Mean-square approximation of complex variable functions by Fourier series in the weighted Bergman space

In this paper we consider the problem of mean-square approximation of functions of a complex variable by Fourier series in orthogonal system. The functions f under consideration are assumed to be regular in some simply connected domain D⊂C and square integrable with a nonnegative weight function γ:=γ(|z|) which is integrable in D, that is, when f∈L2,γ:=L2(γ(|z|),D). Earlier, V. A. Abilov, F. V. Abilova and M. K. Kerimov investigated the problems of finding exact estimates of the rate of convergence of Fourier series for functions f∈L2,γ [9]. They proved some exact Jackson type inequalities and found the values of the Kolmogorov's n-width for certain classes of functions. In doing so, a special form of the shift operator was widely used to determine the generalized modulus of continuity of mth order and classes of functions defined by a given increasing in R+:=[0,+∞) majorant. The article continues the research of these authors, namely, the exact Jackson-Stechkin type inequality between the best approximation of a functions f∈L2,γ by algebraic complex polynomials and Lp norm of generalized module of continuity is proved; àpproximative properties of classes of functions are studied for which the Lp norm of the generalized modulus of continuity has a given majorant. Under certain assumptions on the majorant,the values of Bernstein, Kolmogorov, linear, Gelfand, and projection n-widths for classes of functions in L2,γ were calculated. It was proved that all widths are coincide and an optimal subspace is the subspace of complex algebraic polynomials.