A STUDY OF APPROXIMATION OF FUNCTIONS OF BOUNDED VARIATION BY FABER-SCHAUDER PARTIAL SUMS

The Faber-Schauder system of functions was introduced in 1910 and became the first example of a basis in the space of continuous on [0, 1] functions. A number of results are known about the properties of the Faber-Schauder system, including estimations of errors of approximation of functions by polynomials and partial sums of series in the Faber-Schauder system. It is known that obtaining new estimates of errors of approximation of an arbitrary function by some given class of functions is one of the important tasks in the theory of approximation. Therefore, investigation of the approximation properties of polynomials and partial sums in the Faber-Schauder system is of considerable interest for the modern approximation theory. The problems of approximation of functions of bounded variation by partial sums of series in the Faber-Schauder system of functions are studied. The estimate of the error of approximation of functions from classes of functions of bounded variation C p (1≤p<∞) in the space metric L p using the values of the modulus of continuity of fractional order ϖ 2-1/p (f, t) is obtained. From the obtained inequality, the estimate of the error of approximation of continuous functions in terms of the second-order modulus of continuity follows. Also, in the class of functions C p (1<p<∞), the estimate of the error of approximation of functions in the space metric L p using the modulus of continuity of fractional order ϖ 1-1/p (f, t) is obtained. For classes of functions of bounded variation KCV (2,p) (1≤p<∞), the estimate of the error of approximation of functions in the space metric L p by Faber-Schauder partial sums is obtained. Thus, several estimates of the errors of approximation of functions of bounded variation by their partial sums of series in the Faber-Schauder system are obtained. The obtained results are new in the theory of approximation. They generalize in a certain way the previously known results and can be used for further practical applications.


Introduction
The Faber-Schauder system of functions was introduced in the paper [1] and became the first example of a basis of the space of functions continuous on [0, 1]. Approximate properties of the Faber-Schauder system regarding approximation of individual functions and classes of functions are studied, for example, in [2][3][4][5]. In those studies, the upper bounds of errors of approximation of the function f continuous on [0, 1] by Faber-Schauder partial sums S f x n , ( ) in the uniform metric are obtained.
Particularly, in [2] an estimate of the error of approximation of a continuous function by its Faber-Schauder partial sum is obtained. This result is specified in [4] using the second-order modulus of continuity.
In [6][7][8], the exact estimates of errors of approximation of functions from some function classes by Faber-Schauder partial sums in uniform and integral metrics are obtained.
However, the problems of approximation of functions of bounded variation by their Faber-Schauder partial sums have been investigated in few papers. In particular, in [9] considering the approximation of functions f from the classes Thus, investigation of approximation of functions of bounded variation by their Faber-Schauder partial sums and obtaining new results are of current interest not only to the modern theory of approximation but also to the wavelet theory actively used in modern signal processing. It is also appropriate to use moduli of continuity of fractional orders ω δ for obtaining estimates of errors of approximation of functions by series in the Faber-Schauder system.

Literature review and problem statement
Although the Faber-Schauder system of functions was introduced in 1910 [1], investigation of the properties of the system, including approximate properties, began only in the 1950s with [2,3]. Thus, investigating [2] the approximate properties of the Faber-Schauder system for an arbitrary continuous function, an upper bound of the value ε n C f ( ) , in terms of the second-order modulus of continuity is obtained. Later, in [4] that result is specified and the following estimate of the error of approximation of an arbitrary continuous function by its Faber-Schauder partial sum is obtained: In [3], an estimate of the error of approximation of an arbitrary continuous function by its partial Faber-Schauder sum using the first-order modulus of continuity is obtained.
, . 4 1 ϖ Subsequently, that result is specified in [5] and the validity of the following relation is shown: It should be noted that in [2][3][4][5] only the questions of approximation of continuous functions in uniform metrics are considered and the obtained estimates are not exact in the sense of the final character of the estimates.
The first exact estimates of the errors of approximation of functions by partial sums in the Faber-Schauder system are obtained in [6][7][8]. In [6], the estimates of the errors of approximation of differentiable functions by their partial Faber-Schauder sums on classes of functions C 1 and W H 1 ω are obtained in integral metrics φ L ( ). Moreover, the estimates obtained in [6] can't be improved in case of a convex upward modulus of continuity.
In [7], the following unimprovable estimate of the error of approximation of differentiable functions from class L ∞ 2 by Faber-Schauder partial sums in the metric L ∞ is obtained: Further studies in this direction are continued in [8] where a number of exact estimates of errors of approximation of the classes of differentiable functions L p 1 by Faber-Schauder partial sums in integral metrics L p are obtained.
However, the questions of approximation of functions of bounded variation by either polynomials or partial sums of series in the Faber-Schauder system aren't considered in the foregoing papers.
Only the work [9] is known, where the problems of approximation of functions of bounded variation by Faber-Schauder polynomials are studied with obtaining a number of estimates of approximation errors. Particularly, in [9] an upper bound of the error of the best approximation of functions f of bounded variation from the class C p 1≤ <∞ ( ) p by polynomials in the Faber-Schauder system in the space metric L p is obtained: However, the questions of approximation of functions by Faber-Schauder partial sums aren't addressed in [9].
It should be also noted that studying the approximate properties of the Faber-Schauder system, the moduli of continuity of fractional orders ω δ , are used only in [9]. This is despite the fact that in connection with problems of approximation theory, the moduli of continuity of fractional orders ω δ , were first studied in [19] and used in several papers, for instance, [19][20][21][22], devoted to investigation of some questions of approximation theory, particularly to approximation of functions of bounded p-variation.
Application of the Faber-Schauder system in the theory of nonlinear approximation of functions is considered in [10]. In particular, some issues of the behavior of a greedy algorithm in the Faber-Schauder system in the space of continuous functions are examined [10].
As an example of a piecewise linear wavelet system that has been actively studied and used in recent decades in signal processing, the study of properties of the Faber-Schauder system is of considerable interest for the modern theory of functions, the theory of signal processing and wavelet theory.
In [11,12], the behavior of the coefficients of decomposition of a continuous function in the Faber-Schauder series is investigated. The questions of convergence of series in the Faber-Schauder system are studied in [13][14][15][16]. In [17,18], some questions of decomposition of functions in the Faber-Schauder system of functions are considered.
Consequently, taking into account the abovementioned, the properties of the Faber-Schauder system require further rigorous research. In particular, studying the approximation properties of the Faber-Schauder system and obtaining new results on estimates of errors of approximation of functions by polynomials and partial sums in the Faber-Schauder system are of importance for further investigations.
Using the moduli of continuity of fractional orders ω δ is also of significance for obtaining new results on estimation of approximation errors in case of the Faber-Schauder system.

The aim and objectives of the study
The aim of the study is to consider the issues of approximation of functions of bounded variation by their [ ] for which V f p ( )< ∞ [24]. In case p = 1, V 1 is a usual class of functions of bounded variation. In [23], it is shown that the functions f from the class V p 1≤ <∞ ( )  for the function f V p ∈ is defined in the following way [19]: Let the finite everywhere on 0 1 , [29] if We would like to note that in case m = 1, the class of functions V p 1, ( ) matches with the class of functions of bounded p-variation V p . In case p = 1, the class V m, 1 ( ) was considered in [30]. In case m = 1 and p = 1, the class V 2 1 , ( ) is considered in [31]. [ ] Using the Haar system of functions (3), the system of functions ψ n n Z { } ∈ + is defined in [1] in the following way: It is shown in [1] that every continuous function f C ∈ can be represented by the series: The integral in (5) is understood in the Lebesgue-Stieltjes sense. The result (4) is replicated in [34] using the system of functions  ψ n n Z { } ∈ + that differ from ψ n n Z { } ∈ + by constant factors only. For the n-th partial sum ( ) n N ∈ , we write the expression (4) as: The sum (6) is called the Faber-Shauder partial sum of the function f C ∈ . We introduce the quantity: that is called the error of approximation of the function f by its Faber-Schauder partial sum S f n ( ) in the space metric X.

Results
Proof. Let there given the arbitrary function f C ∈ . We consider the following function on some interval α β , , : We have:  [ ] [ ] Using the definition (9) and the fact that We introduce the following notation: It is known (see, for example, [3], [8]) that the partial sum S f x n ( ; ) defined in (6)  ) and equality (14), we obtain from (12) the following: For the arbitrary function f x C p ( )∈ and any n k m = + 2 ( , , ) m N k m ∈ = − 1 2 1 from (12) and the notations (19) above, we have the following inequality: