Basis of the properties of weighted exponential systems with excess Текст научной статьи по специальности «Математика»

14

YAK 517.982

Вестник Самарского университета. Естественнонаучная серия. Том 24 № 1 2018

DOI: 10.18287/2541-7525-2018-24-1-14-19

A.Sh. Shukurov1

BASIS OF THE PROPERTIES OF WEIGHTED EXPONENTIAL SYSTEMS

WITH EXCESS

The main aim of this paper is the determination of a class of such functions for which a weighted exponential system becomes complete and minimal in appropriate space when exactly one of its terms is eliminated. It is shown that the system, obtained in this way cannot be a Schouder basis in this space. The last fact shows that Muckenhoupt-type criterion for the exponential system to be the Schauder basis in Lebesgue spaces after elimination of an element does not exist. This paper generalizes the results of the paper by E.S. Golubeva.

Key words: system of weighted exponentials, Muckenhoupt condition.

1. Introduction

The basis properties (completeness, minimality and Schauder basicity) of systems of the form {"(t)^n(t)}, where {^n(t)} is an exponential or trigonometric (cosine or sine) systems have been investigated in several papers (see, for example, [1-16]). To our knowledge, first result in this direction is [1] in which Babenko gave an example {|t|a • eint}neZ, where |a| < 1 and a = 0, answering in the affirmative a question of Bari ([17]) on the existence of normalized basis for L2(-n,n) that is not a Riesz basis. The result of Babenko ([1]) was then extended by V.F.Gaposhkin in his famous paper [14], where, in particular, some sufficient condition (on the weight function "(t)) for the system {"(t) • emt}neZ to be a basis in L2(-n,n) was found. And eventually, necessary and sufficient condition on the weight function "(t) whcih ensures the Schauder basicity of the exponential system {eint}neZ in weighted Lebesgue space Lp^^(-n,n) has been obtained (see, for example [16]); such a condition is the Muckenhaupt condition with respect to the weight function "(t).

SUrP(W\ h "(t)dt) (II h —(t)dt) <

where sup is taken over all intervals I and \I\ is the length of the interval I. Note that study of basicity properties of a system in weighted Lebesgue spaces Lpa,q is equivalent to the study of analogous properties of this system with corresponding degenerate coefficient in the "ordinary"Lebesgue space Lp. Therefore, the mentioned criterion can also be considered as a necessary and sufficient condition for the Schauder basicity in Lp of the exponential system with degenerate coefficients {"(t)eint}neZ.

There are concrete examples of weight function "(t) for which the system {"(t)eint}neZ itself is not complete and minimal but becomes so when some of its terms are eliminated. For example, it is proved in [8] that, a system {temt}neZ is not complete and minimal (and hence Schauder basis) in L2(—n,n) but becomes complete and minimal when one of its elements is eliminated from the original system; more precisely, a system {teint}neZ/{0y is complete and minimal system in L2(-n,n), but it is not a Schauder basis in it. It is also mentioned in [8] that the indicated statement about the system remains valid when any one of its terms is eliminated.

We generalize the indicated result of the paper [8] to the most general case of the system {"(t)emt}neZ, where "(t) is any function.

The aim of this note is the determination of the class of all functions "(t) for which the system {"(t)eint}Z becomes complete and minimal in Lp(-n,n), 1 <p < m, space when exactly one of its terms is eliminated. It is also shown that the system, obtained from the system {"(t)eint}Z in this way (by elimination of an element) cannot be a Schauder basis in Lp(-n,n) space.

!© Shukurov A.Sh., 2018

Shukurov Aydin Shukur ([email protected]), Institute of Mathematics and Mechanics, NAS of Azerbaijan, 9, B. Vahabzade, Baku, Az1141, Azerbaijan.

2. Auxiliary facts

We will use some auxiliary facts which are of some interest in their own too.

Lemma 1. If the system {w(t)eint}neZ/ko is a minimal system in Lp(-n ,n), then it has a biorthogonal system {bn(t)}z/k0 which is of the following form:

eint + p eikot

bn(t) = -^-, (1)

where pn are some complex numbers.

Proof. The fact that {u(t)eint}neZ/ko has a biorthogonal system follows from its minimality. Denote the biorthogonal system by {bn(t)}Z/ko. Take arbitrary natural number n = k0. By the definition of the biorthogonal system

f bn (t)u(t)eiktdt = 0, yk = n,k0 (2)

J —n

and

i bn(t)u(t)eintdt = 1. (3)

J —n

The relations (2), along with the fact that the Fourier coefficients of a summable function with respect to an exponential system is unique, imply that there are some complex numbers an and pn such that

eint + p eikot bn(t) '""' "

aneint + pn e%

u(t)

Substitutung it into (3) and taking into account that {eint} is an orthonormal system, we find that an = 1 for all n. This proves the relation (1). The Lemma is proved.

Let {pn} be any sequence of numbers and k0 any integer. For simplicity, we will make use of the following denotation:

&n(t) = eint - pneik0t.

Lemma 2. A function &n(t), where n = k0, may have only simple zeros. Proof. Assume that there is a point t0 for which

$n(to) = 0, $n(t0) =0.

These equalities can be written in the following form:

eint0 - pneik0t0 = 0,

and

neint0 - k0pneik0t0 = 0,

accordingly. These equalities imply that n = k0 which is impossible by the condition of the lemma. The lemma is proved.

Lemma 3. Let A C \-n,n] be such that &n(t) = 0 for all n € Z/{k0}, t £ A. Then the set A consists of at most two points and it is two-element set only if A = {-n,n}. Proof. Let ti,t2 € [—n,n] be such that

e1 - pneik = 0,

and

eint2 - pneikot2 = 0

for all n = k0. These two relations are possible if and only if

(n - k0)(t2 - tl) z 2n

for all n = k0. Taking, in particular, n = k0 + 1, we obtain from the last relation that t2 -11 is an integer multiply of 2n. But this is possible if and only if t1 = -n,t2 = n or t2 = -n,t1 = n. Thus, the set A may contain at most two points and it contains two elements only if A = {-n,n}. The lemma is proved

Lemma 4. Let p be any complex number and n,m any integers such that n = m. Then the function eint - p , eimt may have only finite number of zeroes in the segment [-n,n].

Proof. Assume the contrary: the function eint - p ■ eimt has an infinite number of zeroes. Let {zn}^=1 C C [-n,n] be its zeroes. By the Bolzano-Weierstrass theorem, the sequence {zn}^=1 has a limit point in [-n,n]. Therefore, since the function einz - p ■ eimz is an entire function on the whole complex plane, the uniqueness theorem for analytic functions implies that eint - p ■ eimt = 0 on the segment [-n,n]. This means that the system of functions {einz,eimz} is linearly independent system. Contradiction: since it is orthonormal, it cannot be linearly independent. The Lemma is proved.

3. Main result and its proof

The aim of this paper is to prove the following

Main Theorem. Let k0 be any integer. The system {"(t)eint}neZ/{koy is complete and minimal i Lp(-n,n), 1 <p < x>, space if and only if "(t) £ Lp(-n,n), -^J) £ Lq(-n,n) and besides

in

1) there is a (unique) point t0 G \-n,n] such that G Lq(-n,n);

or

2) £ Lq(-n,n).

Proof. Necessity. The validity of "(t) £ Lp(-n,n) is evident.

Let the system {"(t)eint}neZ/{ko} be complete and minimal in Lp(-n,n). Then

"(t)

Assume the contrary:

1 ' Lq(-n,n). (4)

1 G Lq (-n,n).

"(t)

Then eikot £ Lq(-n,n) and besides this, it is evident that the function eikot is not trivial (is not equivalent to zero) and

"(t)eint — eik0tdt = 0 "(t)

for all n £ Z,n= k0. These observations show that the system {"(t)eint}neZ/{koy is not complete. Thus, our assumption is false - (4) is valid.

Since the system {"(t)eint}neZ/{koy is minimal, by Lemma 1 it has a biorthogonal that is of the following form:

eint p eik0t

bn(t) = -pn-, yn = k0. (5)

"(t)

If the function eint -pneikot has no zeros on the segment [-n,n] for some index n, then the representation (5), the fact that bn(t) £ Lq(-n,n) and the continuity of the function eint - pneikot imply that ^y £ £ Lq(-n,n) which contradicts to (4).

Let n0 be any natural number satisfying the condition n0 = k0. By Lemma 4, the function einot - pneikot has finite number of zeros z1 ,...,zm on the segment [-n,n]. With this in mind, using the representation (5) of the biorthogonal system, the fact that bn(t) £ Lq(-n,n) for all n = k0 and the condition £ Lq(-n,n), we obtain that there is a nonempty subset A C [-n,n] (consisting of the points z1 ,...,zm ) such that

eint - pneik0t = 0

for all n £ Z/{k0}. By Lemma 3, the set A consists of at most two points and it contains two elements only if A = {-n,n}.

We treat the single point and two-point cases separately.

First, consider the case A = {t0}, where t0 is some number in {z1,..., zm}. Write the function bn0 (t) in the following form

b (t) = t-0 einot - pneik0t bn0 ( ) = "(t) ^ t - t0 ,

and consider an auxiliary function

6 6 . if t = to'

K0 (to), if t = to.

Then, using the definition of the set A, taking into account that bn (t) e Lq(—n,n) for all n = k0 and using Lemma 2, it can be derived from here that

tt—) e Lq (-n,n).

Now, consider the case A = {—n,n}. Write the function bno(t) in the following form

kot

b (t — n)(t + n) einot — Ueikot w = k

no (t) = -T----------, Vn = ko.

w(t) (t — n)(t + n)

and consider a function

not—C eiko

(t—^t+n) , if t e (—n,n);

m = { — , if t = —.

o (n) if t = n.

2n

Then, again, using the definition of the set A, taking into account that bn(t) e Lq(—n,n) for all n = k0 and applying Lemma 2, it can be shown that

(t — n)(t + n) T

—^— e Lq (—n>n)

Sufficiency. First, consider the case 1) : there is a unique point t0 e [—n,n] such that e Lq(—n,n). Let k0 be any integer. If a function f (t) is orthogonal to the system {^(t) ■ eint}neZ/{ko},

i f (t)u(t)eintdt = 0, Vn = k0, (6)

>J —n

then f (t) = c'6(t) for some constant c (it is a consequence of the fact that Fourier coefficients of a summable

function with respect to an exponential system is unique). Since e Lq(—n,n), f (t) = ^(to e Lq(—n,n) if and only if c = 0, i.e. f (t) = 0. Thus, the relations (6 ) imply f (t) = 0. This means that the system {u(t)eint]nez/{ko} is complete in Lp(—n,n).

Consider a function bn(t) defined by (5 ), where = -¡jinotto. Writing the function bn(t) in the form

, . . t — to eint — &eikot

bn(t)

w(t) t — to '

taking into accout the relation t—|y e Lq(—n,n) and the fact that t0 is a root of the function eint — £neikot, we obtain that bn(t) e Lq(—n,n) for all n e Z,n = k0; besides it, it is easy to see that

f bn(t)u(t)eimtdt = 5nm 'J —n

for all n,m = k0, where Snm is a Kronecker symbol. Therefore, the system {bn(t)} is biorthogonal to {u(t)eint]neZ/{ko} and hence, the system {u(t)eint}neZ/{ko} is minimal in Lp(—n,n) space. The case 2) is treated similarly. The theorem is proved.

This theorem immediately implies the following

Corollary. If the system {w(t)eint} z becomes complete and minimal in Lp(—n,n) space when one of its terms is eliminated, then it also becomes complete and minimal in Lp(—n,n) when any one of its terms is eliminated.

6

4. Schauder basicity

The result of the previous section characterizes the class of all functions u(t) for which the system {w(t)eint}z becomes complete and minimal in Lp(—n,n), 1 < p < space when exactly one (actually, by Corollary of the previous section, any one) of its terms is eliminated. Therefore it is natural to ask for

condition on u(t) which ensures Schauder basicity in Lp(-n,n) of the system, obtained from the system {u(t)eint}z by elimination of exactly one of its terms. It turns out that such a condition does not exist:

Theorem. Let w(t) be any function and k0 be any integer. Then the system {u(t)eint} z/ {k0} is not a Schauder basis in Lp(-n,n), 1 <p < ж, space.

Proof. Assume the contrary: the system {^(t)eint}Z/{koy is a Schauder basis in Lp(-n,n). Then the function u(t)eikot has an expansion (in Lp norm) of the form

u(t)eikot =J2 cnu(t)eint. (7)

n=ko

Take arbitrary natural number n = k0. Applying the biorthogonal system (5) to both sides of (7), we obtain that

Cn = in, Vn = ко

Thus, the series

E inu(t)eint

n=ko

int I

is convergent. Therefore, by the necessary condition for the convergence of the series, \\£nu(t)e- --\\Lp =

= 0. This equality and the identity \\£nu(t)eint\\Lp = |£n| • ||w(t)\\Lp imply that limn^TO = 0. But, on the other hand, as a consequence of the basicity, the system {^(t)eint}Z/{koy is complete and minimal in Lp(—n,n) and a closer look at the proof of Main Theorem from the previous section shows that the number £n in the definition of the biorthogonal system (5) satisfies the equality £n = for all n e Z,n = k0, for some

t0 e [-п,п]. This yields a contradiction. The Theorem is proved.

Remark. It should be noted that this fact also follows from the more general result obtained in [18-20].

Note that negative results on Schauder basicity of some systems of a certain form were also studied earlier in papers [21-25].

Acknowledgement. The author is grateful to Professor B.T.Bilalov for encouraging discussion. The author also thanks N.J. Guliyev for his useful assistance.

References

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А.Ш. Шукюров2

О БАЗИСНЫХ СВОЙСТВАХ ВЗВЕШЕННЫХ ЭКСПОНЕНЦИАЛЬНЫХ

СИСТЕМ С ИЗБЫТКОМ

Целью настоящей работы является обобщение результатов Э.С. Голубева на основе свойств взвешенных экспоненциальных систем, опубликованных ранее в этом журнале, в наиболее общем случае.

Ключевые слова: система взвешенных экспоненциалов, условие МискепЬоир^

Статья поступила в редакцию 28/Л/2018. The article received 28/Л/2018.

2Шукюров Айдын Шукюр оглы ([email protected]), Институт Математики и Механики, Национальная Академия Наук Азербайджана, Az1141, Азербайджан, Баку, Б. Вахабзаде, 9.