Mean-Square Approximation of Complex Variable Functions by Fourier Series in the Weighted Bergman Space

The problem of mean-square approximation of complex variables functions regularly in some simply connected domain D ⊂ C with Fourier series by orthogonal system when the weighted function γ: = γ(|z|) is nonnegative integrable in D, was considered. An exact convergence rate of Fourier series by orthogonal system of functions on some class of functions given by special module of continuity of m-th order were obtained. An exact values of n-widths for specified class of functions were calculated.


Introduction and Preliminary Results
In this paper, the quadratic approximation of functions with Fourier series by orthogonal system over complex variable domain in the presence of weight was considered.
In the domain of ⊂ ℂ is given a nonnegative measurable and not equivalent zero function (| |), such that there is exists a finite integral where the integral is understood in sense of Lebesgue and the element of area. Function = (| |) satisfying the above condition we call a weight function. We will consider the problems of mean square approximation by Fourier sum of complex function regularly in the simply connected domain and is belong to the space 2, : = 2 ( (| |), ) with finite norm ∥ ∥ 2, : =∥ ∥ 2, = (∬ ( ) where (| |) weighted function in the domain . Where the domain is disk | | < (0 < < ∞) the 2, space is a Bergman space 2, introduced in [1,2]. An extremal problems of analytic functions and the problem of calculation of different values of -widths in the space 2, are considered in works (see, e.g., [3][4][5][6][7]).
The results which are obtained in this paper are the generalization and continuation of work [9]. We shall indicate some other papers that are close to our work in which the analogy questions for other orthogonal system of functions [10][11][12][13] were studied, and therefore, we bring the necessary definitions and facts from it for further studying.
Let { ( )} =0 ∞ be complete orthonormal system in domain of a system of complex functions in the space 2, : are the Fourier series of function ∈ 2, under this system, are the partial sums of order. Let be the subspace of generalized complex polynomials of form where ∈ ℂ. Then, as it is well known (see, e.g., [8], p.263): where ( ) are the Fourier coefficients of function defined in (1). Now consider the function where ℎ ∈ (0,1), ( , ) ∈ × , and the series in the right side of (3) is understood in the meaning of convergence in the space 2 ( × ; (| |) (| |)). Just note that in some cases we can show the explicit form for the function ( , ; ℎ). Thus, for example, if = { ∈ ℂ; | | < 1}, (| |) = 1, then the system of functions ( ) = √( + 1)/ , = 0,1, . .. is orthonormalized (see, e.g., [8, p.208]). In this case, we have (see, e.g., [9]): We return to the general case of the domain . In 2, space, we shall consider an operator (4) which is called generalized translation operator. The operator ℎ ( ) has the following properties: Using the generalized translation operator ℎ ( ) for an arbitrary function ∈ 2, , we define the finite-difference of first and higher order by the equations where ℎ 0 ( ) = ( ) = ( ), ℎ ( ) ( ) = ℎ ( ℎ ( −1) ( )), = 1, , ∈ ℕ, -unit operator in the space 2, . The magnitude Ω ( ; ) 2, = sup{∥ Δ ℎ ∥ 2, : 0 < ℎ ≤ }(5) we call a generalized module of continuity of m-th order of function ∈ 2, . Further, we need the following simple lemma. Lemma 1. For an arbitrary function ∈ 2, is hold Proof. First, it is observed that an operator (4) with respect of (3) is representable in form using which we find consecutively Then using the obtained formula for any ∈ ℕ and ℎ ∈ (0,1) we find: Applying the Parseval equality for (7) and because of system of functions which afford to obtain (6) because of (5). Lemma is proved. We must note that the last equation was stated in [9, pp.1001] without proof. In work [9] it was proved that for any arbitrary function ∈ 2, for each ∈ (0,1) is hold an estimate (8) and for each fixed the constant in the right side of inequality (7) cannot be reduced. Indeed, on one hand for any function ∈ 2, we find:

Main Results
Further and everywhere by weighted function in segment [0, ℎ] we shall understand a nonnegative measurable and summable in [0, ℎ] function ( ) that is not equal to zero. The following theorem is valid.

.(16)
For obtaining the bellow estimate of the same magnitude we assume that 0 ( ): = ( ) ∈ 2, . Since for this function Comparing the above estimate (16) and below estimate (17) we obtain an equation (12). This completes the proof of Theorem 1.
The proved Theorem 1 implies the following corollaries.
Theorem 2 implies the following statement.

Corollary 3.
In theorem 2 when ℎ = 1/ , ∈ ℕ there is hold an asymptotic equation Theorem 3. Let ∈ ℕ, 0 < ≤ 2, ℎ ∈ (0,1), ≥ 0 is a weighted function in (0, ℎ). Then for any ∈ ℕ are valid the equations Raising both side of inequality (30) to the power , multiplying them by weighted function and integrating both side with respect in the limits from = 0 to = ℎ we have We obtain the required equation (28) by comparing the above (29) and below (31) estimates. Theorem 3 is proved.