On Generalized Approximative Properties of Systems

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Volume 2012, Article ID 729235,7pages doi:10.1155/2012/729235

Research Article

On Generalized Approximative

Properties of Systems

Turab Mursal Ahmedov and Zahira Vahid Mamedova

Institute of Mathematics and Mechanics, National Academy of Sciences of Azerbaijan, 9 B. Vaxabzade, Baku AZ1141, Azerbaijan

Correspondence should be addressed to Zahira Vahid Mamedova,zahira eng13@hotmail.com

Received 5 January 2012; Accepted 15 February 2012

Academic Editor: Siegfried Gottwald

Copyrightq2012 T. Mursal Ahmedov and Z. Vahid Mamedova. 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.

Generalized concepts of b-completeness, b-independence, b-minimality, and b-basicity are

introduced. Corresponding concept of a space of coeﬃcients is defined, and some of its properties

are stated.

1. Introduction

Theory of classical basisincluding Schauder basisis suﬃciently well developed, and quite a good number of monographs such as Day1, Singer2, 3, Young 4, Bilalov, Veliyev

5, Ch. Heil6, and so forth have been dedicated to it so far. There are diﬀerent versions and generalizations of the concept of classical basisfor more details see2,3. One of such generalizations is proposed in 5, 7. In these works, the concept of bY-invariance is first

introduced and then used to obtain main results. It should be noted that the condition of

bY-invariance implies the fact that the corresponding mapping is tensorial.

In our work, we neglect thebY-invariance condition. We give more detailed

consider-ation to the space of coeﬃcients. We also state a criterion of basicity.

2. Needful Notations and Concepts

Let X, Y, Z be some Banach spaces, and let · _X, · _Y, · _Z be the corresponding norms. Assume that we are given some bounded bilinear mappingb : X ×Y → Z, that is,bx;y_Z ≤cx_Xy_Y, for allx∈X, for ally∈Y, wherecis an absolute constant. For simplicity, we denotexy≡ bx;y. LetM⊂Y be some set. ByLbMwe denote theb-span

ofM. So, by definition,LbM≡ {z∈Z:∃{xk}n1 ⊂X,∃{yk}n1 ⊂M, z

_n

k1xkyk}. By·we

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System{yn}_n_∈N⊂Yis calledb-linearly independent if∞n1xnyn0 inZimplies that

xn0,for alln∈N.

In the context of the above mentioned, the concept of usual completeness is stated as follows.

Definition 2.1. System{y_n}_n_∈N⊂Y is calledb-complete inZifLb{y_n}_n_∈N ≡Z. We will also

need the concepts ofb-biorthogonal system andb-basis.

Definition 2.2. System{y_n∗}_n_∈N⊂LZ;Xis calledb-biorthogonal to the system{yn}_n_∈N⊂Y if

y∗

nxyk δnkx, for alln, k∈N, for allx∈X, whereδnkis the Kronecker symbol.

Definition 2.3. System{yn}_n_∈N ⊂ Y is calledb-basis inZif for for allz ∈ Z,∃!{xn}n∈N ⊂ X :

z∞_n₁xnyn.

To obtain our main results we will use the following concept.

Definition 2.4. System{yn}_n_∈N⊂Y is called nondegenerate if∃cn>0 : xX ≤cnxyn_Z, for

allx∈X, for alln∈N.

3. Main Results

3.1. Space of Coefficients

Let{yn}n∈N⊂Y be some system. Assume that

Ky≡

{xn}n∈N ⊂X: the series

∞

n1

xnyn converges inZ

. 3.1

With regard to the ordinary operations of addition and multiplication by a complex number,

Kyis a linear space. We introduce a norm · Ky inKyas follows:

x_Ky sup

m

n1

xnyn

Z

, 3.2

wherex≡ {x_n}_n_∈N∈ K_y. In fact, it is clear that

λ x_Ky |λ| x_Ky, ∀λ∈C;

x1 x2_Ky ≤ x1_Ky x2_Ky, ∀xk∈ Ky, k1,2.

3.3

Assume thatx_Ky 0 for somex ≡ {xn}n∈N ∈ Ky .Letn0 inf{n : xk 0, for allk ≤

n−1}. In what follows we will suppose that the system{y_n}_n_∈Nis nondegenerate. Letn0< ∞.

We have

sup

m

n1

xnyn

Z

≥n0

n1

xnyn

Z

xn0yn0Z≥

1

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But this is contrary tox_Ky 0. Consequently,n0 ∞, that is,x 0. Thus,Ky; · Ky

is a normed space. Let us show that it is complete. Let{xn}n∈N ⊂ Kybe some fundamental

sequence withx_n≡ {x_kn}_k_∈N⊂X. For arbitrary fixed numberk∈N. We have

x_kn−x_kn p

X≤ck

x_kn−x_kn pyk

Z ck k i1

x_in−x_in pyi− k−1

i1

x_in−x_in pyi

Z

≤2cksup

m m i1

x_in−x_in pyi

2ckxn−xn p_Ky −→0, asn, p−→ ∞.

3.5

As a result, for all fixed k ∈ N the sequence {x_kn}_n_∈N is fundamental in X. Let x_kn → xk, n → ∞. Let us take arbitrary positive ε. Then ∃n0: for all n ≥ n0, for all p ∈ N, we

havex_n−x_n _p_Ky < ε. Thus

m k1

x_kn−x_kn pyk

Z

< ε, ∀n≥n0, ∀p, m∈N. 3.6

Passing to the limit asp → ∞yields

m k1

x_kn−xk

yk Z

≤ε, ∀n≥n0, ∀m∈N. 3.7

It is easy to see that

mp

km

x_kn−xk

yk Z

≤2ε, ∀n≥n0, ∀m, p∈N. 3.8

Since the series∞_k₁x_knykconverges inZ, it is clear that∃m0n: for allm≥m

n

0 , for allp∈N,

we have

mp

km

x_knyk

Z

< ε. 3.9

Then it follows from the previous inequality that

m_k_mpxkyk

Z

≤ mp

km

x_kn−xk

yk Z

mp

km

x_knyk

Z

≤ 3ε,

∀m≥m₀n, ∀p∈N.

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Consequently, the series∞_k₁xkyk converges inZ, and thereforex ≡ {xk}k∈N ∈ Ky. From

3.7it follows directly thatx_n−x_Ky → 0, n → ∞. Thus,K_yis a Banach space. Let us consider the operatorK: Ky → Z, defined by the expression

Kx∞

n1

xnyn, x≡ {xn}n∈N. 3.11

It is obvious thatKis a linear operator. LetzKx. We have

K x_Zz_Z∞

n1

xn yn

Z

≤ sup

m

n1

xn yn

Z

x_Ky. 3.12

It follows that K ∈ LKy;Z and K ≤ 1. Let x0 ≡ {x; 0;· · · }, x ∈ X. It is clear that Kx0_Zx0_Ky. Consequently,K1. It is absolutely obvious that if the system{y_n}_n_∈N⊂

Y isb-linearly independent, then KerK {0}. In this case,∃K−1 : Z → Ky. If ImKis closed,

then, by the Banach’s theorem on the inverse operator, we obtain thatK−1 ∈ LImK;Ky.

The same considerations are valid in the case when the system{y_n}_n_∈Nhas ab-biorthogonal

system. We will call operatorKa coeﬃcient operator. Thus, we have proved the following.

Theorem 3.1. Every nondegenerate systemSy ≡ {yn}n∈N ⊂ Y is corresponded by a Banach space

of coeﬃcientsK_y and coeﬃcient operatorK ∈ LK_y;Z,K 1. If the system S_y isb-linearly

independent or has a b-biorthogonal system, then ∃K−1. Moreover, if ImK is closed, thenK−1 ∈

LImK;Ky.

In what follows, we will need the concept of b-basis in the space of coeﬃcientsK_y.

Definition 3.2. System {Tn}n∈N ⊂ LX; Ky is called b-basis in Ky if for for allx ∈

Ky,∃!{xn}n∈N⊂X:x ∞n1Tnxnconvergence inKy.

Consider the operatorsE_n : X−→ K_y:E_nx{δ_nkx}_k_∈N, n∈N. We have

EnxKy xyn_Z≤yn_YxX, ∀x∈X. 3.13

Thus,En ∈LX;Ky, for alln∈N. Takex≡ {xn}n∈N∈ Ky. Then

x−

m

n1

Enxn

Ky

⎧ ⎨

⎩0; · · ·; 0;

m

xm 1;xm 2;· · ·

⎫ ⎬ ⎭

Ky

sup

p

mp

nm 1

xnyn

z

−→0, 3.14

as m → ∞, since the series ∞_n₁xnyn converges in Z. As a result, we obtain that x

_∞

n1Enxn. Consider the operatorsPn :Ky → X:Pn xxn, n∈N. We have

PnxX xnX≤cnxnyn_Z≤cnsup

m

k1

xkyk

Z

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Consequently,Pn∈LKy;X,n∈N. Let us show that the expansionx∞n1Enxnis unique.

Let∞_n₁E_nx_n 0. We have 0 P_k∞_n₁E_nx_n ∞_n₁P_kE_nx_n x_k,for allk ∈ N. As a result, we obtain that the system{En}n∈Nforms ab-basis forKy. We will call this system a

canonical system. So we have proved the following.

Theorem 3.3. Let Ky be a space of coeﬃcients of nondegenerate system {yn}n∈N ⊂ Y. Then the

canonical system{En}n∈Nforms ab-basis forKy.

Let the nondegenerate system{yn}_n_∈N⊂Yform ab-basis forZ. Consider the coeﬃcient

operatorK: K_y → Z. By definition of b-basis, the equationK xzis solvable with regard tox ∈ Ky for for allz ∈ Z. It is absolutely clear thatKerK {0}. Then it follows from Theorem 3.1and Banach theorem thatK−1 ∈LZ;Ky. Consequently, operatorK performs

isomorphism betweenK_y and Z.

Vice versa, let{yn}_n_∈N⊂Y be a nondegenerate system and letKybe a corresponding

space of coeﬃcients. Assume that a coeﬃcient operatorK ∈ LKy;Zis an isomorphism.

Take for allz ∈Z. It is clear that∃x≡ {xn}n∈N∈ Ky : Kxz, that isz ∞n1xnyn inZ.

Consequently,zcan be expanded in a series with respect to the system{y_n}_n_∈N. Let us show that such an expansion is unique. Let∞_n₁x0

nyn0 for somex0 ≡ {x0n}n∈N∈ Ky. This means

thatKx0 0⇒ x0 0⇒ x_n0 0, for for alln∈N. Thus, the system{yn}_n_∈N forms a b-basis

for Z. So the following theorem is proved.

Theorem 3.4. LetKy be a space of coeﬃcients of nondegenerate system{yn}n∈N and letK be a

corresponding coeﬃcient operator. Then this system forms a b-basis for Z if and only if K is an

isomorphism inLKy; Z.

3.2. Criterion of Basicity

Let the systems{yn}_n_∈N⊂Y and{yn∗}n∈N ⊂LZ; Xbe b-biorthogonal. Take for allz∈Zand

consider the partial sums

Snz n

k1

y∗

kzyk, n∈N. 3.16

We have

SnSmz n

k1

y∗

kSmzyk n

k1

y∗

k

_m

i1

y∗

izyi

yk

n

k1

m

i1

δkiyi∗zyk

min{n;m}

k1

y∗

kzykSmin{n;m}z, ∀n, m∈N.

3.17

Hence,S2

n Sn, for for alln ∈N, that is,Snis a projector inZ. It follows directly from the

estimate

_y∗

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thatSnis a continuous projector. Suppose that the nondegenerate system{yn}_n_∈N⊂Y forms

ab-basis forZ. Then for for allz∈Zhas a unique expansionz∞_n₁x_ny_ninZ. We denote

the correspondencez → xnbyyn∗ :y∗nz xn, for alln∈N. It is obvious thaty∗n:Z → Xis

a linear operator. LetKybe a space of coeﬃcients of basis{yn}_n_∈N, and letK:Ky → Zbe the

corresponding coeﬃcient operator. ByTheorem 3.4,Kis an isomorphism. We have

_y∗

nzX xnX≤cnxnyn≤cnsup m

m

k1

xkyk

Z

cnxKy

cnK−1z

Ky ≤cn

K−1z_Z,

3.19

wherex≡ {xn}n∈N. Consequently,{yn∗}n∈N⊂LZ;X. It follows directly from the uniqueness

of the expansion thaty∗_nxy_k δ_nkx; for alln, k∈N, for for allx∈X. As a result, we obtain that the system{y∗_n}_n_∈Nisb-biorthogonal to{yn}_n_∈N. Let us consider the projectorsSm∈LZ

for for allz∈Z:

Smz m

n1

y∗

nzyn, m∈N. 3.20

As the series3.20converges for for allz∈Z, it follows from Banach-Steinhaus theorem that

Msup

m Sm< ∞. 3.21

It is absolutely obvious that the system{yn}n∈Nis b-complete inZ. Thus, if the system{yn}n∈N

forms ab-basis forZ, then1it isb-complete inZ;2it has ab-biorthogonal system;3the corresponding family of projectors is uniformly bounded.

Vice versa, let the system{yn}n∈Nbeb-complete inZand have ab-biorthogonal system

{y∗

n}n∈N. Assume that the corresponding family of projectors{Sm}m∈Nis uniformly bounded,

that is, relation3.21holds. Letz∈Zbe an arbitrary element. Let us take arbitrary positive ε. It is clear that∃{xn}m_n₁0 ⊂X: z−m_n₁0 xnyn_Z< ε. Letz0m_n₁0 xnyn. We have

y∗

nz0

m0

k1

y∗

nxkykxn, n1, m0;y∗nz0 0,∀n > m0. 3.22

As a result, we obtain form≥m0that

z−SmzZ≤ z−z0Z z0−SmzZ≤ε

m

n1

y∗

nz−z0yn

Z

≤M 1ε. 3.23

From the arbitrariness of ε we get limm→ ∞Smz z. Consequently, for for all z ∈ Z can

be expanded in a series with respect to the system {yn}_n_∈N. The existence ofb-biorthogonal

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Theorem 3.5. Nondegenerate system{yn}_n_∈N⊂Y forms ab-basis forZif and only if the following

conditions are satisfied:

1the system isb-complete inZ;

2it has ab-biorthogonal system{y∗_n}_n_∈N⊂LZ;X;

3the family of projectors3.21is uniformly bounded.

References

1 M. M. Day, Normed Linear Spaces, Izdat. Inostr. Lit., Moscow, Russia, 1961.

2 I. Singer, Bases in Banach Spaces, vol. 1, Springer, New York, NY, USA, 1970.

3 I. Singer, Bases in Banach Spaces, vol. 2, Springer, New York, NY, USA, 1981.

4 R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, NY, USA, 1980.

5 B. T. Bilalov and S. G. Veliyev, Some problems of Basis, Elm, Baku, Azerbaijan, 2010.

6 Ch. Heil, A Basis Theory Primer, Springer, New York, NY, USA, 2011.

7 B. T. Bilalov and A. I. Ismailov, “Bases and tensor product,” Transactions of National Academy of Sciences

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