**Applied Mathematics** Vol.4 No.11A(2013), Article ID:38837,7 pages DOI:10.4236/am.2013.411A1002

Representation of Functions in Weighted Spaces by Series with Monotone Coefficients in the Walsh Genrealized System^{*}

Department of Physics, Yerevan State University, Yerevan, Armenia

Email: gmarting@ysu.am, artavazdminasyan@gmail.com

Copyright © 2013 Martin Grigoryan, Artavazd Minasyan. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received June 28, 2013; revised July 28, 2013; accepted August 5, 2013

**Keywords:** Orthonormal System; Convergence; Functional Series

ABSTRACT

Let be the Walsh generalized system. In the paper constructed a weighted space, and series in the Walsh generalized system with monotonically decreasing coefficient such that for each function in the space one can find a subseries that converges to in the weighted and almost everywhere on.

1. Introduction

In the present paper we study the following natural question: does there exist a weighted space, with, such that for every function in the space

one can find a series in the Walsh generalized system of the form

that possess the following property: for any function there exists a growing sequence of natural numbers such that the subseries converges to in the norm and a.e.

Note that the problem of representing a function by a series in classical and general orthonormal systems has a long history. Of course the problem of the representation of functions was studied before Luzin’s work. It goes back to D. Bernoulli, L. Euler and many others.

A question posed by Lusin in 1915 asks whether it is possible to find for every measurable function a trigonometric series, with coefficient sequence converging to zero, that converges to the function almost everywhere. For real-valued functions, this question was given an affirmative answer by Men’shov [1] in 1941.

There are many other works (see [2-11]) devoted to representations of functions by series in classical and general orthonormal systems and the existence of different types of universal series in the sense of convergence almost is everywhere and by measure.

Since the trigonometric and Walsh systems have many properties in common, one would think that there should be a corresponding result for the Walsh system. This is, indeed, the case, and, in fact, the same sort of result holding for a multitude of Walsh subsystems, many of them are quite sparse and far from complete.

In this paper we prove the following theorem:

**Theorem 1.** For any there exists a measurable function with

such that for any and any function there exists a series in the Walsh generalized system of the following form

(1)

which converges to in the—metric and almost everywhere.

Note that there exist functions in the space that can not be represented by series in the Walsh system (see [8], pp. 124-125).

Theorem 1 is a consequence of the more general Theorem 2, which is stated as follows:

**Theorem 2.** For any there exists a measurable function with

and a series in the Walsh generalized system of the form

that possess the following property: for any function there exists a growing sequence of natural numbers such that the subseries

converges to in the -norm and a.e.

Recall the following definition: a series is said to be universal with respect to subseries in the space, if for each function, one can select a subseries which converges to in norm .

The above-mentioned definitions are given not in the most general form and only in the generality, in which they will be applied in the present paper.

Note that the result of the Theorem 2 is definitive in a certain sense: one can not replace by because no orthonormal system of bounded functions does there exist a series universal in with respect to subseries. This is almost obvious.

The following problems remain open.

Question 1. Are the theorems 1 and 2 true for the trigonometric system?

Question 2. What kind of necessary and sufficient conditions should be imposed on the weight function in order to construct a Walsh series to be universal in the space with respect to subseries?

2. Proofs of Main Lemmas

Let be a fixed integer and. Recall the following definitions.

The Rademacher system of order is defined inductively as follows. For let

and for let

The Walsh generalized system (see [3] and [13,14]) of order is defined by

and if, where, then

.

We denote the generalized Walsh system of order by. Note that is the classical Walsh system. The basic properties of the generalized Walsh system of order have been obtained by H. E. Chrestenson, J. Fine, C. Vateri, W. Young, N. Vilenkin and others. Next we list some properties of, which will be useful later.

• Each -th Rademacher function has period.

• , , and (mod).

• is a finite product of Rademacher functions with values in.

• if.

• , is a complete orthonormal system in

and it is basic in for.

We put

(2)

and periodically extend these functions on with period 1.

By we denote the characteristic function of the set, i.e.

(3)

Then, clearly

(4)

and let for the natural numbers

(5)

(6)

Hence

(7)

. (8)

**Lemma 1.** Let dyadic interval

and numbers be given. Then there exists a measurable set and a polynomial in the Walsh generalized system of the following form

which satisfy the following conditions:

1) the coefficients are or2)3)4)

where is a constant5).

**Proof.** Let

(9)

We define the polynomial and the numbers, and in the following form:

(10)

(11)

(12)

Taking into consideration the following equation

and having the following relations (5)-(8) and (10)-(12), we obtain that the polynomial has the following form:

(13)

where

(14)

Then let

Clearly that (see (2) and (10)),

(15)

(16)

Hence

where Repeating the arguments in the proof of Lemma 1, we get a proof of the last statement of Lemma 1. Lemma 1 is proved.

**Lemma 2.** Let given the numbers. Then for any function, one can find a set and a polynomial in the Walsh generalized system

satisfying the following conditions:

and the non-zero coefficients in

are in decreasing order2)3)

4) for every measurable subset e of E5).

**Proof. **We choose some non-overlapping binary intervals and a step function

(17)

satisfying the conditions

(18)

(19)

(20)

Successively applying Lemma 1, we determine some sets and polynomials

(21)

where or, if,

(22)

, (23)

, (24)

Then let

(25)

(26)

>From (19), (21), (22) and (25) follows, that

and and the non-zero coefficients in are in decreasing order, i.e. the statements 1) - 3) of Lemma 2 are valid.

To verify the statement 4), for any determine from the condition. Then by (21) and (26)

(27)

Since for any point, (see (17), (23) and (26)), then from the conditions (18), (24), and (27) for every measurable subset e of E.

We have

Repeating the arguments in the proof of Lemma 2, we get a proof of the last statement of Lemma 2. Lemma 2 is proved.

The main tool in the proof of Theorem 2 is the following result.

**Lemma 3.** Let the Walsh generalized system, then for any there exist a weight function

with

such that for any numbers , and evry function, one can find polynomial in the Walsh generalized system

satisfying the following conditions:

2)

3)

.

Proof of Lemma 3

Let

(28)

be the sequence of all algebraic polynomials with rational coefficients. Applying repeatedly Lemma 2, we obtain sequences of sets and polynomials in the Walsh systems

(29)

where

which satisfy the following conditions:

(30)

(31)

(32)

for every measurable subset e of

(33)

Setting

(34)

It is clear (see (33), (34))

We define a function in the following way:

(35)

where

(36)

It follows from (34)-(36) that for all

(37)

In a similar way for all we have

(38)

By the conditions (31), (35)-(38) for all we obtain

(39)

Taking relations (32), (34)-(36) into account we obtain that for all, and

(40)

From the sequence (28) we choose a function such that

(41)

. (42)

Then, we set

,.

Now, it is not difficult to verify (see (30), (39)-(42)) that the function and the polynomials satisfy the requirements of Lemma 3.

Remark: In Lemma 3 polynom can be chosen such that

Lemma 3 is proved.

Proof Theorem 2

Let and let

(43)

be the sequence of all algebraic polynomials with rational coefficients. Applying repeatedly Lemma 3, we obtain a weight function with and, a sequences of polynomials in the Walsh generalized systems

(44)

where

which satisfy the following conditions:

(45)

(46)

(47)

Consider a series

(48)

Clearly (see (45), (48))

let and let. We choose some from sequence (43), to have

Suppose that the numbers and polynomials are already determined satisfying to the following conditions:

(49)

(50)

Let a function, be chosen from the sequence (43) such that

(51)

Hence by (49) we obtain

(52)

From the conditions(46) (47), (52) follows that

(53)

(54)

Then we obtain that the series

where

converges to in the -norm. Repeating the arguments in the proof of Theorem 2 and using Lemma 1, Lemma 2 and remark of Lemma 3 we get the proof of the second statement of Theorem 2.

Theorem 2 is proved.

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NOTES

^{*}This work was supported by State Committee Science MES RA, in frame of the research project No. SCS 13-1A313.