Linear best method for recovering the second derivatives of Hardy class functions

The linear best method for approximating the second derivatives of Hardy class functions defined in the unit circle at zero in accordance with the information about their values in a finite number of points forming a regular polygon is found. The paper is divided into three sections. The first contains the necessary concepts and results from the work of K.Yu. Osipenko. It also recalls some results obtained by S. Ya. Havinson and other authors. In the second section, the error of the best method is calculated, and the corresponding extremal functions are written out. The third proves that the linear best approximation method is unique, and its coefficients are calculated.


Introduction
Let W be some set lying in a linear complex space X and , 1 , … , − linear complex functionals defined on X. If ( 1 , … , ) is a complex function of n complex variables, then the approximation error of the functional L by the values of the functionals 1 , … , by the method S on the set W is the quantity ( ) = sup ∈ | ( ) − ( 1 ( ), … , ( )|.
Complex function 0 ( 1 , … , ) is called the best approximation method if The existence of the linear best method was proved in [1] (under certain conditions on the set W). In addition, it was found that the error of the best approximation method can be calculated by the following formula Be the unit ball in the Hardy space (1≤ p <∞) (see the definition of Hardy classes in [2], [3]). We note that the problems of optimal recovery of functions belonging to certain classes and derivatives of functions with respect to their values in a finite number of points have been studied in many papers (see, for example, [1], [4][5][6][7][8][9][10][11][12]). This work is a continuation of [13].
Let's consider the following system of points forming a regular polygon where − set number; 0 < < 1.
If now > 1. And if ( ) ∈ (G) (here 1 + 1 = 1 ), then the following duality relation holds The extremal function * ( ) is unique on the right side of equality (4) and on the right side of equality (6). The extreme function * ( ) on the left side of equality (6) is unique up to a constant factor ( ∈ ). In addition, in order for the functions * ( ) ∈ 1 и * ( ) ∈ to be extremal in equality (6), it is necessary and sufficient that the following relation holds almost everywhere on the boundary G: where − the total value in equality (6), ∈ . Recall that the extreme function * ( ) satisfies the condition (for all 1 ≤ < ∞) G Further, if ( ) is the boundary value on the circumference G of the function ( ) meromorphic in the circle K with poles 1 , … , (each pole is repeated as many times as its multiplicity is), then the product analytically on the boundary G and has in K zeros. It is obtained in [14]- [15] where | | ≤ 1, = 1, … , − 1; − constant number.
Moreover, if > 1, then the extremal function * ( ) of problem (15) is unique up to a In the case when = 1, the extremal function of problem (15) is not unique. Any of the extremal functions of problem (15) where ∈ ; , − any complex numbers satisfying the conditions: For all 1 ≤ < ∞. After this, we consider separately two cases: the case when > 1 and when = 1. This (see (18)) implies equality (15). The extremal function of problem (15) is unique for > 1 (up to a factor equal to unity in modulo) and has the form (16) (see (19) and the introduction).
Therefore, the extremal function of problem (15) in the case when = 1 has the form (17) (see (21)). Conversely, it is easy to verify that a function of the form (17) is an extremal function of problem (15) for = 1. The lemma is proved. It is obvious that where ( ) ∈ 1 . Conversely, any of the functions f(z) having the form (26) belongs to the family E. Since (see (12), (13) It is clear that the extremal function of problem (3) for > 1 has the form (23), and for = 1, any of the extremal functions of problem (3) has the form (24).

Finding the coefficients of the linear best approximation method
Let ∑ ( ) − =1 linear best method of recovering the value of ′′(0) by the values of ( 1 ), … , ( ), where ( ) ∈ 1 , and points 1 , … , has the form (2), and * ( ) is an extremal function of problem (3). Then, it is easy to verify that * ( ) is an extremal function of the problem of recovering ′′(0) by values of functions at points 1 , … , is unique ( ( ) ∈ ; 1 ≤ < ∞), and its coefficients are found by the formulas for all values of = 1, … , .

Conclusion
Thus, the problem of the optimal recovery of the second derivatives of the functions of Hardy class at zero by their values in a finite number of points forming a regular polygon centered at zero is solved. It would be interesting to solve the problem of optimal recovery of derivatives ( ) (0), where − any natural number by values ( 1 ), … , ( ) ( ( ) ∈ ), and points 1 , … , form a regular polygon.