Deconvolution problem for indicators of segments

  • Volchkova Natalia P., volchkova.n.p@gmail.com Donetsk National Technical University, 58 Artyom Street, Donetsk 83000, Ukraine
  • Volchkov Vitaly V., volna936@gmail.com Donetsk National University, 24 Universitetskaya Street, Donetsk 83001, Ukraine
Keywords: convolution equations, inversion formulas, two-radii theorem, compactly supported distributions

Abstract

Let $\mu_1,\dots,\mu_n$ be a family of compactly supported distributions on real axis. Reconstruction of a function (distribution) $f$ by given convolutions $f\star\mu_1,\dots,f\star\mu_n$ is called deconvolution. We consider the deconvolution problem for $n=2$ and $\mu_j=\chi_{r_j},$ $j=1,2,$ where $\chi_{r_j}$ is the indicator of segment $[−r_j, r_j].$ This problem is correctly settled only under the condition of incommensurability of numbers $r_1$and $r_2$. The main result of the article gives an inversion formula for the operator $f\rightarrow(f\star\chi_{r_1},f\star\chi_{r_2})$ in the indicated case.

References


[1]
Berenstein C. A. and Yger A., “Le probleme de la deconvolution,” J. Funct. Anal., 54, 113–160 (1983).

[2]
Casey S. D. and Walnut D. F., “Systems of convolution equations, deconvolution, Shannon sampling, and the wavelet and Gabor transforms,” SIAM Rev., 36, No. 4, 537–577 (1994).

[3]
Volchkov V. V. and Volchkov Vit. V., Harmonic Analysis of Mean Periodic Functions on Symmetric Spaces and the Heisenberg Group, Springer-Verl., London (2009).

[4]
Volchkov V. V., Integral Geometry and Convolution Equations, Dordrecht, Kluwer (2003).

[5]
Borodin A. I., Number Theory, Vysha Shkola, Kiev (1992).

[6]
Berenstein C. A. and Yger A., “Analytic Bezout identities,” Adv. Appl. Math., 10, 51–74 (1989).

[7]
Hormander L., “Generators for some rings of analytic functions,” Bull. Amer. Math. Soc., 73, 943–949 (1967).

[8]
Hormander L., The Analysis of Linear Partial Differential Operators, Vols. I, II, SpringerVerl., New York (1983).

[9]
Volevich L. R. and Gindikin S. G., Generalized Functions and Convolution Equations [in Russian], Nauka, Moscow (1994).

[10]
Levin B. Ya. (in collaboration with Lyubarski˘i Yu., Sodin M., and Tkachenko V.), Lectures on Entire Functions, Amer. Math. Soc., Providence, RI (1996) (Transl. Math. Monogr.; 150).

[11]
Ilyin V. A., Sadovnichiy V. A., and Sendov Bl. Kh., Mathematical Analysis [in Russian], Mosk. Gos. Univ., Moscow (1987).
How to Cite
Volchkova, N. and Volchkov, V. ( ) “Deconvolution problem for indicators of segments”, Mathematical notes of NEFU, 26(3), pp. 3-14. doi: https://doi.org/10.25587/SVFU.2019.47.12.001.
Section
Mathematics