Numerical Solution of Mean-Square Approximation Problem of Real Nonnegative Function by the Modulus of Double Fourier Integral

ABSTRACT

A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are presented.

A nonlinear problem of mean-square approximation of a real nonnegative continuous function with respect to two variables by the modulus of double Fourier integral dependent on two real parameters with use of the smoothing functional is studied. Finding the optimal solutions of this problem is reduced to solution of the Hammerstein type two-dimensional nonlinear integral equation. The numerical algorithms to find the branching lines and branching-off solutions of this equation are constructed and justified. Numerical examples are presented.

KEYWORDS

Mean-Square Approximation, Discrete Fourier Transform, Two-Dimensional Nonlinear Integral Equation, Nonuniqueness and Branching of Solutions, Two-Dimensional Nonlinear Spectral Problem

Mean-Square Approximation, Discrete Fourier Transform, Two-Dimensional Nonlinear Integral Equation, Nonuniqueness and Branching of Solutions, Two-Dimensional Nonlinear Spectral Problem

Cite this paper

nullP. Savenko and M. Tkach, "Numerical Solution of Mean-Square Approximation Problem of Real Nonnegative Function by the Modulus of Double Fourier Integral,"*Applied Mathematics*, Vol. 2 No. 9, 2011, pp. 1076-1090. doi: 10.4236/am.2011.29149.

nullP. Savenko and M. Tkach, "Numerical Solution of Mean-Square Approximation Problem of Real Nonnegative Function by the Modulus of Double Fourier Integral,"

References

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[2] P. Savenko and M. Tkach, “Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Descrete Fourier Transform,” Applied Mathematics, Vol. 1, No. 1, 2010, pp. 65-75. doi:10.4236/am.2010.11008

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[1] A. N. Tikhonov and V. Y. Arsenin, “The Methods of Solution of Incorrect Problems,” Nauka, Moscow, 1979.

[2] P. Savenko and M. Tkach, “Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Descrete Fourier Transform,” Applied Mathematics, Vol. 1, No. 1, 2010, pp. 65-75. doi:10.4236/am.2010.11008

[3] E. G. Zelkin and S. I. Solov’yev, “Methods of Synthesis of Antennas: Phased Antenna Arrays and Antennas with Continuous Aperture,” Sovet radio, Moscow, 1980.

[4] P. O. Savenko, “Nonlinear Problems of Radiating Systems Synthesis (Theory and Methods of the Solution),” Institute for Applied Problems in Mechanics and Mathematics, Lviv, 2002.

[5] A. N. Kolmogorov and S. V. Fomin, “Elements of Functions Theory and Functional Analysis,” Nauka, Moscow, 1968.

[6] М. M. Vainberg and V. А. Trenogin, “Theory of Branching of Solutions of Nonlinear Equations,” Nauka, Moscow, 1969.

[7] V. A. Trenogin, “The Functional Analysis,” Nauka, Moscow, 1980.

[8] М. А. Krasnoselskii, G. М. Vainikko and P. P. Zabreiko, “Approximate Solution of Operational Equations,” Nauka, Moscow, 1969.

[9] L. V. Kantorovich and G. P. Akilov, “The Functional Analysis,” Nauka, Moscow, 1977.

[10] I. I. Liashko, V. F. Yemel’ianov and A. K. Boyarchuk, “Bases of Classical and Modern Mathematical Analysis,” Vysshaya Shkola Publishres, Kyiv, 1988.

[11] S. G. Mikhlin, “The Direct Methods in Mathematical Physics,” Gosudarstvennoje Izdatelstvo Tekhnicheskoy Literatury, Moscow-Leningrad, Leningrad, Moscow, 1950.

[12] G. I. Marchuk, “The Methods of Calculus Mathematics,” Nauka, Moscow, 1977.

[13] P. O. Savenko, “The Branching of Solutions of Antennas Synthesis Problems According to the Given Amplitude Directivity Pattern with Use of Regularization Functionals,” Izvestija Vysshykh Uchebnykh Zavedeniy Radioelektronica, Vol. 39, No. 2, 1996, pp. 35-50.

[14] G. M. Vainikko, “Analysis of Discretized Methods,” Tartuskiy Gosudarstvennyy Universitet, Tartu, 1976.

[15] P. A. Savenko and L. P. Protsakh, “Implicit Function Method in Solving a Two-dimensional Nonlinear Spectral Problem,” Russian Mathematics (Izv. VUZ), Vol. 51, No. 11, 2007, pp. 40-43.

[16] М. M. Vainberg and V. А. Trenogin, “Theory of Branching of Solutions of Nonlinear Equations,” Nauka, Moscow, 1969.

[17] I. G. Petrovskii, “Lectures on the Theory of Ordinary Differential Equations,” Nauka, Moscow, 1970.

[18] V. I. Smirnov, “Course of High Mathematics, vol. 1,” Nauka, Moscow, 1965.

[19] A. Gursa, “Course of Mathematical Analysis, Vol. 1, Part 1,” Gosudarstvennoje Tekhniko-teoreticheskoje Izdatalelstvo, Leningrad, Moscow, 1933.

[20] P. P. Zabreiko, А. I. Koshelev and М. А. Krasnoselskii, “Integral Equations,” Nauka, Moscow, 1968.

[21] S. G. Mikhlin, “Variational Methods in Mathematical Physics,” Nauka, Moscow, 1970.

[22] P. A. Savenko, “Numerical Solution of a Class of Nonlinear Problems in Synthesis of Radiating Systems,” Computational Mathematics and Mathematical Physics, Vol. 40, No. 6, 2000, pp. 889-899.

[23] A. F. Verlan and V. S. Sizikov, “Integral Equations: Methods, Algorithms, Programs. Reference Textbook,” Naukova Dumka, Kyiv, 1986.

[24] V. A. Morozov, “Regularizing Methods of Solving the Incorrect Problems,” Nauka, Moscow, 1987.

[25] A. N. Tikhonov, A. V. Goncharskyi, V. V. Stepanov and A. G. Yagola, “Regularizing Algoritms and a Priori Information,” Nauka, Moscow, 1983.

[26] A. G. Yagola, “About Selection of Regularizing Parameter in the Solution of Incorrect Problems in Reflexive Spaces,” Journal of Computational Mathematics and Mathematical Physics, Vol. 20, No. 3, 1980, pp. 586-596.