Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform

Affiliation(s)

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine.

Pidstryhach Institute for Applied Problems of Mechanics and Mathematics, National Academy of Sciences of Ukraine, Lviv, Ukraine.

ABSTRACT

The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear problem on mean-square approximation of a real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on two parameters, are constructed and justified.

The numerical algorithms for finding the lines of branching and branching-off solutions of nonlinear problem on mean-square approximation of a real finite nonnegative function with respect to two variables by the modulus of double discrete Fourier transform dependent on two parameters, are constructed and justified.

KEYWORDS

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

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

Cite this paper

nullP. Savenko and M. Tkach, "Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform,"*Applied Mathematics*, Vol. 1 No. 1, 2010, pp. 65-75. doi: 10.4236/am.2010.11008.

nullP. Savenko and M. Tkach, "Numerical Approximation of Real Finite Nonnegative Function by the Modulus of Discrete Fourier Transform,"

References

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[2] 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.

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[8] S. I. Solov’yev, “Preconditioned Iterative Methods for a Class of Nonlinear Eigenvalue Problems,” Linear Algebra and its Applications, Vol. 41, No. 1, 2006, pp. 210-229.

[9] 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.

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

[11] I. I. Privalov, “Introduction to the Theory of Functions of Complex Variables,” Nauka, Moscow, 1984.

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

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

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

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

[16] E. Zeidler, “Nonlinear Functional Analysis and Its Appli- cations I: Fixed-Points Theorem,” Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985.

[17] P. A. Savenko, “Synthesis of Linear Antenna Arrays by Given Amplitude Directivity Pattern,” Izv. Vysch. uch. zaved. Radiophysics, Vol. 22, No. 12, 1979, pp. 1498-1504.

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

[19] V. V. Voyevodin and Y. J. Kuznetsov, “Matrices and Calcu- lations,” Nauka, Moscow, 1984.

[20] A. Gursa, “Course of Mathematical Analysis, Vol. 1, Part 1,” Moscow-Leningrad, Gos. Technical Theory Izdat, 1933.

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

[1] B. M. Minkovich and V. P. Jakovlev, “Theory of Synthesis of Antennas, ” Soviet Radio, Moscow, 1969.

[2] 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.

[3] 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.

[4] G. M. Vainikko, “Analysis of Discretized Methods,” Таrtus Gos. University of Tartu, Tartu, 1976.

[5] R. D. Gregorieff and H. Jeggle, “Approximation von Eigevwertproblemen bei nichtlinearer Parameterabh?ngi- keit,” Manuscript Math, Vol. 10, No. 3, 1973, pp. 245- 271.

[6] O. Karma, “Approximation in Eigenvalue Problems for Holomorphic Fredholm Operator Functions I,” Numerical Functional Analysis and Optimization, Vol. 17, No. 3-4, 1996, pp. 365-387.

[7] M. A. Aslanian and S. V. Kartyshev, “Updating of One Numerous Method of Solution of a Nonlinear Spectral Problem,” Journal of Computational Mathematics and Mathe- matical Physics, Vol. 37, No. 5, 1998, pp. 713-717.

[8] S. I. Solov’yev, “Preconditioned Iterative Methods for a Class of Nonlinear Eigenvalue Problems,” Linear Algebra and its Applications, Vol. 41, No. 1, 2006, pp. 210-229.

[9] 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.

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

[11] I. I. Privalov, “Introduction to the Theory of Functions of Complex Variables,” Nauka, Moscow, 1984.

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

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

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

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

[16] E. Zeidler, “Nonlinear Functional Analysis and Its Appli- cations I: Fixed-Points Theorem,” Springer-Verlag, New York, Berlin, Heidelberg, Tokyo, 1985.

[17] P. A. Savenko, “Synthesis of Linear Antenna Arrays by Given Amplitude Directivity Pattern,” Izv. Vysch. uch. zaved. Radiophysics, Vol. 22, No. 12, 1979, pp. 1498-1504.

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

[19] V. V. Voyevodin and Y. J. Kuznetsov, “Matrices and Calcu- lations,” Nauka, Moscow, 1984.

[20] A. Gursa, “Course of Mathematical Analysis, Vol. 1, Part 1,” Moscow-Leningrad, Gos. Technical Theory Izdat, 1933.

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