Summability of a Tchebysheff system of functions

We consider a special type of Tchebysheff systems of functions {ui(·)}i=0 and {vi(·)}i=0 defined on the intervals (0, 1] and [1,+∞) , respectively, such that ui(t) = t α0 1 ∫

Abstract.We consider a special type of Tchebysheff systems of functions {ui(•)} n i=0 and {vi(•)} n i=0 defined on the intervals (0, 1] and [1, +∞), respectively, such that and We find necessary and sufficient conditions under which functions from the investigated systems belong to the corresponding Lebesgue spaces Lp(0, 1) and Lp (1, +∞) .In order to prove the main results we obtain lower and upper estimates of these functions that are of independent interest.

Introduction
Let α i and β i , i = 0, 1, . . ., n, be real numbers.On the intervals (0, 1] and [1, +∞), respectively, we consider the following systems of functions: (1.1) These functional systems {u i (•)} n i=0 and {v i (•)} n i=0 form Tchebysheff systems or T -systems on the intervals where they are defined.Moreover, the second system {v i (•)} n i=0 is an extended complete Tchebysheff system or ECT -system.
T -systems are very important in different areas of analysis, the theory of differential equations and statistics, e.g., in the theory of approximation (interpolation methods, cubature formulas), in boundary-value problems and problems with oscillation properties of zeros of the solutions of differential equations, and in the theory of statistical inequalities.In the monographs [1] and [2] we can find an almost complete presentation of applications of Tchebysheff systems.
Let us illustrate at least one classical problem with a solution given in terms of a Tchebysheff system.
Problem definition.Let P n (•) be a generalized polynomial This polynomial is called the polynomial of the best uniform approximation.

Haar Theorem. For any continuous function u(•) there exists a unique polynomial of the best uniform approximation if and only if the system of functions {u
In our turn we study T -systems {u i (•)} n i=0 and {v i (•)} n i=0 in connection with problems of correct posing of some boundary-value problems for (n + 1):th order differential equations, which have singularities at zero and infinity.In the case when a solution of the differential equation and its derivatives do not have traces at these singular points, we consider the following generalized conditions: at zero and at infinity lim It is obvious that the introduced differential operator (1.3) and the systems {u i (•)} n i=0 and {v i (•)} n i=0 have the following close connection: these two systems (here for the second system consider α instead of β ) are the fundamental systems of the solutions of the equation: In order to be able to solve the boundary-value problems for equations with singularities at zero and infinity, we are faced with the problem to find conditions under which the functions from (1.1) and (1.2) belong to the Lebesgue space L p (I), where I = (0, 1) and I = (1, +∞), respectively, with the norm: In this paper we present a complete solution of this problem.In particular, our result can be useful to solve the approximation problem given above or in many other applications of Tchebysheff systems.
If in the system (1.2) we make a change variables t = 1 x , then we get the system of functions { v i (x)} n i=0 , which are defined on the interval (0, 1] and have the same forms as functions from {u i (t)} n i=0 , i.e.: Therefore, we consider the system (1.1) as the main object of our investigation.
The paper consists of three sections.It is organized as follows: In Section 2 we state and prove our main result (Theorem 1) concerning a characterization of the functions in the studied system, which belong to the space L p (0, 1).Moreover, we present some lemmas concerning lower and upper estimates of functions from the investigated system.These lemmas are of independent interest but also necessary for the proof of main theorem.Finally, in Section 3 we present and discuss some further remarks and results.
Conventions.Here and in the sequel we suppose that The symbol X Y means X ≤ cY with some constant c > 0.

The main result
Our main result reads: Remark 1.A corresponding result for the functions v i , i = 0, 1, . . ., n, from the system (1.2) is given in the next Section (see Theorem 2).
Case 3 i 0 = 0 .By changing the order of integration, the function u i can be presented in the following form: (2.22) Since in this case, due to (2.13), s j=1 (α j + 1) ≥ 0 for all 1 ≤ s ≤ i , then in the same way as in the previous case it is easy to prove that This estimate together with (2.22) give that (2.23) | ln t| li .
Remark 2. The assumptions of Lemma 1 and Lemma 2 hold also for the functions u 0 , since in this case i 0 = 0 and l 0 = 0 , and, thus,
Let us now turn to the problem of summability of functions from (1.2).
Theorem 2. The functions v i , i = 0, 1, . . ., n, from the system (1.2) belong to L p (1, +∞), 1 ≤ p < ∞, if and only if It was mentioned above that if in (1.2) we change variables t = 1 x , then we get the system (1.4) of functions defined on (0, 1].Moreover, these functions have the same forms as functions from (1.1).By comparing powers of functions from (1.1) and (1.4), we have that α 0 = −β 0 , α i = −(β i + 2), i = 1, 2, . . ., n.Therefore, all conditions we introduced for (1.1) can easily be rewritten for the system (1.4).Thus, Finally, to prove Theorem 2 we state and prove the corresponding Lemmas (to Lemmas 1 and Lemma 2).Lemma 3.For the functions v i , i = 1, 2, . . ., n, from the system (1.2) and for all λ > 1 there exists λ 1 ≥ λ > 1 such that for any t ∈ [λ 1 , +∞) In this system the first functions w 0 and w 1 coincide with the functions u 0 and u 1 from (1.1), respectively.But, as we can see, the lower and upper integral bounds of the functions w i and u i , i = 2, 3, . . ., n, are correspondingly different.
By changing the order of integration in each function from (3.7), for w i we have that Now the function w i has the same form as the function u i from (1.1) with the difference in order of powers of the functions in the integrals.Hence, it is easy to formulate lower and upper estimates for w i and conditions of its summability to the power p on the interval (0, 1).
Lemma 5.For the functions w i , i = 1, 2, . . ., n, from the system (3.7) and for all 0 < δ < 1 there exists 0 < δ 1 ≤ δ such that for any t ∈ (0, δ 1 ] the estimate c i (δ)t ≤ w i (t) holds, where c i (δ) → 0 when δ → 1 , i = 1, 2, . . ., n. Lemma 6.For the functions w i , i = 1, 2, . . ., n, from the system (3.7) for any t ∈ (0, 1] the following estimate Remark 4. The assumptions of Lemma 5 and 6 hold also for the functions w 0 , since in this case l 0 = 0 , and, thus, | ln t| l0 .Theorem 3. The functions w i , i = 0, 1, . . ., n, from the system (3.7)belong to L p (0, 1), It is possible to formulate two main lemmas and a theorem as above also for this case.However, these results can be derived by only making a change of variables as discussed before so we leave out both the formulations and the proofs.