Document Type : Research Paper
Authors
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.
Keywords
Main Subjects
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[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.
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[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.
[40] C.T. Zorko, Morrey space, Proc. Amer. Math. Soc. 98 (1.4) (1986) 586-592.
[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.
[40] C.T. Zorko, Morrey space, Proc. Amer. Math. Soc. 98 (1.4) (1986) 586-592.
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