Hilbert transform on $W_{\sigma}^1$

V. Dilnyi; Kh. Voitovych

doi:10.30970/ms.52.1.32-37

We obtain a boundedness criterion for the Hilbert transform on the Paley-Wiener space in the terms of decomposition. Since we have a simple method of evaluation of the Hilbert transform.

1. N. Sengupta, Md. Sahidullah, G. Saha, Lung sound classification using cepstral-based statistical features, Conputer in Biology and Medicine, 75 (2016), 118–126.

2. F.W. King, Hilbert transforms, V.1-2, Cambridge University Press, Cambridge, UK, 2009.

3. S. Salih, Fourier transform applications, InTech, 2012.

4. J. Garnett, Bounded analytic functions, Academic Press, USA, 1981.

5. A.M. Sedleckii, An equivalent definition of Hp spaces in the half-plane and some applications, Mathematics of the USSR-Sbornik. 25, (1975), 69–76.

6. B. Levin, Yu. Ljubarskii, Interpolation by means of special classes of entire functions and related expanstions in series of exponentials, Mathematics of the USSR-Sbornik, 9 (1975), 621–662.

7. V.M. Dilnyi, T.I. Hishchak, On splitting functions in Paley-Wiener space, Mat. Stud., 45 (2016), 137– 148.

8. V.M. Dilnyi, Equivalent definition of some weighted Hardy spaces, Ukrainian Math. J., 60 (2008), 1477– 1482.

9. R.S. Yulmukhametov, Solution of the Ehrenpreis factorization problem, Sbornik: Mathematics, 190 (1999), 597–629.

10. Yu. Ljubarskii, Representation of function in Hp on half-plane and some application, Teor. Funktsii, Funkts. Anal. Pril., 38 (1982), 76–84.

11. I.E. Chyzhykov, Growth of p-th means of analytic and subharmonic functions in the unit disc and angular distribution of zeros, 1509.02141.v2/arXiv.org. (2015), 1–19.

12. V.M. Dilnyi, Splitting of some spaces of analytic functions, Ufa Math. J., 6 (2014), 25–34. 13. R. Boas, Entire function, New York: Academic Press, 1954.

14. C.Eoff, The discrete nature of the Paley–Wiener spaces, Proc. Amer. Math. Soc., 123 (1995), 505–512.

15. G.Z. Ber, On interferention phenomenon in integral metric and approximation of entire functions of exponential type, Teor. Funktsii, Funkts. Anal. Pril., 34 (1980), 11–24.

16. I.I. Privalov, Boundary Properties of Analytic Functions, 2nd ed., GITTL, Moscow-Leningrad, 1950.

	Hilbert transform on $W_{\sigma}^1$
Author	V. Dilnyi¹, Kh. Voitovych² dilnyi@ukr.net¹, khrystyna.huk2711@gmail.com² 1) University of Agriculture in Krakow, Cracow, Poland; Drohobych Ivan Franko State Pedagogical University Drohobych, Ukraine; 2) Drohobych Ivan Franko State Pedagogical University Drohobych, Ukraine
Abstract	We obtain a boundedness criterion for the Hilbert transform on the Paley-Wiener space in the terms of decomposition. Since we have a simple method of evaluation of the Hilbert transform.
Keywords	Hilbert transform; Paley–Wiener space; weighted Hardy space
DOI	doi:10.30970/ms.52.1.32-37
Reference	1. N. Sengupta, Md. Sahidullah, G. Saha, Lung sound classification using cepstral-based statistical features, Conputer in Biology and Medicine, 75 (2016), 118–126. 2. F.W. King, Hilbert transforms, V.1-2, Cambridge University Press, Cambridge, UK, 2009. 3. S. Salih, Fourier transform applications, InTech, 2012. 4. J. Garnett, Bounded analytic functions, Academic Press, USA, 1981. 5. A.M. Sedleckii, An equivalent definition of Hp spaces in the half-plane and some applications, Mathematics of the USSR-Sbornik. 25, (1975), 69–76. 6. B. Levin, Yu. Ljubarskii, Interpolation by means of special classes of entire functions and related expanstions in series of exponentials, Mathematics of the USSR-Sbornik, 9 (1975), 621–662. 7. V.M. Dilnyi, T.I. Hishchak, On splitting functions in Paley-Wiener space, Mat. Stud., 45 (2016), 137– 148. 8. V.M. Dilnyi, Equivalent definition of some weighted Hardy spaces, Ukrainian Math. J., 60 (2008), 1477– 1482. 9. R.S. Yulmukhametov, Solution of the Ehrenpreis factorization problem, Sbornik: Mathematics, 190 (1999), 597–629. 10. Yu. Ljubarskii, Representation of function in Hp on half-plane and some application, Teor. Funktsii, Funkts. Anal. Pril., 38 (1982), 76–84. 11. I.E. Chyzhykov, Growth of p-th means of analytic and subharmonic functions in the unit disc and angular distribution of zeros, 1509.02141.v2/arXiv.org. (2015), 1–19. 12. V.M. Dilnyi, Splitting of some spaces of analytic functions, Ufa Math. J., 6 (2014), 25–34. 13. R. Boas, Entire function, New York: Academic Press, 1954. 14. C.Eoff, The discrete nature of the Paley–Wiener spaces, Proc. Amer. Math. Soc., 123 (1995), 505–512. 15. G.Z. Ber, On interferention phenomenon in integral metric and approximation of entire functions of exponential type, Teor. Funktsii, Funkts. Anal. Pril., 34 (1980), 11–24. 16. I.I. Privalov, Boundary Properties of Analytic Functions, 2nd ed., GITTL, Moscow-Leningrad, 1950.
Pages	32-37
Volume	52
Issue	1
Year	2019
Journal	Matematychni Studii
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