Bases of the Perturbed System of Exponents in Weighted Lebesgue Space with a General Weight

Download PDF


DOI: 10.46793/KgJMat2203.477S


The weighted Lebesgue and Hardy spaces with a general weight are considered. Basicity of a part of exponential system is proved in Hardy classes, where the weight satisfies the Muckenhoupt condition. Using these results the basicity of the perturbed system of exponents in the weighted Lebesgue space is studied. Some special cases are considered.


Weighted space, system of exponents, basicity.


[1]   B. T. Bilalov, On uniform convergence of series with regard to some system of sines, Differentsial’nye Uravneniya 24(1) (1988), 175–177.

[2]   B. T. Bilalov, Basicity of some systems of functions, Differentsial’nye Uravneniya 25 (1989), 163–164.

[3]   B. T. Bilalov, Basicity of some systems of exponents, cosines and sines, Differentsial’nye Uravneniya 26(1) (1990), 10–16.

[4]   B. T. Bilalov, On basicity of systems of exponents, cosines and sines in Lp, Dokl. Math. 365(1) (1999), 7–8.

[5]   B. T. Bilalov, On basicity of some systems of exponents, cosines and sines in Lp, Dokl. Math. 379(2) (2001), 7–9.

[6]   B. T. Bilalov, The basis properties of some systems of exponential functions, cosines, and sines, Sibirskii Matematicheskii Zhurnal 45(2) (2004), 264–273.

[7]   I. I. Danilyuk, Non-Regular Boundary Value Problems in the Plane, Nauka, Moscow, 1975.

[8]   G. J. Garnett, Bounded Analytic Functions, Mir, Moscow, 1984.

[9]   Z. A. Kasumov and A. Basicity, The basis property of a system of exponents in weighted Lebesgue space, Estestv. i Tekhnicheskiye Nauki, Moscow 6(50) (2010), 35–41.

[10]   K. S. Kazaryan and P. I. Lizorkin, Multipliers, bases and unconditional bases in the weighted spaces B and SB, Proc. Steklov Inst. Math. 187 (1989), 111–130.

[11]   E. I. Moiseev, Some boundary value problems for mixed-type equations, Differ. Equ. 28(1) (1992), 105–115.

[12]   E. I. Moiseev, Solution of the Frankl problem in a special domain, Differentsial’nye Uravneniya 28(4) (1992), 721–723.

[13]   E. I. Moiseev, On existence and uniqueness of solution a classical problem, Dokl. Math. 336(4) (1994), 448–450.

[14]   E. I. Moiseev, On basicity of systems of sines and cosines, DAN SSSR 275(4) (1984), 794–798.

[15]   E. I. Moiseev, On basicity of a system of sines, Differentsial’nye Uravneniya 23(1) (1987), 177–179.

[16]   E. I. Moiseev, On basicity of systems of cosines and sines in weight space, Differentsial’nye Uravneniya 34(1) (1998), 40–44.

[17]   E. I. Moiseev, The basicity in the weight space of a system of eigen functions of a differential operator, Differentsial’nye Uravneniya 35(2) (1999), 200–205.

[18]   S. M. Ponomarev, On an eigenvalue problem, DAN SSSR 249(5) (1979), 1068–1070.

[19]   I. I. Privalov, Boundary Properties of Analytic Functions. M-L, Gostekhizdat, 1950.

[20]   S. S. Pukhov and A. M. Sedletskii, Bases of exponents, sines and cosines in weight spaces on finite interval, Dokl. Math. 425(4) (2009), 452–455.

[21]   S. R. Sadigova, The general solution of the homogeneous Riemann problem in the weighted Smirnov classes, Proceedings of the Institute of Mathematics and Mechanics 40(2) (2014), 115–124.

[22]   A. M. Sedletskii, Biorthogonal expansions of functions in series of exponents on intervals of the real axis, Uspekhi Matematicheskikh Nauk 37(5) (1982), 51–95.