K-FUNCTIONALS AND EXACT VALUES OF n-WIDTHS IN THE BERGMAN SPACE

In this paper, we consider the problem of mean-square approximation of complex variables functions which are regular in the unit disk of the complex plane. We obtain sharp estimates of the value of the best approximation by algebraic polynomials in terms of K-functionals. Exact values of some widths of the specified class of functions are calculated.


Introduction and preliminary facts
We consider the problem of mean-square approximation by Fourier sums of complex functions f which are regular in a simply connected domain D ⊂ C and belong to the space L 2 := L 2 (D) with the finite norm , where the integral is understood in the Lebesgue sense and dσ is an element of area.
The study of the mean-square approximation of functions in the domain D ⊂ C is closely related to the theory of orthogonal functions.A sequence of complex functions {ϕ k (z)} (k = 0, 1, 2, ...) is called an orthogonal system on the domain D if Such a sequence of functions is called orthonormal system if 1 π where δ k,l = 0, k = l, and are called the Fourier coefficients of the function f with respect to the orthonormal system {ϕ k (z)} (k = 0, 1, 2, ...).We associate with a given function f its Fourier series with respect to the specified orthogonal system: Let be the partial sum of order n of the series (2).We form a linear combination of the first n functions of the system {ϕ k (z)}: where d k ∈ C are arbitrary complex coefficients.We call this linear combination a generalized polynomial.It is well known (see, for example, [1], p.263) that where a k (f ) are the Fourier coefficients of the function f defined by (1).
In the case of the mean approximation of complex functions in a simply connected domain D ⊂ C by Fourier series with respect to an orthogonal system of functions {ϕ k (z)} ∞ k=0 on D, the problem of finding the exact constant in the Jackson-Stechkin inequality was studied in [2].Recall that Jackson-Stechkin inequalities are inequalities in which the value of the best approximation of a function by a finite dimensional subspace of a given normed space is estimated by the modulus of smoothness of the function itself or some its derivative.In this paper, we use the same methods as in [2,3,5,15].
We study in more detail the case where D is the unit disk U := {z ∈ C : |z| < 1}.In this case, it is clear that the system of functions However, this system is not orthonormal, since Therefore, the system of functions ϕ * k (z) = √ k + 1z k (k = 0, 1, 2, ...) is orthonormal.We denote by A(U ) the set of all functions f analytic in U .The Maclaurin series of such a function has the form where c k (f ) are the Maclaurin coefficients of f .We note that It was proved in the monograph [1] that the Fourier series of a function f with respect to the orthonormal system ϕ * k (z) = √ k + 1z k , k = 0, 1, 2, ..., coincides with the series (4) for f ∈ A(U ); i.e., Therefore, the series ( 6) can be differentiated term by term any number of times and, according to the Weierstrass theorem [6, p.107], for any r ∈ N, we get where We denote by L (r) 2

Sharp estimates of the value of the best approximation by means of K-functionals
In this section, we prove some sharp inequalities relating the value E n−1 (f ) of the best approximation of functions in the class L (r) 2 and Peetre K-functionals.The definition and some properties of Peetre K-functionals are given in [7].The direct and inverse theorems of the theory of approximation by means of K-functionals were proved in [8,9].We define the K-functional constructed by the spaces L 2 and L (m) 2 as follows: where m ∈ N and 0 < t ≤ 1.We note that a weak equivalence of the K-functional defined by ( 8) and a special generalized modulus of continuity of order m was established in [8].
Theorem 1.Let n, m ∈ N and r ∈ Z + be arbitrary numbers such that n ≥ r + m.Then the following equality holds: P r o o f.Using (7), we easily find that Taking into account equality (10), we obtain Now, for an arbitrary function f ∈ L (r) 2 , we write where S n−r−1 (g) is the partial sum of order n − r of the Fourier series of an arbitrary function g ∈ L (m) 2 .In view of ( 2) and ( 11), we get It follows from inequalities (12) and ( 13) that Now, we note that the left-hand side of inequality ( 14) does not depend on g ∈ L (m) 2 .Therefore, passing to the infimum over all functions g ∈ L (m) 2 on the right-hand side of ( 14) and using the definition (8) of K, we get This implies the following upper bound: where P r is the subspace of complex algebraic polynomials of degree at most r.
To obtain a lower bound of the extremal characteristic on the left-hand side of (15), in (8), we put f (z) := p n (z), where p n (z) is an arbitrary complex algebraic polynomial in P n .Since the function g(z) ≡ 0 belongs to the class L (m) 2 , we obtain from (8) the upper bound K m (p n ; t m ) 2 ≤ p n .
Since the function g(z) := p n (z) also belongs to the class L (m) 2 , we find from (8) that Thus, the last two relations imply that, for any element p n (z) ∈ P n , We consider the function f 0 (z) = z n .Since according to (16), we have Using the obtained inequality and the second equality in (5), we establish that We obtain equality ( 9) by comparing the upper bound (15) with the lower bound (17).The theorem is proved.

Exact values of n-widths of a class of functions
We assume that S is the unit ball in the space L 2 , Λ n ⊂ L 2 is an n-dimensional subspace, and Λ n ⊂ L 2 is a subspace of codimension n.Let L : L 2 → Λ n be a continuous linear operator, let L ⊥ : L 2 → Λ n be a continuous linear projection operator, and let M be a convex centrally symmetric subset of L 2 .The quantities are called, respectively, the Bernstein, Kolmogorov, linear, Gelfand, and projection n-widths of the subset M in the space L 2 .These widths are monotone with respect to n, and the following relation holds (see, for example, [10,11]): We recall (see, for example, [12, p. 25]) that a nondecreasing function In this definition, Ψ is a majorant, L 2 ≡ L 2 , and We note that, in the Bergman space, values of widths of some classes of analytic functions in a disk were calculated, for example, in [13][14][15][16][17][18][19].
Theorem 2. Let Ψ be the majorant defining the class W (r) 2 (K m , Ψ), m ∈ N, and r ∈ R + .Then, for any natural number n ≥ m + r, we have 2 (K m , Ψ), relations (15) and (18) imply that To find the corresponding lower bound, in view of (18), it suffices to estimate the Bernstein n-width of the class W (r) 2 (K m , Ψ).On the set P n ∩ L 2 , we define the ball .
Now, we note that, in view of formula (7) and the identity α k,r+m = α k,r α k−r,m , for an arbitrary p n (z) = n k=0 a k (p n )z k ∈ P n , the following equality holds: Hence, using the Parseval equality and the inequality α k,r ≤ α n,r , k ≤ n, we obtain the Bernstein type inequality By definition, for the majorant Ψ and for any 0 < τ 1 ≤ τ 2 ≤ 1, we have the inequality τ 1 Ψ(τ 2 ) ≤ τ 2 Ψ(τ 1 ).Therefore, for any 0 < t 1 ≤ t 2 ≤ 1, setting τ 1 = t m 1 and τ 2 = t m 2 , we obtain We now show that M n+1 ⊂ W (r) 2 (K m , Ψ).Thus, we need to prove that, for any polynomial Since, by assumption, m, n ∈ N, r ∈ Z + , and n ≥ m + r, we consider two cases: In this case, using inequality (22) with and applying ( 12) and ( 21), for any p n ∈ M n+1 , we obtain Now, let Then using (16) and the Bernstein type inequality p (r) n ≤ α n,r • p n and taking into account that the majorant Ψ is nondecreasing, we find that The definition of the class W Comparing the upper bound (20) and the lower bound (25), we get the required equality (19).The theorem is proved.
where λ n (•) is any of the n-widths b n (•), d n (•), d n (•), δ n (•), and Π n (•).P r o o f.Let n be a natural number such that n ≥ m + r.In view of the definition of the class W (r) m , Ψ) along with (23) and (24) implies thatM n+1 ⊂ W (r) 2 (K m , Ψ).Then, taking into account the definition of the Bernstein n-width and (18), we obtain