Advances in Pure Mathematics, 2013, 3, 438-441
http://dx.doi.org/10.4236/apm.2013.34062 Published Online July 2013 (http://www.scirp.org/journal/apm)
Bilinear Mappings and the Frame Operator
Enrico Au-Yeung
Pacific Institute for the Mathematical Sciences, Vancouver, Canada
Received March 28, 2013; revised April 30, 2013; accepted May 26, 2013
Copyright © 2013 Enrico Au-Yeung. 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.
ABSTRACT
The theory of frames has been actively developed by many authors over the past two decades, both for its applications
to signal processing, and for its deep connections to other areas of mathematics such as operator theory. Central to
the study of frames is the frame operator. We initiate an investigation that extends the frame operator to the bilinear
setting.
Keywords: Harmonic Analysis; Frames
1. Introduction
The theory of frames was initiated by Duffin and Scha-
effer [1] to study some deep problems in non-harmonic
Fourier series. For more than three decades, their ideas
did not seem to generate much interest outside of non-
harmonic Fourier series. Finally in 1986, Daubechies,
Grossman, and Meyer [2] in their groundbreaking paper
observed that frames can be used for painless nonor-
thogonal expansions for functions. Since then, frames
have been used in signal processing, image processing,
and data compression, as well as being studied for their
deep connections to operator theory [3]. Frames are im-
portant in signal processing because they can be used to
provide stable reconstruction of signals. For background
in the theory of frames, see [4-6]. Central to the study of
frames is the frame operator.
We initiate an investigation that extends the frame op-
erator to the bilinear setting. Bilinear operators in har-
monic analysis have been studied by many authors, see,
for example, [7-10]. The conjecture that the bilinear Hil-
bert transform can be extended to a bounded operator has
remained open for some 30 years before it was settled in
the celebrated work of Lacey and Thiele [11]. The results
in our current work extend the results concerning a class
of bilinear operators known as paraproducts; these ope-
rators are better behaved than the usual products of func-
tions, see [12]. The results in this article indicate that
there is a rich underlying theory that awaits to be devel-
oped. The present work only touches on certain aspects
of that theory.
Let H be a separable Hilbert space. A sequence
1
nn

x


of elements in
H
is a frame for
H
if there
exist positive constants
A
and such that B
2
22
1
,,.
n
n
f
HAffx Bf


 

For the rest of this article, the Hilbert space
H
is
taken to be


2d
L

d

d


2d
fL
. Let be the Schwartz
space of rapidly decreasing smooth functions on .
The Fourier transform of a function is de-
fined by




2πi
ˆed
dx
fx x



.
f
2. Main Results
We begin with a useful lemma that will simplify our
calculations later.
Lemma 2.1 (Convolution with a radial function is a
self-adjoint operator )

Let

d
 ,
d
x

be a radial function, i.e.

x
x  
 
22
:dd TL L
.
Define an operator by


 
2,
d.
d
d
fL
Tfxf xyxyfx 

 



T Then is a self-adjoint operator. That means,


2
,,,,.
d
f
gLTfg fTg 


2
,.
d
fg L

2d
Tf L
Proof. Let
We first note that since
221
ff.

 
C
opyright © 2013 SciRes. APM

E. AU-YEUNG 439



 


 

  
   
 
 
 
 
,d
dd
dd,
dd,
dd
dd,
dd
d
,.
dd
dd
dd
dd
dd
dd
dd
d
Tfgfxyxy gxx
fyx y ygxx
fy yxygxx
d
x x
f
yyxygxyxy
fyxgx yxy
fy xgyxxyx
fy xgyxxy
fyg yy
fTg


































xxy
x





Consider a :.BH HH bilinear operator

,,,
f
gHBfg H , and B is linear in each of the two variables separately.
with ac

Let ,  

d compt support, and
  
d0,d1.is radial,..
dd .
x
xxx  

 iexx
 
alize We norm

so that

2
0ˆt


d1.
t
t Write

1.
d
x
x
tt
t




  
Define

,Bfg by:
 
0tt
Bfg xf 


d
,.
tt
g xt
  (1)



0
d
Consider,,d ,
d ntttn
t
Bfg efgex
t
 



where .
n
e H
  
   
    

 
 
0
d
,, d
d ,by
d
d,
d
d
d
d
ntttn
tt tn
tt tn
ttn
t
Bfg efgex
t
fgxexx
fgxexx
fxgex x
 
 
 
 













Lemma 2.1
here the last line follows from w


,,, withtn

.
t
f
ThTf hhe  
We are now re
g  
nstruct the frame operator. Let ady to co

e

1
nn be a sequence in
H
.
Fix
g
H. Define the alysis operator :an 2
M
Hl
by
  

3
,
,, , .
H
MfBfg eBfg eBfge


The frame operator :SH H is given by
S
nt operator 2
12
,,,,, ,
f
.M M


adjoiof M. :. First, calculate the
M
Hl
2
:
2,
nn
l
,, ,
H
H
McfcMfcB fge

Using ,SMM

 we obtain

n


M
lH
.
12
,,,,cc be a sequence in Let

3
cc2 l. Then,


123
,,, ,,, ,,,
Sf Mf
MBfg eBfgeBfg e


By the above calculations on

,Bfg, we see that:
M




,,
n
Bfg e
   





0
dd
,d
,
d
d
tt tn
n
t
f
xgexx
t
fxBeg xx
 







E. AU-YEUNG 440
where
     
0
ntt
Be g xe



d
,.
nt
t
gxt
 
Therefore,

 
,
n
Beg H 


, ,
nn
2
1
1
,,
,,
,,,where
l
H
nn
n
nn
n
Mcf cMf
cBegf
cBeg f









1
So,
n
M
ccBeg
2
, :Ml H

.




123
,,cccc

,
Hence,



1
,, ,
nn
H
n
SfBfgeB eg



 
,:.S HH

i.e.

, ,
nn
HBeg. This is our frame
We have constructed a framwith a bilinear
. Let us summarize all our calculations in the
g lemma.
Lemma 2.2 (Frame operator with bilinear mapping)
1, n
SffB eg



operator.
e operator
mapping
followin
Let
H
be the

2d
L
 compact support, and
Hilbert space

d with

.
Let ,
 

d0,
dxx
 


d1.
dxx




is radial, i.e.
 
x
x  
We normalize

so that

2dt

0t
ˆ1.t
 
 Write

1.
d
t
x
xt
t




Consider a BH HH defined
by Equation (1). H H is given
bilinear operator :
Then the frame operator :S by
 
 
,, ,
,.
nn
n
SfBfgeB eg
Be g


tablished th
To prove that this bilinear operator is bounded, we
need some preparation.
Let
1

1n
Proof. Our calculations ese lemma.
,,
nn
fBeg


Lemma 2.3
H
be a separable Hilbert space (or
a seble Banacace). Let parah sp
M
be a dense sub-
space of
H
. Let :BM HH be a bilinear ope-
rator such that

,,,.
H
H
H
Then the above inality holds for all
fMgHB fgCf g  (2)
equ
f
in
H
, for
all
g
in
H
, and B extends to a bounded bilinear ope-
rator from
H
H

t
H
o .
ach Proof. For e
f
H
in , there exists a sequence

1
nnM
 such that

f
0,
nH
ff as .n Since for each
g
H

,






,,
,
b
y2
mn
Cf f g 
mn
H
m
BfgBf g
f g

nH
HH
Bf

So, for each
 
,Bf uchy
quence in
 
1
,
nn
gH g


 is a Case-
H
.
Hence,


,
n
Bf g converges in
H
to an element in
H
, and we can define a bounded bilinear operator



,:BfgH HH by
 
,lim ,.
nn
BfgBf g


Definition 2.4 (BMO) If
f
is a locally integrable
functio d say that n in , we
f
BMO if there exists a
constant
A

, suce d Q, h that for any cub
1

d.
Q
Q
f
xfxA
Q



Here, Q
f
is the average of
f
over the cube. The
integration is over the cube. Te smallest bound h
A
for
which the above inequality is satisfied is taken to be the
norm of
f
, and is denoted by .
B
MO
sFor background on BMO functions,Chapter 4 of
[13], as well as the seminal paper by C Fefferman and
St. n
f
see
.
ein [14]The next theorem oBMO functions, to-
gether with Lemma 2.3, will allow us to establish the
boundedness of the bilinear operator.
Theorem 2.5 Let


d
  be such that


 d0.
dxx

 Let

 be a bounded, integrable func-
tion that is positive, radial, and decreasing. Write

1.
td
x
xt
t




Let bBMO

. Then
 

22
0
2
2
dd
d.
d
d
tt
BMO
x
t
bxt
Cbf xx
 





fx
Definition 2.6 BMOL then

If

2d
f,


2d
BMO
fL.
Remark 2.7 The space

2d
BMO
L is a dense sub-
space of


2d
L in the topology of

2d
L
Theorem 2.8 (Boundedness of the bilinear perator)
Assume the hypothesis of Lemma 2.2. Define
.
o


,g by: Bf


     
2d
0tt
,,
d
,.
fgL
g xg 



tt t
Bf f x 

 

Then there exists 0C such that for each n eH

,

E. AU-YEUNG 441
with 1,
n
e
 

22
22
,,
,, .
dd
BMO
n
fL gL
Bfg eCfg
 
 

2
BMO

2. d Let
Proof. Let fLgL
n
eL w

.
d
 Let

2d
, ith21.
n
e

   
   
 

0
0
22
0
12
2
0
12
,,
dd
d
d
d
d
n
ttt
12
dd
dd
dd
n
t tn
tt
tn
Bfg e
t
t
f
gxex x
tt
f
gxexx
t
t
fxgx x
t
t
ex x
t
II
 




























Let

2d. t
t Then
0t
Gx gx 


22
1
22
GCg
by Plancherel Theorem. Hence,
 

22 dd
d.
t
2
10
2
2
1
d
d
tt
BMO
I
fx
Cf gx








gxx
t
x

In the last inequality, we used Theorem 2.5. Another
m gives the following: application of Plancherel Theore

2
2
2
0
d
dtn t
Iex
t




2
2
d.
n
xCe
Hence,

22
,, .
nn
BMO
Bfg eCfge 
Therefore B is a bounded operator on

22dd
LL
BMO . Recall that the space



e
2d
BMO
L

2d
La 2.3, B

22dd
. This
3. Acknowledgments
The author is gratefdetto from
teaching him the subject of
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Fourier Seriesmerican Mathema-
tical Society, V341-366.
University of Maryland for
harmonic analysis and the theory of frames.
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