Comparison theorems and some of their applications

Analogues of Kolmogorov comparison theorems and some of their applications were established.

For natural r let L r ∞ (R) denote the space of functions x : R → R such that the derivative x (r−1) , x (0) = x, is locally absolute continuous and x (r) ∈ L ∞ (R). Set L r ∞,∞ (R) := L r ∞ (R) L ∞ (R). For r ∈ N by ϕ r (t) we will denote the Euler prefect spline of the order r (i. e. r-th periodic integral of the functions sgn sin t with zero mean value on the period). For λ > 0 set ϕ λ,r (t) := λ −r ϕ r (λt).
To prove his outstanding inequality (see [1] [3]) Kolmogorov proved a statement, known as a comparison theorem.
Theorem A. Let r ∈ N and a function x ∈ L r ∞,∞ (R) are given. Let numbers a ∈ R and λ > 0 be such, that λ,r , k ∈ {0, r}.
Both the Kolmogorov comparison theorem and its proof played important role in exact solutions of many extremal problems in approximation theory (see. [4,5,6,7]).
The goal of this paper is to prove several analogues of Kolmogorov comparison theorems.
In the next paragraph we will introduce a family of splines, which will play the same role, as Euler perfect splines play in the theorem A, and study some of their properties. In § 3 we will prove 3 analogues of Kolmogorov comparison theorem for the cases when the norms of a function and its derivatives of orders r − 1 and r are given; the norms of a function and its derivatives of orders r − 2 and r are given; the norms of a function and its derivatives of orders r − 2, r − 1 and r are given. In § 4 we will give several applications of the obtained comparison theorems.
We will list several properties of the function ψ r (a 1 , a 2 ; t), r ∈ N, which can easily be proved either directly from definition, or similar to the corresponding properties of Euler splines ϕ r (see, for example, [4,Chapter 5], [5,Chapter 3]). Note, that the function ψ r is 2 · T -periodic and for all r ≥ 1 Moreover, the function ψ 2 (a 1 , a 2 ; t) has exactly two zeroes on the period the points a 1 + a2 2 + 1 and 2a 1 + 3a2 2 + 3. Hence the functions ψ r (a 1 , a 2 ; t) for r ≥ 2 also have exactly two zeroes on the period: for any k ∈ N ψ 2k+1 (a 1 , a 2 ; 0) = ψ 2k+1 (a 1 , a 2 ; a 1 + a 2 + 2) = 0, ψ 2k a 1 , a 2 ; a 1 + a 2 2 Note, that for a 1 = 0 the equality (2) is true for k = 0 too. Hence, in turn, we have that for r ≥ 3 (in the case a 1 = 0 for r ≥ 2) the function ψ r (a 1 , a 2 ; t) is strictly monotone between zeroes of its derivative and the plot of the function ψ r (a 1 , a 2 ; t) is strictly convex at the intervals of constant sign. Moreover, it is easy to see, that the plot of the function ψ r (a 1 , a 2 ; t) is symmetrical with respect to its zeroes and the lines t = t 0 , where t 0 -is the zero of ψ ′ r (a 1 , a 2 ; t). At last note, that ψ r (0, 0; t) = ϕ π/2,r (t). For Note, that the function Ψ a1,a2,b,λ (t) is λ -periodic.
Theorem 1 Let r ∈ N and x ∈ L r ∞,∞ (R) be given. Then a) there exist a 2 ≥ 0, λ > 0 and b ∈ R such, that The truth of the theorem 1, actually, follows from Kolmogorov comparison theorem. We will prove the statement a), the rest of the statements can be proved analogously.

Comparison theorems.
The next theorem contains three analogues of Kolmogorov comparison theorem.
Below we count that N is chosen enough big, so that the property 3 holds. Set and Then and Moreover, for k = 1, . . . , r Hence, in virtue of property 3 of the function µ N and the choice of the number N , we get , and hence the function ∆ N (t) has at least three sign changes on the interval [τ 1 , τ 2 ]. At each of the rest monotonicity intervals of the function Ψ the function ∆ N has at least one sign change. Hence on the interval τ 1 − N · λ 2 , τ 1 + N · λ 2 the function ∆ N (t) has at least 2N + 2 sign changes. Moreover, in virtue of (2),(3) and (10) for all i = 1, 2, . . . , r−1 2 the following equalities hold All of the arguments above are true if any of the condition a) − c) hold. Let now condition a) of the theorem holds.
Applying Rolle's theorem and counting (11) we have that the function ∆ (r−1) N (t) has at least 2N + 2 zeroes on the interval Hence on some monotonicity interval changes the sign at least three times on some monotonicity interval of the function Ψ

(t).
If the condition b) of the theorem holds, then applying similar arguments we will get contradiction with Kolmogorov comparison theorem.
If the condition c) of the theorem holds, then applying similar arguments we will get contradiction with already proved case when condition a) holds. The theorem is proved.
From the theorem 2 we immediately get Lemma 1 Let r ∈ N, x ∈ L r ∞,∞ (R) and one of the conditions a) − c) of the theorem 2 holds. Then on each monotonicity interval of the function Ψ a1,a2,b,λ (t) the difference Ψ a1,a2,b,λ (t) − x(t) has at most one sign change.
For 1-periodic non-negative integrable on period function x(·) denote by r(x, ·) the decreasing rearrangement of the function x (see, for example [4, Chapter 6]).
As a corollary of the theorem 2 and the results of the Chapter 3 of the monograph [5] we get the following theorem.
Theorem 3 Let r ∈ N and 1-periodic function x ∈ L r ∞,∞ (R) are given. Let one of the conditions a) − c) of the theorem 2 holds. Then for all t > 0 From the theorem 3 and general theorems about rearrangements comparison (see, for example, [6, Statement 1.3.10]) we get the following analogue of the Ligun inequality [9] (see also [7], Chapter 6).
Theorem 4 Let r ∈ N and 1-periodic function x ∈ L r ∞,∞ (R) are given. Let one of the conditions a) − c) of the theorem 2 holds. Then for all 1 ≤ p < ∞ and natural k < r − 2 ( if condition a) holds, then for all natural The following lemma is an analogue to the Bohr-Favard inequality, see, for example, [10], Chapter 6.
Lemma 2 Let r ∈ N and 1-periodic function x ∈ L r ∞,∞ (R) are given. Let for λ = 1 one of the following conditions holds. a) Numbers a 1 = 0, a 2 ≥ 0 and b = 0 are such that We will proceed by induction on r. The basis of the induction easily follows. We will dwell on the induction step. Assume the contrary, let E 0 (x) > Ψ a1,a2,b,1 . Let c be a constant of the best uniform approximation of the function x. We can count that max This means that However the last inequality together with the induction hypothesis and (12) contradicts the lemma 1.
Using the theorem 2, lemma 2 and ideas from [11] (see also § 6.4 of the monograph [7]) we get the following analogue of Babenko, Kofanov and Pichugov inequality.
Theorem 5 Let r ∈ N and 1-periodic function x ∈ L r ∞,∞ (R) are given. Let for some λ > 0 one of the conditions a) − c) of the lemma 2 holds and E 0 (x) = Ψ a1,a2,b,λ . Then For a function x ∈ L ∞ (R) let c(x) denote the constant of the best approximation for the function x in L ∞ (R).
From theorem 5, theorem 2 and ideas from [12] (see also § 6.7 of the monograph [7]) we get the following analogue of Nagy type inequality (see [13]), that was obtained by Babenko, Kofanov and Pichugov.