Document Type: Research Paper
Author
- Fidan Seyidova ^{}
Ganja State University, Ganja, Azerbaijan.
Abstract
In this work systems of sines $\sin \left(n+\beta \right)t,\, \, n=1,2, \ldots,$ and cosines $\cos \left(n-\beta \right)t,\, \, n=0,1,2, \ldots,$ are considered, where $\beta \in R-$is a real parameter. The subspace $M^{p,\alpha } \left(0,\pi \right)$ of the Morrey space $L^{p,\alpha } \left(0,\pi \right)$ in which continuous functions are dense is considered. Criterion for the completeness, minimality and basicity of these systems with respect to the parameter $\beta $ in the subspace $M^{p,\alpha } \left(0,\pi \right)$, $1<p <+\infty, $ are found.
Keywords
[1] D.R. Adams, Morrey spaces, Switzherland, Springer, 2016.
[2] B.T. Bilalov, A system of exponential functions with shift and the Kostyuchenko problem, Sib. Math. J., 50(2) (2009), pp. 223-230.
[3] B.T. Bilalov, Bases of a system of exponentials in $L_p $, Dokl. Akad. Nauk, 392(5) (2003), pp. 583-585.
[4] B.T. Bilalov, Basis Properties of Some Systems of Exponents, Cosines and Sines, Sib. Math. J., 45 (2004), pp. 264-273.
[5] B.T. Bilalov, On solution of the Kostyuchenko problem, Sib. Math. J., 53(3) (2012), pp. 404-418
[6] B.T. Bilalov, The basis property of a perturbed system of exponentials in Morrey-type spaces, Sib. Math. J., 60:2 (2019), pp. 249-271.
[7] B.T. Bilalov, T.B. Gasymov, A.A. Guliyeva, On solvability of Riemann boundary value problem in Morrey-Hardy classes, Turkish J. Math., 40(50) (2016), 1085-1101.
[8] B.T. Bilalov, A.A. Guliyeva, On basicity of the perturbed systems of exponents in Morrey-Lebesgue space, Internat. J. Math., 25:1450054, (2014), pp. 1-10.
[9] B.T. Bilalov, Z.G. Guseynov, Basicity criterion for perturbed systems of exponents in Lebesgue spaces with variable summability, Dokl. Akad. Nauk, 436(5) (2011), pp. 586-589.
[10] B.T. Bilalov, Z.G. Guseynov, Basicity of a system of exponents with a piece-wise linear phase in variable spaces, Mediterr. J. Math, 9(3) (2012),
pp. 487-498.
[11] B.T. Bilalov, A.A. Huseynli, S.R. El-Shabrawy, Basis Properties of Trigonometric Systems in Weighted Morrey Spaces, Azerb. J. Math., 9(2) (2019), pp. 200-226
[12] B.T. Bilalov, F.Sh. Seyidova, Basicity of a system of exponents with a piecewise linear phase in Morrey-type spaces, Turkish J. Math. 43 (2019), pp. 1850-1866
[13] R.E. Casillo, H. Rafeiro, An introductory course in Lebesgue spaces, Springer, 2016.
[14] D.V. Cruz-Vrible, A. Fiorenza, Variable Lebesgue spaces, Springer-Verlag, Basel, 2013.
[15] L. Diening, P. Harjuleto, P. Hasto, M. Ruzicka, Lebesgue and Sobolev spaces with variable exponents, Springer, 2011, 509 p.
[16] M.I. Kadets, The exact value of the Paley-Wiener constant, Dokl. Akad. Nauk, 155 (1964), pp. 1253-1254.
[17] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, Volume 1: Variable Exponent Lebesgue and Amalgam Spaces, Springer, 2016.
[18] V. Kokilashvili, A. Meskhi, H. Rafeiro, S. Samko, Integral Operators in Non-Standard Function Spaces, Volume 2: Variable Exponent H"older, Morrey-Campanato and Grand Spaces, Springer, 2016.
[19] B.Y. Levin, Distribution of Roots of Entire Functions, Moscow, GITL, 1956 (in Russian).
[20] E.I. Moiseev, On basicity of a system of sines, Differ. Equ. 23(1) (1987), pp. 177-179.
[21] E.I. Moiseev, On basicity of systems of sines and cosines, Dokl. Akad. Nauk, 275(4) (1984), pp. 794-798.
[22] E.I. Moiseev, On solution of Frankle's problem in special domain, Differ. Equ., 28(4), (1992) pp. 682-692.
[23] E.I. Moiseev, On some boundary value problems for mixed type equations, Differ. Equ., 28(1), (1992) pp. 123-132.
[24] E.I. Moiseev, N.O. Taranov, Solution of a Gellerstedt problem for the Lavrent'ev-Bitsadze equation, Differ. Equ., 45 (2009), pp. 543-548.
[25] T.R. Muradov, On basicity of double and unitary systems in generalized Lebesgue spaces, Math. AE terna, 5(4) (2015), pp. 657-666.
[26] N.P. Nasibova, F.Sh. Seyidova, On the basicity of a perturbed system of exponents with a unit in Morrey-type spaces, Trans. Natl. Acad. Sci. Azerb. Ser. Phys.-Tech. Math. Sci. Mathematics, 39(1) (2019), pp. 151-161.
[27] R. Paley, N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ., 19 (Amer. Math. Soc., Providence, RI, 1934)
[28] S.M. Ponomarev, On one eigenvalue problem, Dokl. Akad. Nauk,, 249 (1979), pp. 1068-1070. (in Russian)
[29] S.M. Ponomarev, On the theory of boundary value problems for equations of mixed type in three-dimensional domains, Dokl. Akad. Nauk,, 246 (6) (1979) pp. 303-1304.
[30] A.M. Sedletskii, Biorthogonal expansions in series of exponents on intervals of real axis, Uspekhi Mat. Nauk, 37(5) (1982), pp. 51-95.