waset.org/committees">Committees</a></li> <li><a href="https://publications.waset.org">Publications</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Research <li><a href="https://publications.waset.org/abstracts">Abstracts</a></li> <li><a href="https://publications.waset.org">Periodicals</a></li> <li><a href="https://publications.waset.org/archive">Archive</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Open Science <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Philosophy.pdf">Open Science Philosophy</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Science-Award.pdf">Open Science Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Open-Society-Open-Science-and-Open-Innovation.pdf">Open Innovation</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Postdoctoral-Fellowship-Award.pdf">Postdoctoral Fellowship Award</a></li> <li><a target="_blank" rel="nofollow" href="https://publications.waset.org/static/files/Scholarly-Research-Review.pdf">Scholarly Research Review</a></li> </ul> </div> <div class="col-md-2"> <ul class="list-unstyled"> Support <li><a href="https://waset.org/page/support">Support</a></li> <li><a href="https://waset.org/profile/messages/create">Contact Us</a></li> <li><a href="https://waset.org/profile/messages/create">Report Abuse</a></li> </ul> </div> </div> </div> </div> </div> <div class="container text-center"> <hr style="margin-top:0;margin-bottom:.3rem;"> <a href="https://creativecommons.org/licenses/by/4.0/" target="_blank" class="text-muted small">Creative Commons Attribution 4.0 International License</a> <div id="copy" class="mt-2">© 2024 World Academy of Science, Engineering and Technology</div> </div> </footer> <a href="javascript:" id="return-to-top"><i class="fas fa-arrow-up"></i></a> <div class="modal" id="modal-template"> <div class="modal-dialog"> <div class="modal-content"> <div class="row m-0 mt-1"> <div class="col-md-12"> <button type="button" class="close" data-dismiss="modal" aria-label="Close"><span aria-hidden="true">×</span></button> </div> </div> <div class="modal-body"></div> </div> </div> </div> <script src="https://cdn.waset.org/static/plugins/jquery-3.3.1.min.js"></script> <script src="https://cdn.waset.org/static/plugins/bootstrap-4.2.1/js/bootstrap.bundle.min.js"></script> <script src="https://cdn.waset.org/static/js/site.js?v=150220211556"></script> <script> jQuery(document).ready(function() { /*jQuery.get("https://publications.waset.org/xhr/user-menu", function (response) { jQuery('#mainNavMenu').append(response); });*/ jQuery.get({ url: "https://publications.waset.org/xhr/user-menu", cache: false }).then(function(response){ jQuery('#mainNavMenu').append(response); }); }); </script> </body> </html>

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Numerical Applications of Tikhonov Regularization for the Fourier Multiplier Operators