Yaremko Oleg Emanuilovich, Doctor of physical and mathematical sciences, professor, sub-department of computer techologies, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: firstname.lastname@example.org
Yaremko Nataliya Nikolaevna, Doctor of pedagogical sciences, professor, sub-department of mathematical education, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: email@example.com
Mogileva Elena Sergeevna, Postgraduate student, Penza State University (40 Krasnaya street, Penza, Russia), E-mail: firstname.lastname@example.org
Background. The method of integral transforms is one of the most important analytical methods of mathematical modeling. Numerical methods and computational algorithms are developed on its basis. Image properties indirectly reflect the properties of the originals. Sometimes, for example, for Fourier images, these properties contain new information about the original. The article is devoted to the study of the logarithmic convexity of the image for a non-negative original.
Materials and methods. Methods of information geometry allowed us to establish the properties of integral Fourier transforms for the first time by studying the corresponding Fisher information matrix. Methods of Laplace, Mellin, Weierstrass and others integral transforms theory were also used in obtaining the results.
Results. Formula for the Fischer information matrix and stress tensor for randomized families of distributions associated with Laplace, Mellin, and Weierstrass integral transforms is found. The logarithmic image’s convexity for a non-negative original is established. A new proof of the logarithmic convexity of the Gammafunction and the moments inequality of distribution is proposed.
Conclusions. The proposed methods can be useful in the study of special functions of mathematical physics, in the theory of fractional-order integrals. Having an explicit expression of the information matrix is important for statistical applications.
1. Bavrin I. I., Pan'zhenskiy V. I., Yaremko O. E. Chebyshevskiy sbornik [Peer-reviewed theoretical journal Chebyshevskii Sbornik]. 2015, vol. 16, no. 4, pp. 28–40. [In Russian]
2. Rao S. R. Lineynye statisticheskie metody i ikh primeneniya [Linear statistical methods and their applications]. Moscow: Nauka, 1968, 548 p. [In Russian]
3. Barndorff N. O. Information and exponential families in statistical theory. Wiley Series in Probability and Mathematical Statistics. Chichester: John Wiley & Sons, 1978, 238 p.
4. Brychkov Y. A., Prudnikov A. P. Integral Transforms of Generalized Functions, Chapter 5. New York-London, CRC Press, 1989, 342 p.
5. Ahmed I. Z. Handbook of Function and Generalized Function Transformations, Chapter 18. Taylor & Francis, CRC Press, 1996, 672 p.
6. Feller V. Vvedenie v teoriyu veroyatnostey i ee prilozheniya: v 2 t. [Introduction to probability theory and its applications: in 2 volumes]. Moscow: Mir, 1984, vol. 1, 511 p. [In Russian]
7. Podlubny I. Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, Some Methods of Their Solution and Some of Their Applications, Mathematics in Science and Engineering. Academic Press, 1998, 340 p.
8. Bilodeau G. G. University of North Carolina, Duke Mathematical Journal. 1962, vol. 129, pp. 293–308.
9. Carslaw H. S., Jaeger J. C. Conduction of Heat in Solids. 2nd ed. Oxford: Oxford University Press, 1959, 510 p.
10. Courant R., Hilbert D. Methods of Mathematical Physics, vol. II. New York, Inter science (Wiley), 1962, 575 p.