On the coefficients of the expansion of elements from C [ 0 , 1 ] space by the Faber-Schauder system

Elements (functions) of continuous on [0, 1] functions space (C[0, 1]) are described which have 1 2 -monotone coefficients of the expansion by the Faber-Schauder system.

Definition 1.Let t ∈ (0, 1].We say that coefficients of f (x) ∈ C[0, 1] function's expansion by the Faber-Schauder system are t-monotone if where {n k } ∞ k=1 is the spectrum of function f (x), i.e. the set of integers where A n (f ) is non zero.
We consider the following question: let t ∈ (0, 1].Is it possible to modify a continuous on [0, 1] function on a "small" set to make coefficients of expansion (1.2) t-monotone?
In the present work we were able to prove the following Theorem 1.For every ∈ (0, 1) there exists a measurable set E ⊂ [0, 1] with measure |E| > 1 − , such that to each function f (x) ∈ C[0, 1] one can find a function g(x) ∈ C[0, 1] that coincides with f (x) on E and coefficients of which's expansion by the Faber-Schauder system are t-monotone for all t ∈ (0, 1  2 ]. From Theorem 1 follows Theorem 2. For every ∈ (0, 1) there exists a measurable set

. are partial sums of the series (1.2).
Note that from this theorem immediately follows that the sequence {G m=1 uniformly converges to g(x).Analogous questions are interesting to consider in various functional spaces with respect to other bases.Note that in particular in [6] it is proved the following theorem for the Walsh system: Theorem (Grigoryan).For every ∈ (0, 1), p ≥ 1 and each function of which's expansion by the Walsh system are 1monotone.
Note that the idea of modification of a function improving its properties belongs to N.N.Luzin.The following famous result was obtained by him in 1912 (see [14]).

Theorem (N.N. Luzin's C -property). Each measurable and a.e. finite function can be made continuous after a modification on a subset of arbitrarily small Lebesgue measure.
In 1939, Men'shov [15] proved the following fundamental theorem.
The following questions are open: Question 1.Is Theorem 1 true for t ∈ ( 1 2 , 1]? Question 2. Do Theorems 1 and 2 hold for other bases in the space C[0, 1] (in particular, for the Franklin system)?Question 3. Is Theorem 1 true for the trigonometric system for any t ∈ (0, 1]? Question 4. Let t ∈ (0, 1).Is there a continuous function on [0, 1] such that Weak Thresholding Greedy Algorithm by the Faber-Schauder system with weakness parameter t diverges by measure (or at least almost everywhere) in [0, 1]?
Concerning with Question 4 note that in [5] it is proved that the answer is positive for t = 1.

Proofs of basic lemmas
A n ϕ n (x) , by the system (1.1), such that Proof.At first we consider the case when N 0 ≤ 2 p + i .We take natural number q > log 2 1 + 1 (0 < < |Δ|) and put 3), it is not hard to see that where presents a polynomial by Faber-Schauder system, satisfying the conditions of the lemma.Now we consider the case N 0 > 2 p + i .In this case some points x n , n < N 0 (see (1.1)) belong to Δ.We denote them by ).We take a natural number q > log 2 L+1 + 1 (0 < < |Δ|) and put is linear and continuous on [ It is clear that 2 q > N 0 , g(x) C = |γ| and |E| > |Δ| − .Taking into account (1.3) we obtain that in the expansion of the function g(x) ∈ C [0,1] by the system (1.1), coefficients of functions {ϕ n (x)} with numbers n < N 0 and n > 2 q and coefficients of functions {ϕ n (x)} with numbers N 0 ≤ n ≤ 2 q , whichs supports don't belong to Δ or Δ n ⊂ E , are equal to 0 .From (1.3) and (2.1) it follows that coefficients of functions , are equal to either γ or γ 2 , depending on the fact that function g(x) takes value 0 in both endpoints of Δ n or only in one endpoint.So we obtain that the function g(x) presents a polynomial by the system (1.1) satisfying the conditions of the lemma.
) and numbers γ = 0, N 0 ∈ N, 0 < < |Δ| be given.Then one can find a measurable set E ⊂ Δ and a polynomial Q(x) by the Faber-Schauder system of the form satisfying the following conditions: Proof.Let ν 0 > |γ| be some natural number.By virtue of Lemma 1, for every natural number ν ∈ [1, ν 0 ] one can find a measurable set E ν ⊂ Δ and a polynomial of the form satisfying the conditions: We put where 3 ) it follows that the set E and the polynomial Q(x) satisfy conditions 1)-3) of the lemma and that all {A k } K k=k0 have the same sign.
Hence, taking into account (2.2) we obtain and the assertion follows. .

Lemma 3. Let dyadic intervals {Δ
be given.Then one can find a measurable set E ⊂ [0, 1] and a polynomial Q(x) by the Faber-Schauder system of the form satisfying the following conditions: Proof.By virtue of Lemma 2, there are a measurable set E 1 ⊂ Δ (1) and a polynomial of the form satisfying the conditions: Similarly, there are a measurable set E 2 ⊂ Δ (2) and a polynomial of the form Proceeding thus, inductively, for each ν ∈ [1, 2 p ] one determines a measurable set E ν ⊂ Δ (ν) and a polynomial of the form satisfying the conditions: (2.4) We put (see (2.9)).From here , from (2.5), (2.7) and from the fact that supports of different intervals from {Δ (ν) } 2 p ν=1 are pairwise-disjoint, we have i.e. conditions 3) and 5) of Lemma 3 hold.Similarly, using (2.5), (2.8) and (2.9), we obtain

Proof of the Theorem 1.
Assume that 0 < < 1 .By numbering all step functions of the form where γ ν , ν = 1, 2, ..., 2 p are non-zero rational numbers, and Δ Using Lemma 3 repeatedly we can find sequences of sets {E m } ∞ m=1 and polynomials that satisfy the following conditions for all m ∈ N: Consider the function g(x) defined as follows: (3.10) where {c k } ∞ k=1 is the sequence a x ∈ E and the result is proved.