On the nonsymmetric approximation of continuous functions in the integral metric

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $H^\omega[a,b]$ is proved. Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes H ω [a, b] is proved.
Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P.Korneichuk, fnd the exact estimate for the best one-sided approximation obtained by V.G.Djrjnin fnd A.A.Ligun.
Key words: modulus of continuity, nonsymmetric approximations.
Here ω(f, t) is the modulus of continuity of a function f and ω(t) is given modulus If f ∈ L p and H ⊂ L p , then the quantity is called (see [1]) the best (α, β)-approximation of a function f by a set H in the space L p . For α = β = 1 we obtain the usual best L p -approximation of a function f by a set H, which we denote by E(f, H) p .
For f ∈ L p we set The value is called the best L p -approximation from below (+) and above (-) of a function f by the set H.
Theorem 1. For any α, β > 0 and any f ∈ H ω [a; b] the following estimate is valid: If ω(t) is concave modulus of continuity and p = 1, or ω(t) = t and p > 1, then inequlity (2) is the best possible.
Proof. We will use ideas from [2] (see also [3, §4.2]). Taking into account the criterion of the element of the best (α, β)-approximation [1] we see that to prove the theorem it is sufficiently to prove the inequality where H ω 0 [a; b] is the set of piecewise linear functions from H ω [a; b] without horizontal links, and such thatˆb where C is an arbitrary constant. Let f (t) > 0(< 0) almost everywhere on (a; γ), and f (t) < 0(> 0) almost everywhere on (δ; b), Let also F (a) = F (b), F (γ) = F (δ), and e = (a; γ) ∪ (δ; b). Then Proof of Lemma 1. Let, for definiteness, f (t) > 0, t ∈ (a; γ), f (t) < 0, t ∈ (δ; b). We define on strictly decreasing function ρ(t) by the equality We will have also The function ρ(t) and its inverse function ρ −1 (t) are absolutely continuous. Differentiating the equalities (4) and (5) we obtain that almost everywhere Consider e (αf + (t) + βf − (t)) p dt = α pˆγ Using (7) and substituting ρ −1 (u) = t we will havê and therefore Analogously using (6) we obtain From the last two equalities, we derive Applying Hölder inequality, we get Taking into account the sign of the function f (t) on the intervals (a; γ) and (δ; b), we getˆe Consequently,ˆe Lemma 1 is proved. Let us finish the proof of Theorem 1. Choose C in definition of the function F (t) such that F (a) = F (b) = 0. Write a representation of F (t) as sum of simple function (see [6, §6.3]): Suppose that functions F k are ordered in such a way that lengths |b k − a k | decrease. Applying Lemma 1 to F = F k we will have that for any k Inequality (2) is proved.