^{1}Institute of Mathematics and Mechanics of NAS of Azerbaijan, Az1141, Baku, Azerbaijan.

^{2}Math teacher at the school No 297, Baku, Azerbaijan.

Abstract

Double system of exponents with complex-valued coefficients is considered. Under some conditions on the coefficients, we prove that if this system forms a basis for the Morrey-Lebesgue type space on $\left[-\pi , \pi \right]$, then it is isomorphic to the classical system of exponents in this space.

[1] A.N. Barmenkov and A.Y. Kazmin, Completeness of a system of functions of special type. In the book: theory of mappings, some its generalization and applications, Kiev, Naukova Dimka 1982, p.29-43.

[2] B.T. Bilalov, On uniform convergence of series in a system of sines, Diff. func., 1988, Vol. 24, No 1, p. 175-177.

[3] B.T. Bilalov, Basicity of some systems of functions, Diff. func., 1989, Vol. 25, p. 163-164.

[4] B.T. Bilalov, Basicity of some systems of exponents, cosines and sines, Diff. func., 1990, Vol. 26, No 1, p. 10-16.

[5] B.T. Bilalov, Basis problems of some systems of exponents, cosines and sines, Sibirskiy matem. zhurnal, 2004, Vol. 45, No 2 , p. 264-273.

[6] B.T. Bilalov, On completeness of exponent system with complex coefficients in weight spaces, Trans. of NAS of Azerbaijan, XXV, No 7, 2005, p. 9-14.

[7] B.T. Bilalov, On isomorphism of two bases in textbf{$L_{p} $}, Fundam. Prikl. Mat., 1: 4 (1995), 1091-1094.

[8] B.T. Bilalov and Z.G. Guseynov, Basicity of a system of exponents with a piece-wise linear phase in variable spaces, Mediterr. J. Math. Vol. 9, No 3 (2012), 487-498.

[9] B.T. Bilalov and Z.G. Guseynov, Basicity criterion of perturbed system of exponents in Lebesgue spaces with variable summability index, Dokl. Akad. Nauk, 2011, Vol. 436, No 5, p. 586-589.

[10] B.T. Bilalov and A. A. Quliyeva, On basicity of exponential systems in Morrey-type spaces, International Journal of Mathematics. Vol. 25, No. 6 (2014) 1450054 (10 pages).

[11] A.V. Bitsadze, On a system of fuctions, UMN, 1950, Vol.5, issue 4 (38), p. 150-151.

[12] Y. Chen, Regularity of the solution to the Dirichlet problem in Morrey space, J. Partial Differ. Eqs. 15 (2002) 37-46.

[13] G.G. Devdariani, Basicity of a system of sines, Trudy Inst. Prikl. mat. I.N. Vekua, 1987, Vol. 19, p. 21-27.

[14] G.G. Devdariani, On basicity of a system of functions, Diff. func., 1986, Vol. 22, No 1, p. 170-171.

[15] G.G. Devdariani, On basicity of a system of functions, Diff. func., 1986, Vol. 22, No 1, p. 168-170.

[16] G.M. Goluzin, Geometrical theory of a complex variable functions, M., Nauka, 1966, p. 626.

[17] D.M. Israfilov and N.P. Tozman, Approximation by polynomials in Morrey-Smirnov classes, East J. Approx. 14(1.3) (2008) 255-269.

[18] D.M. Israfilov and N.P. Tozman, Approximation in Morrey-Smirnov classes, Azerbaijan J. Math. 1(1.1) (2011) 99-113.

[19] M.I. Kadets, On exact value of Paley-Wiener constant, DAN SSSR, 1964.

[20] V. Kokilashvili and A. Meskhi, Boundedness of maximal and singular operators in Morrey spaces with variable exponent, Govern. College Univ. Lahore 72 (2008) 1-11.

[21] N.X. Ky, On approximation by trigonometric polynomials in L p u -spaces, Studia Sci. Math. Hungar 28 (1993) 183-188.

[22] B. Ya. Levin, Distribution of the roots of entire functions. M., GITL, 1956.

[23] A.L. Mazzucato, Decomposition of Besov-Morrey spaces, in "Harmonic Analysis at Mount Holyoke'', American Mathematical Society Contemporary Mathematics, 320 (2003) 279-294.

[24] E.I. Moiseev, On basicity of the system of sines and cosines, DAN SSSR, 1984, Vol. 275, No 4, p. 794-798.

[25] E.I. Moiseev, On some boundary value problems for mixed equations, Diff. func., 1992, Vol. 28, No 1, p. 123-132.

[26] E.I. Moiseev, On the solution of Frankles problem in a special domain, Diff. func., 1992, Vol. 28, No 4, p. 682-692.

[27] E.I. Moiseev, On the existence and uniqueness of the solution of a classic problem, Dokl. RAN, 1994, Vol. 336, No 4, p. 448-450.

[28] E.I. Moiseev, On basicity of a system of sines, Diff. func., 1987, Vol. 23, No 1, p. 177-179.

[29] E.I. Moiseev, On basicity of a system of sines, cosines in weight space. Diff. func., 1998, Vol. 34, No 1, p. 40-44.

[30] E.I. Moiseev, On differential properties of expansions in the system of sines and cosines, Diff. func., 1996, Vol. 32, No 1, p.117-126.

[31] R. Paley and N. Wiener, Fourier Transforms in the Complex Domain, Amer. Math. Soc. Colloq. Publ., 19 (Amer. Math. Soc., Providence, RI, 1934).

[32] J. Peetre, On the theory of spaces, J. Funct. Anal. 4 (1964) 71-87.

[33] S.M. Ponomarev, On an eigenvalue problem, DAN SSSR, 1979, Vol. 249, No 5, p. 1068-1070.

[34] S.M. Ponomarev, To theory of boundary value problems for mixed type equations in three-dimensional domains, DAN SSSR, 1979, Vol. 246, No 6, p. 1303-1304.

[35] S.S. Pukhov and A.M. Sedletskiy, Bases of exponents, sines and cosines in weighs spaces on a finite interval, Dokl. RAN, 2009, Vol. 425, No 4, p. 452-455.

[36] D.L. Russel, On exponential bases for the Sobolev spaces over an interval, Journ. of Math. Anal. and Appl., 87, 528-550 (1982).

[37] N. Samko, Weight Hardy and singular operators in Morrey spaces, J. Math. Anal. Appl. 35 (1.1) (2009) 183-188.

[38] A.M. Sedletskiy, Biorthogonal expansions in series of the exponents on the intervals of a real axis, Usp. Mat. Nauk, 1982, Vol. 37, issue 5 (227), p. 51-95.

[39] A.M. Sedletskiy, Approximate properties of a system of exponents in Sobolev spaces, Vestnik Mosc. Univ., ser. 1, math.-mech., 1999, No 6, p. 3-8.