R E S E A R C H
Open Access
Bochner
pg
-frames
Mohammad H Faroughi
1,2and Morteza Rahmani
1**Correspondence:
[email protected] 1Department of Pure Mathematics,
Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran Full list of author information is available at the end of the article
Abstract
In this paper we introduce the concept of Bochnerpg-frames for Banach spaces. We characterize the Bochnerpg-frames and specify the optimal bounds of a Bochner
pg-frame. Then we define a Bochnerqg-Riesz basis and verify the relations between Bochnerpg-frames and Bochnerqg-Riesz bases. Finally, we discuss the perturbation of Bochnerpg-frames.
MSC: 42C15; 46G10
Keywords: Banach space; Hilbert space; frame; Bochner measurable; Bochner
pg-frame; Bochnerpg-Bessel family; Bochnerqg-Riesz basis
1 Introduction and preliminaries
The concept of frames (discrete frames) in Hilbert spaces has been introduced by Duffin and Schaeffer [] in to study some deep problems in nonharmonic Fourier series. After the fundamental paper [] by Daubechies, Grossmann and Meyer, the frame theory began to be widely used, particularly in the more specialized context of wavelet frames and Gabor frames. Frames play a fundamental role in signal processing, image and data com-pression and sampling theory. They provided an alternative to orthonormal bases and have the advantage of possessing a certain degree of redundancy. A discrete frame is a countable family of elements in a separable Hilbert space which allows for a stable, not necessarily unique, decomposition of an arbitrary element into the expansion of frame elements. For more details about discrete frames, see []. Resent results show that frames can provide a universal language in which many fundamental problems in pure mathematics can be formulated: the Kadison-Singer problem in operator algebras, the Bourgain-Tzafriri con-jecture in Banach space theory, paving Toeplitz operators in harmonic analysis and many others. Various types of frames have been proposed, for example,pg-frames in Banach spaces [], fusion frames [], continuous frames in Hilbert spaces [], continuous frames in Hilbert spaces [], continuousg-frames in Hilbert spaces [], (p,Y)-operator frames for a Banach space [].
This paper is organized as follows. In Section , we introduce the concept of Bochnerpg -frames for Banach spaces. Actually, continuous -frames motivate us to introduce this kind of frames and analogous to continuous frames which are a generalized version of discrete frames, we want to generalizepg-frames in a continuous sense. Like continuous frames, these frames can be used in the areas where we need generalized frames in a continuous aspect. Also, we define corresponding operators (synthesis, analysis and frame operators) and discuss their characteristics and properties. In Section , we define a Bochnerqg -Riesz basis and verify its relations by Bochnerpg-frames. Finally, Section is devoted to perturbation of Bochnerpg-frames.
Throughout this paper,XandHwill be a Banach space and a Hilbert space respectively, and{Hω}ω∈is a family of Hilbert spaces.
Suppose that (,,μ) is a measure space, whereμis a positive measure. The following definition introduces Bochner measurable functions.
Definition . A functionf :–→Xis called Bochner measurable if there exists a se-quence of simple functions{fn}∞n=such that
lim
n→∞fn(ω) –f(ω)= , a.e. [μ].
Definition . Ifμis a measure on (,) thenXhas the Radon-Nikodym property with respect toμif for every countably additive vector measureγ on (,) with values inX which has bounded variation and is absolutely continuous with respect toμ, there is a Bochner integrable functiong:–→Xsuch that
γ(E) =
E
g(ω)dμ(ω)
for every setE∈.
A Banach spaceXhas the Radon-Nikodym property ifXhas the Radon-Nikodym prop-erty with respect to every finite measure. Spaces with Radon-Nikodym propprop-erty include separable dual spaces and reflexive spaces, which include, in particular, Hilbert spaces.
Remark . Suppose that (,,μ) is a measure space andX∗has the Radon-Nikodym property. Let ≤p≤ ∞. The Bochner space ofLp(μ,X) is defined to be the Banach space
of (equivalence classes of )X-valued Bochner measurable functionsF fromto Xfor which the norms
Fp=
F(ω)pdμ(ω)
p
, ≤p<∞,
F∞=ess sup ω∈
F(ω), p=∞
are finite. In [, ] and [, p.] it is proved that if ≤p<∞andqis such that p+
q= ,
thenLq(μ,X∗) is isometrically isomorphic to (Lp(μ,X))∗if and only ifX∗has the Radon-Nikodym property. This isometric isomorphism is the mapping
ψ:Lqμ,X∗→Lp(μ,X)∗,
g→ψ(g),
where the mappingψ(g) is defined onLp(μ,X) by
ψ(g)(f) =
g(ω)f(ω)dμ(ω), f ∈Lp(μ,X).
So for allf ∈Lp(μ,X) andg∈Lq(μ,X∗), we have
f,ψ(g) =
In the following, we use the notation f,ginstead of f,ψ(g), so for allf ∈Lp(μ,X) and
g∈Lq(μ,X∗),
f,g=
f(ω),g(ω) dμ(ω).
Particularly, if H is a Hilbert space, then (Lp(μ,H))* is isometrically isomorphic to
Lq(μ,H). So, for allf ∈Lp(μ,H) andg∈Lq(μ,H),
f,g=
f(ω),g(ω) dμ(ω),
in which f(ω),g(ω)does not mean the inner product of elementsf(ω),g(ω) inH, but
f(ω),g(ω) =νg(ω)f(ω),
whereν:H–→H*is the isometric isomorphism betweenHandH*, for more details refer
to [, p.].
We will use the following lemma which is proved in [].
Lemma . If U:X–→Y is a bounded operator from a Banach space X into a Banach space Y , then its adjoint U*:Y*–→X*is surjective if and only if U has a bounded inverse
on RU.
Note that for a collection{Hβ}β∈Bof Hilbert spaces, we can suppose that there exists a Hilbert spaceKsuch that for allβ∈B,Hβ⊆K, whereK=
β∈BHβis the direct sum of
{Hβ}β∈B, see .. in [, p.].
2 Bochnerpg-frames
Bochner spaces are often used in a functional analysis approach to the study of partial differential equations that depend on time,e.g., the heat equation: if the temperatureg(t,x) is a scalar function of time and space, one can write (f(t))(x) :=g(t,x) to makef a function of time, with f(t) being a function of space, possibly in some Bochner space. Now, we intend to use this space to define a new kind of frames which contain all of continuous and discrete frames; in other words, we will generalize theg-frames to a continuous case that is constructed on the concept of Bochner spaces. Of course, this new frame can be useful in function spaces and operator theory to gain some general results that are achieved by g-frames or discrete frames.
2.1 Bochnerpg-frames and corresponding operators
We start with the definition of Bochnerpg-frames. Then we will give some characteriza-tions of these frames.
Definition . Let <p<∞. The family{ ω∈B(X,Hω) :ω∈}is a Bochnerpg-frame forXwith respect to{Hω}ω∈if:
(ii) there exist positive constantsAandBsuch that
Ax ≤
ω(x)pdμ(ω)
p
≤Bx, x∈X. (.)
AandBare called the lower and upper Bochnerpg-frame bounds respectively. We call that{ ω}ω∈is a tight Bochnerpg-frame ifAandBcan be chosen such thatA=B, and a Parseval Bochnerpg-frame ifAandBcan be chosen such thatA=B= . If for eachω∈, Hω=H, then{ ω}ω∈ is called a Bochnerpg-frame forX with respect toH. A family
{ ω∈B(X,Hω) :ω∈}is called a Bochnerpg-Bessel family forXwith respect to{Hω}ω∈ if the right inequality in (.) holds. In this case,Bis called the Bessel bound.
Example . Let{fi}i∈Ibe a frame for Hilbert spaceH,=Iandμbe a counting measure
on. Set
i:H–→C,
i(h) = h,fi, h∈H.
Then{ i}i∈Iis a Bochnerpg-frame forHwith respect toC.
Example . Let={a,b,c},={∅,{a,b},{c},}andμ:–→[,∞] be a measure such thatμ(∅) = , μ({a,b}) = ,μ({c}) = andμ() = . Assume thatX=Lp() and {Hω}ω∈ is a family of arbitrary Hilbert spaces and consider a fixed family {hω}ω∈⊆
{Hω}ω∈such thathω= ,ω∈. Suppose that
ω:Lp() –→Hω, ω(ϕ) =ϕ(c)hω.
It is clear that ω’s are bounded and for eachϕ∈Lp(),ω–→ ω(ϕ) is Bochner
mea-surable. Also,
ω(x)pdμ(ω) =
ϕ(c)pdμ(ω) =ϕ(c)pμ() = ϕ(c)p.
So,{ ω}ω∈is a Bochnerpg-frame forLp() with respect to{Hω}ω∈.
Now, we state the definition of some common corresponding operators for a Bochner pg-frame.
Definition . Let{ ω}ω∈be a Bochnerpg-Bessel family forXwith respect to{Hω}ω∈,
andqbe the conjugate exponent ofp. We define the operatorsTandU, by
T:Lq
μ, ω∈
Hω
–→X*, (.)
x,TG=
ω(x),G(ω)dμ(ω), x∈X,G∈Lq
μ, ω∈
Hω
U:X–→Lp
μ, ω∈
Hω
, (.)
Ux,G=
ω(x),G(ω) dμ(ω), x∈X,G∈Lq
μ, ω∈
Hω
.
The operatorsTandUare called the synthesis and analysis operators of{ ω}ω∈ respec-tively.
The following proposition shows these operators are bounded. It is analogous to Theo-rem .. in [].
Proposition . Let{ ω∈B(X,Hω) :ω∈}be a Bochner pg-Bessel family for X with respect to{Hω}ω∈and with Bessel bound B. Then the operators T and U defined by (.) and (.) respectively, are well defined and bounded withT ≤B andU ≤B.
Proof Suppose that{ ω}ω∈is a Bochnerpg-Bessel family with boundB, andqis the con-jugate exponent ofp. We show that for allx∈Xand allG∈Lq(μ,ω∈Hω), the mapping ω–→ ω(x),G(ω)is measurable. For allx∈X andG∈Lq(μ,
ω∈Hω),ω–→ ω(x) andGare Bochner measurable, so there are sequences of simple functions{λn}∞n= and
{gn}∞n=such that
lim
n→∞λn(ω) – ω(x)= , a.e. [μ], lim
n→∞gn(ω) –G(ω)= , a.e. [μ].
For eachn, λn,gnis a simple function and
ω(x),G(ω) –λn(ω),gn(ω)
≤ ω(x) –λn(ω),G(ω) +λn(ω),gn(ω) –G(ω) ≤ ω(x) –λn(ω)G(ω)+λn(ω)gn(ω) –G(ω).
So
lim
n→∞ ω(x),G(ω) –
λn(ω),gn(ω) =
andω–→ ω(x),G(ω)is measurable. For eachx∈XandG∈Lq(μ,
ω∈Hω), we have
x,TG=
ω(x),G(ω) dμ(ω)
≤
ωxG(ω)dμ(ω)
≤
ωxpdμ(ω)
p
G(ω)qdμ(ω)
q
Thus T is well defined, and T ≤B. By a similar discussion, U is well defined and
U ≤B.
The following proposition provides us with a concrete formula for the analysis operator.
Proposition . If{ ω}ω∈is a Bochner pg-Bessel family for X with respect to{Hω}ω∈ then for all x∈X,(Ux)(ω) = ωx, a.e.[μ].
Proof Letqbe the conjugate exponent ofpandx∈X. For allG∈Lq(μ,
ω∈Hω), we have
Ux,G=
ω(x),G(ω) dμ(ω)
={ ωx}ω∈,G.
So Ux–{ ωx}ω∈,G= , for allG∈Lq(μ,
ω∈Hω). There existsG∈Lq(μ,
ω∈Hω) such thatGq= and
Ux–{ ωx}ω∈,G =Ux–{ ωx}ω∈p,
which impliesUx–{ ωx}ω∈p= . Therefore, (Ux)(ω) = ωx, a.e. [μ].
The following proposition shows that it is enough to check the Bochnerpg-frame con-ditions on a dense subset. The discrete version of this proposition is available in [, Lemma ..].
Proposition . Suppose that(,,μ)is a measure space whereμisσ-finite. Let{ ω∈ B(X,Hω) :ω∈}be a family such that for each x∈X,ω–→ ω(x)is Bochner measurable, and assume that there exist positive constants A and B such that (.) holds for all x in a dense subset V of X. Then{ ω}ω∈is a Bochner pg-frame for X with respect to{Hω}ω∈ with bounds A and B.
Proof Let{n}∞n=be a family of disjoint measurable subsets ofsuch that=
∞
n=n
withμ(n) <∞for eachn≥. Letx∈Xand assume, without loss of generality, ωx = ,ω∈. Let
xm=ω∈|m– < ωx ≤m
, m= , , , . . . .
It is clear that for eachm= , , , . . . ,x
m⊆is measurable and=
∞
m=,n=(xm∩n),
where{xm∩n}∞n=,∞m=is a family of disjoint and measurable subsets of. If{ ω}ω∈is not a Bochnerpg-Bessel family forX, then there existsx∈Xsuch that
ω(x)pdμ(ω)
p
>Bx.
So
m,n
xm∩n
ω(x)p
and there exist finite setsIandJsuch that
m∈I
n∈J
xm∩n
ω(x)pdμ(ω) >Bpxp. (.)
Let{xk}∞k=be a sequence inV such thatxk→xask→ ∞. The assumption implies that
m∈I
n∈J
xm∩n
ω(xk)pdμ(ω)≤Bpxkp,
which is a contradiction to (.) (by the Lebesgue’s Dominated Convergence Theorem). So{ ω}ω∈is apg-Bessel family forXwith respect to{Hω}ω∈and Bessel boundB. Now, we show that
ω(xk)p
dμ(ω)
p →
ω(x)p
dμ(ω)
p
.
Since{ ω}ω∈is apg-Bessel family forX, the operatorUdefined by (.) is well defined and bounded. Assume thatqis the conjugate exponent ofpand let
G:–→ ω∈
Hω, G(ω) = ωx
p–q q
ωx, ω∈,
and
Gk:–→
ω∈
Hω, Gk(ω) = ωxk p–q
q
ωxk, ω∈.
It is obvious that for eachk∈N,GkandGbelong toLq(μ,
ω∈Hω), and we have
Ux,G=
ω(x),G(ω) dμ(ω) =
ω(x)pdμ(ω),
Uxk,Gk=
ω(xk),Gk(ω)dμ(ω) =
ω(xk) p
dμ(ω).
Sincelimk→∞(Gk–G)(ω)q= and
(Gk–G)(ω) q
≤Gk(ω)+G(ω) q
≤q–Gk(ω) q
+G(ω)q,
so by the Lebesgue’s Dominated Convergence Theorem,
lim
k→∞
(Gk–G)(ω) q
dμ(ω) = .
Therefore,limk→∞Gk–Gq= , hence
ω(xk) p
dμ(ω) –
ω(x)p
dμ(ω)
≤
ω(xk–x),Gk(ω) dμ(ω)
+
ω(x), (Gk–G)(ω) dμ(ω)
≤
ω(xk–x)pdμ(ω)
p
Gk(ω)qdμ(ω)
q
+
ω(x)pdμ(ω)
p
(G–Gk)(ω) q
dμ(ω)
q
≤Bxk–xGkq+Bx|Gk–Gq.
By lettingxk→x, the proof is completed.
2.2 Characterization of Bochnerpg-frames
Now we give some characterizations of Bochnerpg-frames in terms of their corresponding operators.
At first, we show the next lemma that is very useful in the case of complex valuedLp
-spaces.
Lemma . Let(,,μ)be a measure space whereμisσ-finite. Let <p<∞and q be its conjugate exponent. If F:–→H is Bochner measurable and for each G∈Lq(μ,H), | F(ω),G(ω)Hdμ(ω)|<∞, then F∈L
p(μ,H).
Proof Let{n}∞n=be a family of disjoint measurable subsets ofsuch that for eachn≥,
μ(n) <∞and=
∞
n=n. Without loss of generality, we can assumeF(ω) = ,ω∈.
Let
m=
ω∈|m– <F(ω)≤m, m= , , , . . . .
It is clear that for eachm= , , , . . . ,m⊆is measurable and=∞m=,n=(m∩n),
where{m∩n}∞n=,∞m=is a family of disjoint and measurable subsets of. We have
F(ω)pdμ(ω) = ∞
m=
∞
n=
m∩n
F(ω)pdμ(ω)
and
m∩n
F(ω)pdμ(ω)≤mpμ(n) <∞.
Suppose thatF(ω)pdμ(ω) =∞, then there exists a family{Ek}∞k= of disjoint finite
subsets ofN×Nsuch that
(m,n)∈Ek
m∩n
LetE=∞k=(m,n)∈Ek(m∩n). ConsiderG:–→Hdefined by
G(ω) =
⎧ ⎨ ⎩ c p q kF(ω)
p–q
q F(ω) ifω∈
(m,n)∈Ek(m∩n),k= , , . . . ,
ifω∈\E,
where
ck:=
k q p
(m,n)∈Ek(m∩n)
F(ω)pdμ(ω)
–p
.
ThenGis Bochner measurable, and
G(ω)qdμ(ω) =
E
G(ω)qdμ(ω)
= ∞
k=
(m,n)∈Ek
m∩n
G(ω)qdμ(ω)
= ∞ k=
(m,n)∈Ek(m∩n)
cpkF(ω)pdμ(ω) = ∞
k=
kq <∞.
Therefore,G∈Lq(μ,H). But
F(ω),G(ω)Hdμ(ω)= ∞ k= c p q k
(m,n)∈Ek(m∩n)
F(ω)pdμ(ω)
= ∞ k= k
(m,n)∈Ek(m∩n)
F(ω)pdμ(ω)
p > ∞ k= k=∞,
which is a contradiction.
The following theorem characterizes Bochnerpg-Bessel families by the operatorT de-fined by (.).
Theorem . Suppose that(,,μ)is a measure space whereμisσ-finite. Let{ ω∈ B(X,Hω) :ω∈}be a family such that for each x∈X the mappingω–→ ω(x)is Bochner measurable. If the operator T defined by (.) is well defined and bounded, then{ ω}ω∈ is a Bochner pg-Bessel family for X with respect to{Hω}ω∈with Bessel boundT.
Proof Letqbe the conjugate exponent ofpand forx∈X, consider
Fx:Lq
μ, ω∈
Hω
–→C,
Fx(G) = x,TG=
ω(x),G(ω) dμ(ω), G∈Lq
Then Fx∈(Lq(μ,
ω∈Hω))*. So{ ωx}ω∈ ∈Lp(μ,
ω∈Hω) by Lemma .. By Re-mark ., (Lq(μ,
ω∈Hω))* and Lp(μ,
ω∈Hω) are isometrically isomorphic and
{ ωx}ω∈p=Fx. Therefore,
ωxpdμ(ω)
p
=Fx= sup
Gq=
x,TG≤ Tx.
Similar to discrete frames, the analysis operator has closed range.
Lemma . Let{ ω}ω∈be a Bochner pg-frame for X with respect to{Hω}ω∈. Then the operator U defined by (.) has closed range.
Proof By assumption, there exist positive constantsAandBsuch that
Ax ≤
ω(x)pdμ(ω)
p
≤Bx, x∈X.
By Proposition ., we have
Ax ≤ Uxp≤Bx.
HenceUis bounded below. Therefore,Uhas closed range.
The next proposition shows that there is no Bochnerpg-frames for a non-reflexive Ba-nach spaces.
Proposition . Let{ ω}ω∈be a Bochner pg-frame for X with respect to{Hω}ω∈. Then X is reflexive.
Proof By Lemma .,RU is a closed subspace ofLp(μ,
ω∈Hω) andU:X–→RU is
homeomorphism. SinceLp(μ,ω∈Hω) is reflexive, soXis reflexive by Corollary ..
in [].
In the following lemma, we verify the adjoint operators of synthesis and analysis opera-tors.
Lemma . Suppose that{ ω}ω∈is a Bochner pg-Bessel family for X with respect to
{Hω}ω∈with the synthesis operator T and the analysis operator U. Then (i) U*=T .
(ii) If{ ω}ω∈has the lower Bochner pg-frame condition, then T*J=ψ*JU, where
J:X–→X**
and
J:Lp
μ, ω∈
Hω
–→
Lp
μ, ω∈
Hω
**
Proof
(i) For eachG∈Lq(μ,
ω∈Hω) andx∈X, we have
Ux,G=
ω(x),G(ω) dμ(ω) = x,TG,
soU*=T.
(ii) Since X andLp(μ,ω∈Hω) are reflexive,J andJ are surjective. For each G∈
Lq(μ,ω∈Hω) andx∈X,
G,T*Jx = TG,Jx= x,TG= Ux,G,
also
G,ψ*JUx = ψG,JUx= Ux,ψG= Ux,G.
HenceT*J
=ψ*JU.
The following theorem characterizes Bochnerpg-frames by the operatorT defined by (.).
Theorem . Consider the family{ ω∈B(X,Hω) :ω∈}.
(i) Let{ ω}ω∈be a Bochner pg-frame for X with respect to{Hω}ω∈. Then the operator T defined by (.) is a surjective bounded operator.
(ii) Let(,,μ)be a measure space whereμisσ-finite and for each x∈X,ω–→ ω(x)be Bochner measurable. Let the operator T defined by (.) be a surjective bounded operator. Then{ ω}ω∈is a Bochner pg-frame for X with respect to{Hω}ω∈.
Proof
(i) Since { ω}ω∈ is a Bochner pg-frame, by Proposition ., T is well defined and bounded. From the proof of Lemma .,U is bounded below. So, by Lemma . and Lemma .(i),U*=T is surjective.
(ii) SinceT is bounded,{ ω}ω∈is a Bochnerpg-Bessel family, by Theorem .. Since T=U*is surjective,Uhas a bounded inverse onRUby Lemma .. So there existsA>
such that for allx∈X,Uxp≥Ax. By Proposition ., for allx∈X
Ax ≤ Uxp=
ω(x)pdμ(ω)
p
.
Hence{ ω}ω∈is a Bochnerpg-frame.
Corollary . If{ ω}ω∈is a Bochner pg-frame for X with respect to{Hω}ω∈and q is the conjugate exponent of p, then for each x*∈X*, there exists G∈Lq(μ,
ω∈Hω)such that
x,x* =
ω(x),G(ω)dμ(ω), x∈X.
The optimal Bochnerpg-frame bounds can be expressed in terms of synthesis and anal-ysis operators.
Theorem . Let{ ω}ω∈be a Bochner pg-frame for X with respect to{Hω}ω∈. Then
Tand ˜Uare the optimal upper and lower Bochner pg-frame bounds of{ ω}ω∈ re-spectively, whereU is the inverse of U on R˜ U, and T , U are the synthesis and analysis
operators of{ ω}ω∈respectively.
Proof From the proof of Theorem ., for eachx∈X, we have
ωxpdμ(ω)
p
=Fx= sup
Gq=
x,TG.
Therefore,
sup x=
ωxpdμ(ω)
p
=sup x=
Fx=sup
x=
sup Gq=
x,TG
= sup Gq=
sup x=
x,TG= sup Gq=
TG=T.
By Proposition .,Uxp= (
ωx
pdμ(ω))p; consequently,
inf
x=Uxp=infx=
ωxpdμ(ω)
p
.
The operatorU:X–→Lp(μ,
ω∈Hω) is bounded below, so it has bounded inverseU˜ : RU–→X. We have
inf
x=Uxp=xinf=
Uxp x =Uyinf˜=
yp ˜Uy=infy=
yp ˜Uy=
supy= ˜Uyy
p
=
˜U,
henceinfx=( ωxpdμ(ω))
p =
˜U.
3 Bochnerqg-Riesz bases
In this section, we define Bochnerqg-Riesz bases which are the generalization of Riesz bases and characterize their properties.
Definition . Let <q<∞. A family{ ω∈B(X,Hω) :ω∈}is called a Bochnerqg -Riesz basis forX*with respect to{H
ω}ω∈, if: (i){x: ωx= , a.e. [μ]}={},
(ii) for eachx∈X,ω–→ ω(x) is Bochner measurable, and the operatorT defined by (.) is well defined, and there are positive constantsAandBsuch that
AGq≤ TG ≤BGq, G∈Lq
μ, ω∈
Hω
.
Under some conditions, a Bochnerqg-Riesz basis is a Bochnerpg-frame, more precisely:
Proposition . Suppose that(,,μ)is a measure space whereμisσ-finite and consider the family{ ω∈B(X,Hω) :ω∈}.
(i) Assume that for each x∈X,ω–→ ω(x)is Bochner measurable.{ ω}ω∈is a Bochner qg-Riesz basis for X*with respect to{H
ω}ω∈if and only if the operator T defined by (.) is an invertible bounded operator from Lq(μ,ω∈Hω)onto X*.
(ii) Let { ω}ω∈ be a Bochner qg-Riesz basis for X* with respect to{Hω}ω∈ with the optimal upper Bochner qg-Riesz basis bound B. If p is the conjugate exponent of q, then
{ ω}ω∈is a Bochner pg-frame for X with respect to{Hω}ω∈with optimal upper Bochner pg-frame bound B.
Proof
(i) By Theorem . and Proposition . and Lemma . and Theorems ., . and . in [], it is obvious.
(ii) By assumption and (i), the operatorTdefined by (.) is a bounded invertible opera-tor. So by Theorem .(ii),{ ω}ω∈is a Bochnerpg-frame forXwith respect to{Hω}ω∈ with the optimal upper Bochnerpg-frame boundB.
The next theorem presents some equivalent conditions for a Bochnerpg-frame being a Bochnerqg-Riesz basis.
Theorem . Suppose that(,,μ)is a measure space whereμisσ-finite. Let{ ω}ω∈ be a Bochner pg-frame for X with respect to{Hω}ω∈with the synthesis operator T and the analysis operator U, and q be the conjugate exponent of p. Then the following statements are equivalent:
(i){ ω}ω∈is a Bochner qg-Riesz basis for X*. (ii) T is injective.
(iii) RU=Lp(μ,
ω∈Hω).
Proof
(i)→(ii): It is obvious.
(ii)→(i): By Theorem .(i), the operatorT defined by (.) is bounded and onto. By (ii),T is also injective. Therefore,Thas a bounded inverseT–:X*–→Lq(μ,
ω∈Hω), and hence{ ω}ω∈is a Bochnerqg-Riesz basis forX*.
(i)→(iii): By Theorem .,T is invertible, soT*is invertible. Lemma .(ii) implies
thatRU=Lp(μ,
ω∈Hω).
(iii)→(i): Since the operatorUis invertible, by Lemma .,T=U*is invertible.
4 Perturbation of Bochnerpg-frames
The perturbation of a discrete frame has been discussed in []. In this section, we present another version of perturbation for Bochnerpg-frames.
measurable. Assume that there exist constantsλ,λ,γ such that
≤λ< , –λ≤λ< , ≤γ < ( –λ– λ)A
and
( ω–ω)x,G(ω) dμ(ω)
≤λ
ωx,G(ω) dμ(ω)
+λ
ωx,G(ω) dμ(ω)
+γGq, (.)
for all G∈Lq(μ,ω∈Hω)and x∈X. Then{ω}ω∈is a Bochner pg-frame for X with respect to{Hω}ω∈with bounds
A
( –λ– λ) –γA
–λ
and B
+λ+γB
–λ
,
where A and B are the Bochner pg-frame bounds for{ ω}ω∈.
Proof For eachx∈XandG∈Lq(μ,
ω∈Hω), we have
ωx,G(ω)dμ(ω)
≤
( ω–ω)x,G(ω)dμ(ω)
+
ωx,G(ω)dμ(ω)
≤( +λ)
ωx,G(ω) dμ(ω)
+λ
ωx,G(ω) dμ(ω)
+γGq.
So
ωx,G(ω)dμ(ω)
≤ +λ
–λ
ωx,G(ω)dμ(ω)
+ γ
–λ
Gq
≤ +λ
–λ
BGqx+
γ –λ
Gq
=
+λ
–λ
Bx+ γ –λ
Gq.
Now, defineW:Lq(μ,
ω∈Hω) –→X*by
x,WG=
ωx,G(ω)dμ(ω), x∈X,G∈Lq
μ, ω∈ Hω . Since
WG= sup x=
x,WG= sup x=
ωx,G(ω) dμ(ω)
≤
+λ
–λ
B+ γ –λ
soWis well defined and bounded. By Theorem .,{ω}ω∈is a Bochnerpg-Bessel family forXwith boundB[+λ+γB
–λ ].
Now, we show that{ω}ω∈satisfies the lower Bochnerpg-frame condition. LetTand Ube the synthesis and analysis operators of{ ω}ω∈respectively. By Proposition ., for allx∈X,
Ax ≤ Uxp≤Bx.
By Lemma ., RU is a closed subspace ofLp(μ,
ω∈Hω), soQ=U:X–→RU is a
bijective bounded operator, hence (Q–)*:X*–→R*
U is alike. SinceB–≤ Q– ≤A–,
so(Q–)*=Q– ≤A–. Letx*∈X*andS= (Q–)*, thenS ≤A–andS(x*)∈R*U, by Hahn-Banach theorem there existsϕ∈Lp(μ,
ω∈Hω) such thatϕ|RU=S(x
*) andϕ=
S(x*). It follows that
ϕ=Sx*≤ Sx*≤A–x*. (.)
By Remark ., there existsG∈Lq(μ,
ω∈Hω) such thatψ(G) =ϕ, then
Gq=ϕ ≤A–x*. (.)
Sincex*=Q*(Q–)*(x*), we have for eachx∈X,
x,x* =x,Q*Sx* =Ux,Sx* = Ux,ϕ
=Ux,ψ(G) =
ωx,G(ω) dμ(ω).
From (.) and (.), we obtain that
x*–WG
=sup x=
x,x*–WG
=sup x=
( ω–ω)x,G(ω)μ(ω)
≤ sup x=
λx,x* +λx,WG+γGq
≤ sup x=
(λ+λ)x,x* +λx,WG–x* +γA–x*
≤ sup x=
(λ+λ)x*x+λWG–x*x+γA–x*
≤λ+λ+γA–x*+λWG–x*,
which implies
WG–x*≤(λ+λ)A+γ ( –λ)A
For a givenx∈X, there existsx*∈X*such that
x*= , x=x*(x).
Hence
x=x*(x) =x,x* =x,x*–WG + x,WG
≤x,x*–WG +x,WG
≤(λ+λ)A+γ
( –λ)A
x*x+
ωx,G(ω) dμ(ω)
≤(λ+λ)A+γ
( –λ)A
x*x+
ωxpdμ(ω)
p Gq
≤(λ+λ)A+γ
( –λ)A
x*x+A–x*
ωxpdμ(ω)
p
,
therefore
( –λ– λ)A–γ
–λ
x ≤
ωxpdμ(ω)
p
.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally to the manuscript and read and approved the final manuscript.
Author details
1Department of Pure Mathematics, Faculty of Mathematical Sciences, University of Tabriz, Tabriz, Iran.2Islamic Azad
University-Shabestar Branch, Shabestar, Iran.
Acknowledgements
The authors would like to sincerely thank Prof. Dr. Gitta Kutyniok and Dr. Asghar Ranjbari for their valuable comments.
Received: 10 May 2012 Accepted: 20 August 2012 Published: 4 September 2012
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doi:10.1186/1029-242X-2012-196
