Optimal recovery of derivatives of Hardy class functions

The paper considers the best linear method for approximating the values of derivatives of Hardy class functions in the unit circle at zero according to the information about the values of functions at a finite number of points z1,...,zn that form a regular polygon, and also the error of the best method is obtained. The introduction provides the necessary concepts and results from the papers of K.Yu. Osipenko. Some results of the studies of S.Ya. Khavinson and other authors are also mentioned here. The main section consists of two parts. In the first part of the second section, the research method is disclosed, namely, the error of the best method for approximating the derivatives at zero according to the information about the values of functions at the points z1,...,zn is calculated; the corresponding extremal function is written out. It is established that for p>1, the corresponding extremal function is unique up to a constant factor that is equal to one in modulus. For p=1, the corresponding extremal function is not unique. All such corresponding extremal functions are determined here. In the second part of the second section, it is proved that for all p (1≤p<∞), the best linear approximation method is unique, and the coefficients of the best linear recovery method are calculated. The expressions used to calculate the coefficients are greatly simplified. At the end of the paper, the obtained results are described, and possible areas for further research are indicated.


Introduction
The problems of optimal recovery of derivatives of analytic functions of the Hardy class, given in the unit circle in accordance with the information about their values in a finite number of points, have been considered in many papers (see [1] - [5]). In this paper, a particular case is considered, namely, the values of ′(0) are optimally recovered by the values of ( 1 ), … , ( ), where the points 1 , … , form a regular polygon ( ( ) ∈ , 1 ≤ < ∞). In this case, the coefficients of the best linear method that were found as a result of the recovery have a very simple form.
Let's recall some concepts and results from [6]. Let W be some set lying in the linear space X and , 1 , … , are linear functionals defined in X. If ( 1 , … , ) is any complex function n of complex variables, then the approximation error of the method S is called the following value * Corresponding author: 6714543@rambler.ru The method 0 ( 1 , … , ) is called the best approximation method if ( 0 ) = inf ( ).
In [6], the existence of the best linear approximation method 0 = ∑ ( ) =1 was proved (under certain conditions at W) (here − some complex numbers). Moreover, the error of the best approximation method can be found by the formula (1) unit ball in the space (for the definition of the classes of analytic functions in K, see [7], [8]). Let us introduce the following points that form a regular polygon where − some number (0 < < 1; ≥ 3). If we consider that Now let us present some results from [9], [10]. If ( ) is a bounded function on the circumference Γ, then (5) where ( ) − the space of bounded, analytic in K functions. In order for the functions f * ( ) ∈ 1 1 and * ( ) ∈ ( ) to be extreme in the equality (5), it is necessary and sufficient that the following relation would be satisfied almost everywhere on Γ * ( )[ ( ) − * ( )] = | * ( )| ( real, − the total value of both extrema in (5)).
Let > 1. Then if ( ) ∈ (Γ) ( 1 + 1 = 1), then The extremal function * ( ) is unique in the right sides of equalities (5) and (7). For > 1, the extremal function * ( ) on the left side of the equality (7) is unique up to a constant factor ( ∈ ). In order for the functions * ( ) ∈ 1 and * ∈ to be extremal in the  (7), it is necessary and sufficient that the following relation would be satisfied almost everywhere on Γ * ( )[ ( ) − * ( )] = | * ( )| , (8) where − the total value in the equality (7), ∈ . It is also known that if * ( ) − the extremal function on the left side of the equality (7) (or (5)), then And finally, if ( ) is a boundary value on Γ of a ( ) function meromorphic in K with poles 1 , … , (each pole is repeated as many times as its multiplicity), then the product is analytical on Γ and has in K = − 1 (11) zeros. In the papers [9], [10], the following is also established (see (10), (11)) where | | ≤ 1, = 1, … , − 1; − constant number. Note that this paper is a continuation of [11], in which the coefficients of the best linear method for approximating the values of (0) by the values of ( 1 ), … , ( ), where ( ) ∈ ( ), are found. We need the following equality below (see [3]) where − given number (| | < 1). It was proved in [9] that the extremal function * ( ) of the problem (13) is unique up to a constant factor , ∈ and has the form * ( ) = with all p (1 ≤ < ∞). And finally, in the future, we will use the formulas (see [11]) where points 1 , … , form a regular polygon (see (2)).  (2), a finite Blaschke product has the following properties: Proof. First, we prove the equality (18). Indeed, Let us prove the equality (19). Truly, Let us verify the validity of equalities (20). To do this, decompose a finite Blaschke product B(z) into a Maclaurin series. Let and the extremal function of problem (21) with > 1 is unique up to a factor ( ∈ ) and has the form * ( ) = 1 (2 ) 1 .
with all 1 ≤ < ∞. After that, we consider two cases.
Let's assume that > 1 and f(z) is any function ( ( ) ∈ 1 ). Then, applying the Hölder's inequality, we get This implies the equality (21). The extremal function of the problem (21) with > 1 is unique up to a constant factor that is equal to one in modulus (see (25) and the introduction) and has the form (22). Now let's assume that p=1 and f(z) is any function ( ( ) ∈ 1 1 ). Then  (24)). In this case, the extremal function of the problem (21), as we will see later, is not unique. Let * ( ) be any extremal function of the problem (21). Let * ( ) be the extremal function of the dual problem (see (5) Then the duality relation connecting extremal functions has the form (see (6) , e iδ = 1. Since dζ = iζdφ, the equality (26) holds. Therefore, φ * (z) = 0 (z ∈ K ̅ ). Let's apply the relation (12). We will get f * (z) where − some number; | | ≤ 1. Let's find constant number C. To do this, we calculate the following integral: From here it follows that It means (see (9)) . Therefore, the extremal function of problem (21) with p=1 is not unique and has the form (23) (see (27)). The lemma is proved. It is clear from the above that for > 1 , the extremal function * ( ) of the problem (4) has the form (29), and for p=1, it has the form (30)). The lemma is proved.
Let > 1. We obtain (see (19), (28)) The function Q(z) has specific points 0, 1 , … , (poles). Therefore, it can be represented as After that, we find the residues in the simple poles 1 , … , . First, we note the following. Since So (see (41)), For all = 1, … , . After that, we calculate the residues of the function Q(z) at the points .

Results
Thus, as a result of solving the problem, the error of the best approximation method is calculated, the corresponding extremal function is found, the uniqueness of the best linear recovery method is proved, and finally, the coefficients of the best linear approximation method are determined.