SOME INEQUALITIES BETWEEN THE BEST SIMULTANEOUS APPROXIMATION AND THE MODULUS OF CONTINUITY IN A WEIGHTED BERGMAN SPACE

: Some inequalities between the best simultaneous approximation of functions and their intermediate derivatives, and the modulus of continuity in a weighted Bergman space are obtained. When the weight function is γ ( ρ ) = ρ α , α > 0, some sharp inequalities between the best simultaneous approximation and an m th order modulus of continuity averaged with the given weight are proved. For a speciﬁc class of functions, the upper bound of the best simultaneous approximation in the space B 2 ,γ 1 , γ 1 ( ρ ) = ρ α , α > 0, is found. Exact values of several n -widths are calculated for the classes of functions W ( r ) p ( ω m ,q ).


Introduction
Extremal problems of polynomial approximation of functions in a Bergman space were studied, for example, in [8,[13][14][15].Here, we will continue our research in this direction and study the simultaneous approximation of functions and their intermediate derivatives in a weighted Bergman space based on the works [4][5][6]10].Note that the problem of simultaneous approximation of periodic functions and their intermediate derivatives by trigonometric polynomials in the uniform metric was studied by Garkavi [1].In the case of entire functions, this problem was studied by Timan [12].
To solve the problem, we first will prove an analog of Ligun's inequality [2].Let us introduce the necessary definitions and notation to formulate our results.Let be the unit disk in C, and let A(U ) be the set of functions analytic in the disk U .Denote by B 2,γ the weighted Bergman space of analytic functions f ∈ A(U ) such that [8] f 2,γ := 1 2π dσ is an area element, γ := γ(|z|) is a nonnegative measurable function that is not identically zero, and the integral is understood in the Lebesgue sense.It is obvious, that the norm (1.1) can be written in the form In the particular case of γ ≡ 1, B q := B q,1 is the usual Bergman space.The mth order modulus of continuity in B 2,γ is defined as where u+kh) .
Let P n be the set of complex polynomials of order at most n.Consider the best approximation of functions f ∈ B 2,γ : Denote by B belong to the spaces B 2,γ and B 2 , respectively.Define It is well known [7,8] that the best approximation of functions is equal to and the modulus of continuity of f ∈ B 2,γ is the moments of order s of the weight function γ(ρ) on [0, 1].According to notation (1.4), we write equalities (1.2) and (1.3) in compact form: )

Analog of Ligun's inequality
For compact statement of the results, we introduce the following extremal characteristic: , where m, n ∈ N, r ∈ Z + , n > r ≥ s, 0 < p < 2, 0 < h ≤ π/(n − r), and q(t) is a real, nonnegative, measurable weight function that is not identically zero on [0, h].
, and let q(t) be a nonnegative, measurable function that is not identically zero on [0, h].Then where which is hold for all 0 < p ≤ 2 and h ∈ R + .Setting and this yields the inequality To estimate the value in (2.1) from below, consider the function 2,γ .Simple calculation leads to the following relations: using which, we get the lower estimate r,s,p (q, γ, h) . (2.6) Comparing the upper estimate (2.5) and the lower estimate (2.6), we obtain the required two-sided inequality (2.1).This completes the proof of Theorem 1.
Corollary 1.The following two-sided inequality holds for γ 1 (ρ) = ρ α , α ≥ 0, in Theorem 1: where The following problem naturally arises from (2.7): to find an exact upper bound for the extremal characteristic Theorem 2. Let a weight function q(t), t ∈ [0, h], be continuous and differentiable on the interval.If the differential inequality holds for all k ∈ N, r, s ∈ Z + , k > n > r ≥ s, 0 < p ≤ 2, and α ≥ 0, then the following equality holds for all m, n ∈ N and 0 < h ≤ π/(n − r): . (2.10) P r o o f.To prove equality (2.10), it suffices to show that the following equality holds in (2.7): We should note that a similar problem of finding a lower bound in (2.11) for some specific weights for p = 2 was considered in [2].In the general case, this problem was studied in [9], where it was proved that, if the weight function q ∈ C (1) [0, h] for 1/r < p ≤ 2, r ≥ 1, and 0 < t ≤ h satisfies the differential equation (rp − 1)q(t) − tq ′ (t) ≥ 0, then (2.11) holds.
Let us now show that, under all constrains on the parameters k, r, s, m, p, α, and h in Theorem 2, the function increases for n ≤ k < ∞.Indeed, differentiating (2.12) and using the identity we obtain This relation and condition (2.9) imply that ψ(k) > 0, k ≥ n > r ≥ s, and we obtain equality (2.10).Theorem 2 is proved.

Denote by
2,γ 1 whose rth derivatives f (r) satisfy the following condition for all 0 < h ≤ π/(n − r) and n > r: 2,γ 1 , n > r ≥ s, n ∈ N, and r, s ∈ Z + , is of interest.More precisely, it is required to find the value Corollary 2. The following equality holds for all n ∈ N, n > r ≥ s, 0 < p ≤ 2, and 0 < h ≤ π/(n − r): Moreover, there is a function g 0 ∈ W (r) p (ω m , q) on which the upper bound in (2.13) is attained.
P r o o f.Assuming that γ = γ 1 (ρ) = ρ α in (2.4), with respect to (2.8), we can write Using equality (2.11) and the definition of the class W (r) p (ω m , q), we get From (2.14), it follows the upper estimate of the value on the left-hand side of (2.13):

.15)
To obtain the lower estimate for this value, consider the function and show that g 0 belongs to W (r) p (ω m , q).Differentiating this function r times, we obtain Using this equality and formulas (1.3), we get Raising both sides of this inequality to a power p (0 < p ≤ 2), multiplying them by the weight function q(t), and integrating with respect to t from 0 to h, we obtain Thus, the inclusion g 0 ∈ W p (ω m , q) is proved.Since the relation holds for all 0 ≤ s ≤ r < n, n ∈ N, and r, s ∈ Z + , according to (1.5), we have r,s,p,α (q, h) .
Using this equality, we obtain the lower estimate sup

Exact values of n-widths for the classes
Recall definitions and notation needed in what follows.Let X be a Banach space, let S be the unit ball in X, let Λ n ⊂ X be an n-dimensional subspace, let Λ n ⊂ X be a subspace of codimension n, let L : X → Λ n be a continuous linear operator, let L ⊥ : X → Λ n be a continuous linear projection operator, and let M be a convex centrally symmetric subset of X.The quantities are called the Bernstein, Kolmogorov, linear, Gelfand, and projection n-widths of a subset M in the space X, respectively.These n-widths are monotone in n and related as follows in a Hilbert space X (see, e.g., [3,11]): For an arbitrary subset M ⊂ X, we set Theorem 3. The following equalities hold for all m, n ∈ N, r ∈ Z + , n > r, and 0 ≤ h ≤ π/(n − r): p (ω m , q) with s = 0 from (2.14) since Using relations (3.1) between the n-widths, we obtain the upper estimate in (3.2): To obtain the lower estimate on the right-hand side of (3.2) for all n-widths in the (n + 1)dimensional subspace of complex algebraic polynomials , where n > r, n ∈ N, r ∈ Z + , and show that B n+1 ⊂ W (r) p (ω m , q).Indeed, for all p n (z) ∈ B n+1 , from (1.3), we write We have to prove that Consider the function We will show that the function ϕ(k) is monotone increasing for all accepted values k and h.To this end, it suffices to show that ϕ ′ (k) > 0. In fact Raising both sides of this inequality to a power p (0 < p ≤ 2), multiplying them by the weight function q(t), and integrating with respect to t from 0 to h, we obtain h 0 ω p m p (r) , t p (ω m , q).Then, according to the definition of the Bernstein n-width and (3.1), we can write the following lower estimate for all above listed n-widths: λ n (W (r)  p (ω m , q), B 2,γ 1 ) ≥ b n (W (r) p (ω m , q), B ∈ N the class of functions f ∈ A(U ) whose rth order derivatives

)
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.We obtain the upper estimates of all n-widths for the class W (r)