IJMMS 2004:67, 3695–3702
PII. S0161171204312457
http://ijmms.hindawi.com
© Hindawi Publishing Corp.
INFINITE MATRICES, WAVELET COEFFICIENTS AND FRAMES
N. A. SHEIKH and M. MURSALEEN
Received 2 December 2003
We study the action of A on f ∈ L2 (R) and on its wavelet coefficients, where A = (almjk )lmjk
is a double infinite matrix. We find the frame condition for A-transform of f ∈ L2 (R) whose
wavelet series expansion is known.
2000 Mathematics Subject Classification: 42C15, 41A17, 42C40.
1. Introduction. The notation of frame goes back to Duffin and Schaeffer [7] in the
early 1950s to deal with the problems in nonharmonic Fourier series. There has been
renewed interest in the subject related to its role in wavelet theory. For a glance of the
recent development and work on frames and related topics, see [3, 4, 5, 6, 9]. In this
note, we will use the regular double infinite matrices (see [9, 10]) to obtain the frame
conditions and wavelet coefficients.
2. Notations and known results. N is the set of positive integers, Z is the set of
integers, R is the set of real numbers. The space L2 (R) of measurable function f is
defined on the real line R, that satisfies
∞
−∞
f (x)2 dx < ∞.
(2.1)
The inner product of two square integrable functions f , g ∈ L2 (R) is defined as
f , g =
∞
−∞
f (x)g(x)dx,
(2.2)
f 2 = f , f 1/2 .
Every function f ∈ L2 (R) can be written as
f (x) =
Cj,k ψj,k (x).
(2.3)
j,k∈Z
This series representation of f is called wavelet series. Analogous to the notation of
Fourier coefficients, the wavelet coefficients Cj,k are given by
Cj,k =
∞
f (x)ψj,k (x)dx = f , ψj,k ,
−∞
ψj,k = 2j/2 ψ 2j x − k .
(2.4)
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N. A. SHEIKH AND M. MURSALEEN
Now, if we define an integral transform
Wψ f (b, a) = |a|−1/2
∞
−∞
f (x)ψ
x −b
dx,
a
f ∈ L2 (R),
(2.5)
then the wavelet coefficients become
Cj,k = Wψ f
k 1
.
,
2j 2j
(2.6)
A sequence {xn } in a Hilbert space H is a frame if there exist constants c1 and c2 ,
0 < c1 ≤ c2 < ∞, such that
c1 f 2 ≤
f , xn 2 ≤ c2 f 2 ,
(2.7)
n∈Z
for all f ∈ H. The supremum of all such numbers c1 and infimum of all such numbers c2 are called the frame bounds of the frame. The frame is called tight frame when
c1 = c2 and is called normalized tight frame when c1 = c2 = 1. Any orthonormal basis in
a Hilbert space H is a normalized tight frame. The connection between frames and numerically stable reconstruction from discretized wavelet was pointed out by Grossmann
et al. [8]. In 1985, they defined that a wavelet function ψ ∈ L2 (R), constitutes a frame
with frame bounds c1 and c2 , if any f ∈ L2 (R) such that
c1 f 2 ≤
f , ψj,k 2 ≤ c2 f 2 .
(2.8)
j,k∈z
Again, it is said to be tight if c1 = c2 and is said to be exact if it ceases to be frame by
removing any of its elements. There are many examples proposed by Daubechies et al.
[6]. For further details, one can refer to [1, 5, 6]. Chui and Shi [2] proved that {ψj,k }
is a frame for L2 (R) with bounds c1 and c2 , if for some a > 1 and b > 0, the Fourier
transform ψ̂ satisfies
c1 ≤
1 ψ̂ aj w 2 ≤ c2 a.e.,
b j∈Z
(2.9)
for some constants c1 and c2 . By integrating each term in
2
1 ψ̂ aj w c2
c1
≤
≤
|w|
b j∈Z
|w|
|w|
(2.10)
j 2
ψ̂ a w 1 dw ≤ 2c2 log a,
b j∈Z 1≤|w|≤a
|w|
(2.11)
over 1 ≤ |w| ≤ a, we have
2c1 log a ≤
which immediately yields
c1 ≤
1
2b log a
∞ ψ̂ aj w 2
dw ≤ c2 .
|w|
−∞
(2.12)
WAVELET COEFFICIENTS AND FRAMES
3697
The above condition known as compactibility condition was also observed by
Daubechies [4] by using techniques from trace class operators. The above constants
were given by frame bounds, see [2].
Let A = (amnjk ) be a double infinite matrix of real numbers. Then, A-transform of a
double sequence x = (xjk ) is
∞
∞ amnjk xjk ,
(2.13)
j=0 k=0
which is called A-means or A-transform of the sequence x = (xij ). This definition is
due to Móricz and Rhoades [9].
A double matrix A = (amnjk ) is said to be regular (see [10]) if the following conditions
hold:
∞
(i) limm,n→∞ j,k=0 amnjk = 1,
∞
(ii) limm,n→∞ j=0 |amnjk | = 0, (k = 0, 1, 2, . . .),
∞
(iii) limm,n→∞ k=0 |amnjk | = 0, (j = 0, 1, 2, . . .),
∞
(iv) A = supm,n>0 j,k=0 |am,n | < ∞.
Either of conditions (ii) and (iii) implies that
lim amnjk = 0.
m,n→∞
(2.14)
In this note, we establish the frame condition by using A-transform of nonnegative
regular matrix, also we find action of the matrix A on wavelet coefficients.
3. Main results. In this section, we prove the following theorems.
Theorem 3.1. Let A = (ailjk ) be a double nonnegative regular matrix. If
f (x) =
Cj,k ψj,k (x)
(3.1)
j,k∈Z
is a wavelet expansion of f ∈ L2 (R) with wavelet coefficients
∞
f (x)ψj,k (x)dx = f , ψj,k ,
Cj,k =
−∞
(3.2)
then the frame condition for A-transform of f ∈ L2 (R) is
c1 f 2 ≤
Af , ψi,l 2 ≤ c2 f 2 ,
(3.3)
i,l∈Z
where Af is the A-transform of f and 0 < c1 ≤ c2 < ∞.
Theorem 3.2. If Cj,k are the wavelet coefficients of f ∈ L2 (R), that is, Cj,k = f , ψj,k ,
then the dl,m are the wavelet coefficients of Af , where {dl,m } is defined as the Atransform of {Cj,k } by
dl,m =
∞
j,k=−∞
almjk Cjk .
(3.4)
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N. A. SHEIKH AND M. MURSALEEN
Theorem 3.3. Let A = (almjk ) be a double nonnegative matrix whose elements are
ψj,k , ψl,m . Then, {ψj,k } constitutes a frame of L2 (R) if and only if {ψl,m } constitutes
a frame of L2 (R), where Cj,k = f , ψj,k and dl,m = f , ψl,m .
Proof of Theorem 3.1. We can write
f (x) =
f , ψj,k ψj,k .
(3.5)
j,k∈Z
If we take A-transform of f , we get
Af (x) =
Af , ψi,l ψi,l ,
(3.6)
i,l∈Z
and therefore
∞ Af , ψi,l 2 ≤
Af (x)2 ψi,l (x)2 dx
i,l∈Z
i,l∈Z
−∞
≤ A2 f 22
ψi,l 2 .
(3.7)
2
i,l∈Z
Since A is regular matrix and ψi,l 2 = 1, therefore
Af , ψi,l 2 ≤ c2 f 2 ,
2
(3.8)
i,l∈Z
where c2 is positive constant.
Now, for any arbitrarily f ∈ L2 (R), define

f˜ = 
−1/2
Af , ψi,l 2 
f.
(3.9)
i,l∈Z
Clearly,

−1/2
2
˜

Af , ψi,l =
Af , ψi,l ,
Af , ψi,l | 
(3.10)
i,l∈Z
then
Af , ψi,l 2 ≤ 1.
i,l∈Z
(3.11)
WAVELET COEFFICIENTS AND FRAMES
3699
Hence, if there exists α a positive constant, then
Af˜2 ≤ α,
2
−1
2
Af , ψi,l  Af 2 ≤ α.


(3.12)
2
i,l∈z
Since A is regular, we have


−1
Af , ψi,l 2  f 2 ≤ α1 = α
,
2
A2
i,l∈z
(3.13)
where α1 is another positive constant. Therefore,
c1 f 22 ≤
Af , ψi,l 2 ,
(3.14)
i,l∈z
where c1 = α > 0.
Combining (3.8) and (3.14), we have
c1 f 22 ≤
Af , ψi,l 2 ≤ c2 f 2 .
2
(3.15)
i,l∈z
This completes the proof.
Proof of Theorem 3.2. We can write
∞
Af (x)ψl,m (x)dx
Af , ψj,k =
−∞
=
∞
(3.16)
∞
−∞
almjk cj,k ψj,k (x)ψl,m (x)dx.
j,k=−∞
Now,
∞
∞
Af , ψl,m ψl,m =
l,m=−∞
l,m=−∞
∞
=
∞
−∞
∞
almjk cj,k ψj,k (x)ψl,m (x)ψl,m (x)dx
j,k−∞
∞
dl,m ψl,m
l,m=−∞
∞
=
−∞
ψl,m (x)2
2
(3.17)
dl,m ψl,m .
l,m=−∞
Therefore,
∞
l,m=−∞
dl,m ψl,m =
∞
Af , ψl,m ψl,m .
l,m=−∞
This implies that dl,m are wavelet coefficients of Af .
(3.18)
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N. A. SHEIKH AND M. MURSALEEN
Thus,
dl,m = f , ψl,m .
(3.19)
This completes the proof.
Proof of Theorem 3.3. We observe that
almjk Cj,k = ψj,k , ψl,m f , ψj,k
∞
∞
ψj,k (x)ψl,m (x)dx
f (x)ψj,k (x)dx
=
−∞
−∞
∞
∞
=
f (x)ψl,m (x)dx
ψj,k (x)ψj,k (x)dx
−∞
−∞
∞
f (x)ψl,m (x)dx
=
−∞
= f , ψl,m ,
(3.20)
that is, almjk Cj,k = dl,m .
Now,
dl,m 2 =
almjk Cj,k 2 =
f , ψl,m 2
l,m
l,m
l,m
1 fˆ, ψ̂l,m 2 ,
=
2
(2π ) l,m
2
2π ∞
1 ˆ(w + 2π p)ψ̂(w + 2π p)eilmw dw f
(2π )2 l,m 0 p=−∞
=p
1
=
2π
(3.21)
2
2π ∞
ˆ(w + 2π p)ψ̂(w + 2π p)dw ,
f
0 p=−∞
by Parseval’s formula for trigonometric Fourier series.
Now
2 

∞
∞

fˆ(w + 2π p)ψ̂(w + 2π p)
fˆ(w + 2π p)ψ̂(w + 2Πp)
=
p=−∞
p=−∞


∞
fˆ(w + 2π q)ψ̂(w + 2π q).
×
q=−∞
Let f (w) =
∞
ˆ
q=−∞ f (w + 2π q)ψ̂(w + 2π q).
(3.22)
WAVELET COEFFICIENTS AND FRAMES
3701
Therefore,
2
2π ∞
ˆ(w + 2π p)ψ̂(w + 2π p)dw f
0 p=−∞


∞
1  2π ˆ
=
f (w + 2π p)ψ̂(w + 2π p)dw F (w)dw 
2π
0 p=−∞
∞
1
=
fˆ(w)ψ̂(w)F (w)dw
2π
−∞


∞ ∞


1
ˆ
ˆ
=
f (w)ψ̂(w)f (w + 2π q)ψ̂(w + 2π q)dw

2π  q=−∞ −∞
∞
1
fˆ(w)ψ̂(w)fˆ(w)ψ̂(w)dw
=
2π −∞
1 ∞ fˆ(w)2 ψ̂(w)2 dw
=
2π −∞
1 ∞ fˆ(w)2 dw
=
2π −∞
1
p=
2π
(3.23)
= f 22 ,
that is,
dlm 2 = f 2 ,
2
f ∈ L2 (R).
(3.24)
l,m
Therefore, for a regular matrix A = (almjk ), we have
c1 f 22 ≤
dlm 2 ≤ c2 f 2
2
(3.25)
2
cjk ≤ c f 2 ,
2
2
(3.26)
l,m
if and only if
c1 f 22 ≤
j,k
where, 0 ≤ c1 , c2 < ∞. This completes the proof.
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[2]
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[4]
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N. A. Sheikh: Department of Mathematics, National Institute of Technology, Srinagar, Kashmir
190006, Jammu and Kashmir, India
E-mail address: neyaznit@yahoo.co.in
M. Mursaleen: Department of Mathematics, Aligarh Muslim University, Aligarh 202002, Uttar
Pradesh, India
Current address: Department of Mathematics, Faculty of Science, P.O. Box 80203, King Abdul
Aziz University, Jeddah, Kingdom of Saudi Arabia
E-mail address: mursaleen@postmark.net