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Numerical Applications of Tikhonov Regularization for the Fourier Multiplier Operators

Authors: Fethi Soltani, Adel Almarashi, Idir Mechai

Abstract:

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.

Keywords: Fourier multiplier operators, Gauss-Kronrod method of integration, Paley-Wiener space, Tikhonov regularization.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1124367

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References:


[1] N. Aronszajn, Theory of reproducing kernels, Trans. Amer. Math. Soc. 1948, (68):337–404.
[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.
[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.
[4] S. Saitoh, Hilbert spaces induced by Hilbert space valued functions, Proc. Amer. Math. Soc. 1983,89:74–78.
[5] S. Saitoh, The Weierstrass transform and an isometry in the heat equation, Appl. Anal. 1983, (16):1–6.
[6] S. Saitoh, Approximate real inversion formulas of the Gaussian convolution, Appl. Anal. 2004, (83):727–733.
[7] S. Saitoh, Best approximation, Tikhonov regularization and reproducing kernels, Kodai Math. J. 28 (2005) 359–367.
[8] F. Soltani, Littlewood-Paley g-function in the Dunkl analysis on Rd, J. Inequal. Pure Appl. Math. 2005.
[9] F. Soltani, Inversion formulas in the Dunkl-type heat conduction on Rd, Appl. Anal. 2005, (84):541–553.
[10] F. Soltani, Best approximation formulas for the Dunkl L2-multiplier operators on Rd, Rocky Mountain J. Math. 2012, (42):305–328.
[11] F. Soltani, Multiplier operators and extremal functions related to the dual Dunkl-Sonine operator, Acta Math. Sci. 2013, 33B(2):430–442.
[12] F. Soltani, Operators and Tikhonov regularization on the Fock space, Int. Trans. Spec. Funct. 2014, 25(4):283–294.
[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.
[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.
[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.