International Journal of Analysis and Applications
ISSN 2291-8639
Volume 7, Number 1 (2015), 1-15
http://www.etamaths.com

p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE

WALSH FUNCTIONS

F. A. SHAH

Abstract. A generalization of the notion of p-multiresoltion analysis on a

half-line, based on the theory of shift-invariant spaces is considered. In contrast

to the standard setting, the associated subspace V0 of L
2(R+) has a frame, a

collection of translates of the scaling function ϕ of the form {ϕ(·	 k)}k∈Z+ ,
where Z+ is the set of non-negative integers. We investigate certain properties
of multiresolution subspaces which provides the quantitative criteria for the
construction of p-frame multiresoltion analysis (p-FMRA) on positive half-line

R+. Finally, we establish a complete characterization of all p-wavelet frames
associated with p-FMRA on positive half-line R+ using the shift-invariant
space theory.

1. INTRODUCTION

In the early nineties a general scheme for the construction of wavelets was defined.
This scheme is based on the notion of multiresolution analysis (MRA) introduced
by Mallat [9]. In recent years, the concept of MRA has become an important tool in
mathematics and applications. It provides a natural framework for understanding
of wavelet bases, bases that consist of the scaled and integer translated versions of
a finite number of functions. Mathematically, an MRA is an increasing sequence of
closed subspaces {Vj}j∈Z of L

2(R) such that
⋃
j∈ZVj is dense in L

2(R),
⋂
j∈ZVj =

{0} and which satisfies f(x) ∈ Vj if and only if f(2x) ∈ Vj+1. Furthermore, there
should exist an element ϕ ∈ V0 such that the collection of integer translates of
ϕ,{ϕ(·−k) : k ∈ Z} is a complete orthonormal system for V0. The dilation factor
2 can be replaced by any integer M ≥ 2 and in that case one needs M −1 wavelets
to generate the whole space L2(R). A similar generalization of multiresolution
analysis can be made in higher dimensions by considering matrix dilations (see [2]).

On the other hand, there is considerable interest both in mathematics and
its applications in the study of compactly supported orthonormal scaling functions
and wavelets with an arbitrary dilation factor p ∈ N,p ≥ 2. The motivation comes
partly from signal processing and numerical applications, where such wavelets are
useful in image compression and feature extraction because of their small support
and multifractal structure. Farkov [4] has given the general construction of all
compactly supported orthogonal p-wavelets in L2(R+) and proved necessary and

2010 Mathematics Subject Classification. 42C15, 42C40, 42A38, 41A17.
Key words and phrases. Frame; wavelet frame; p-multiresoltion analysis; scaling function; shift

invariant; Walsh-Fourier transform.

c©2015 Authors retain the copyrights of their papers, and all
open access articles are distributed under the terms of the Creative Commons Attribution License.

1



2 SHAH

sufficient conditions for scaling filters with pn many terms (for any integers p,n ≥ 2)
to generate an p-MRA in L2(R+). These studies were continued by Farkov and
his colleagues [5, 6] where they have given some new algorithms for constructing
the corresponding biorthogonal and non-stationary wavelets related to the Walsh
polynomials on the positive half-line R+. On the other hand, Shah and Debnath
[19] have constructed dyadic wavelet frames on the positive half-line R+ using the
Walsh-Fourier transform and have established a necessary condition and a sufficient
condition for the system

{
ψj,k(x) = 2

j/2ψ(2jx	k) : j ∈ Z,k ∈ Z+
}

to be a frame

for L2(R+). Further, wavelet packets related to the Walsh functions are discussed
in a series of papers by the author [14-17]. In his recent, Shah [18] has given
the construction of tight wavelet frames generated by the Walsh polynomials on
a half-line R+ by following the procedure of Daubechies et al. [3] via extension
principles. He also provide a sufficient condition for finite number of functions to
form a tight wavelet frame and established a general principle for constructing tight
wavelet frames on R+. Recently, Meenaski et al. [10] have introduced the notion
of non-uniform multiresolution analysis (NUMRA) on a half-line R+ and have also
established the necessary and sufficient condition for the existence of corresponding
wavelets on R+.

Since the use of multiresolution analysis has proven to be a very efficient tool
in wavelet theory mainly because of its simplicity, it is of interest to try to generalize
this notion as much as possible while preserving its connection with wavelet analysis.
In this connection, Benedetto and Li considered the dyadic semi-orthogonal frame
multiresolution analysis of L2(R) with a single scaling function and successfully
applied the theory in the analysis of narrow band signals [1]. The characterization
of the dyadic semi-orthogonal frame multiresolution analysis with a single scaling
function admitting a single frame wavelet whose dyadic dilations of the integer
translates form a frame for L2(R) was obtained independently by Benedetto and
Treiber by a direct method [2], and by Kim and Lim by using the theory of shift-
invariant spaces [8]. Later on, Xiaojiang [21] extended the results of Benedetto
and Li’s theory of FMRA to higher dimensions with arbitrary integral expansive
matrix dilations, and has established the necessary and sufficient conditions to
characterize semi-orthogonal multiresolution analysis frames for L2(Rn). On the
other hand, Zhang [22] has given the characterization of generalized frame MRA on
R and has provided a general algorithm for the construction of non-MRA wavelets
by means of the Fourier transforms.

In this paper, we introduce the notion of p-frame multiresolution analysis (p-
FMRA) on positive half-real line R+ by extending the above described methods.
We first investigate the properties of multiresolution subspaces, which will provide
the quantitative criteria for the construction of p-FMRAs. We also show that
the scaling property of an p-FMRA also holds for the wavelet subspaces and that
the space L2(R+) can be decomposed into the orthogonal sum of these wavelet
subspaces. Finally, we study the characterization of p-wavelet frames associated
with p-FMRA on positive half-line R+ using the shift-invariant space theory.

The paper is structured as follows. In Section 2, we introduce some notations
and preliminaries related to the operations on positive half-line R+ including the



p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE WALSH FUNCTIONS 3

definition of the Walsh-Fourier transform. The notion of p-FMRA of L2(R+) is
introduced in Section 3 and its quantitative criteria is given by means of the Theo-
rem 3.10. In Section 4, we establish a complete characterization of p-wavelet frames
generated by a finite number of mother wavelets on R+.

2. WALSH-FOURIER ANALYSIS

We start this section with certain results on Walsh-Fourier analysis. We present
a brief review of generalized Walsh functions, Walsh-Fourier transforms and its
various properties.

As usual, let R+ = [0, +∞), Z+ = {0, 1, 2, . . .} and N = Z+ −{0}. Denote
by [x] the integer part of x. Let p be a fixed natural number greater than 1. For
x ∈ R+ and any positive integer j, we set

xj = [p
jx](mod p), x−j = [p

1−jx](mod p), (2.1)

where xj,x−j ∈ {0, 1, . . . ,p− 1}. It is clear that for each x ∈ R+, there exist
k = k(x) in N such that x−j = 0 ∀j > k.

Consider on R+ the addition defined as follows:

x⊕y =
∑
j<0

ζjp
−j−1 +

∑
j>0

ζjp
−j,

with ζj = xj + yj(mod p), j ∈ Z \{0} , where ζj ∈ {0, 1, . . . ,p− 1} and xj, yj are
calculated by (2.1). As usual, we write z = x 	 y if z ⊕ y = x, where 	 denotes
subtraction modulo p in R

+

.

For x ∈ [0, 1), let r0(x) is given by

r0(x) =




1, if x ∈ [0, 1/p)

ε`p, if x ∈
[
`p−1, (` + 1)p−1

)
, ` = 1, 2, . . . ,p− 1,

where εp = exp(2πi/p). The extension of the function r0 to R+ is given by
the equality r0(x + 1) = r0(x), x ∈ R+. Then, the generalized Walsh functions
{wm(x) : m ∈ Z+} are defined by

w0(x) ≡ 1 and wm(x) =
k∏
j=0

(
r0(p

jx)
)µj

where m =
∑k
j=0 µjp

j, µj ∈{0, 1, . . . ,p− 1} , µk 6= 0. They have many properties
similar to those of the Haar functions and trigonometric series, and form a complete
orthogonal system. Further, by a Walsh polynomial we shall mean a finite linear
combination of Walsh functions.

For x,y ∈ R
+

, let

χ(x,y) = exp


2πi

p

∞∑
j=1

(xjy−j + x−jyj)


 , (2.2)



4 SHAH

where xj,yj are given by (2.1).

We observe that

χ

(
x,
m

pn

)
= χ

(
x

pn
,m

)
= wm

(
x

pn

)
, ∀ x ∈ [0,pn), m,n ∈ Z+,

and

χ(x⊕y,z) = χ(x,z) χ(y,z), χ(x	y,z) = χ(x,z) χ(y,z),

where x,y,z ∈ R+ and x ⊕ y is p-adic irrational. It is well known that systems
{χ(α,.)}∞α=0 and {χ(·,α)}

∞
α=0 are orthonormal bases in L

2[0,1] (See Golubov et
al.[7]).

The Walsh-Fourier transform of a function f ∈ L1(R+) ∩ L2(R+) is defined
by

f̂(ξ) =

∫
R+
f(x) χ(x,ξ) dx, (2.3)

where χ(x,ξ) is given by (2.2). The Walsh-Fourier operator F : L1(R+)∩L2(R+) →
L2(R+), Ff = f̂, extends uniquely to the whole space L2(R+). The properties
of the Walsh-Fourier transform are quite similar to those of the classic Fourier

transform (see [7, 13]). In particular, if f ∈ L2(R+), then f̂ ∈ L2(R+) and∥∥∥f̂∥∥∥
L

2
(R+)

=
∥∥f∥∥

L2(R+).

Definition 2.1 Let H be a separable Hilbert space. A sequence {fk}k∈Z in H is
called a frame for H if there exist constants A and B with 0 < A ≤ B < ∞ such
that

A‖f‖2 ≤
∑
k∈Z

∣∣〈f,fk〉∣∣2 ≤ B‖f‖2 (2.4)
for all f ∈ H. The largest constant A and the smallest constant B satisfying (2.4)
are called the upper and the lower frame bound, respectively. A frame is said to be
tight if it is possible to choose A = B and a frame is said to be exact if it ceases to
be a frame when any one of its elements is removed. An exact frame is also known
as a Riesz basis.

The following theorem gives us an elementary characterization of frames.

Theorem 2.2. A sequence {fk}k∈Z in a Hilbert space H is a frame for H if and
only if there exists a sequence a = {ak} ∈ l2(Z) with ‖a‖l2(Z) ≤ C‖f‖,C > 0 such
that

f(x) =
∑
k∈Z

akfk(x)

and
∑
k∈Z

∣∣〈f,fk〉∣∣2 < ∞, for every f ∈ H.
For any integer p ≥ 2, let Dp : L2(R+) → L2(R+) be the unitary operator

defined via Dpf(x) = p
1/2f(px). For k ∈ Z+, let τk : L2(R+) → L2(R+) denotes

the unitary translation operator such that τkf(x) = f(x	k).



p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE WALSH FUNCTIONS 5

Our study uses the theory of shift-invariant spaces developed in [11, 12] and
the references therein. A closed subspace S of L2(R+) is said to be shift-invariant if
τkf ∈ S whenever f ∈ S and k ∈ Z+. A closed shift-invariant subspace S of L2(R+)
is said to be generated by Φ ⊂ L2(R+) if S = span{τkϕ(.) := ϕ(.	k) : k ∈ Z+,ϕ ∈ Φ}.
The cardinality of a smallest generating set Φ for S is called the length of S which
is denoted by |S|. If |S| = finite, then S is called a finite shift-invariant space (FSI)
and if |S| = 1, then S is called a principal shift-invariant space (PSI). Moreover, the
spectrum of a shift-invariant space is defined to be σ(S) =

{
ξ ∈ [0, 1] : Ŝ(ξ) 6= {0}

}
,

where Ŝ(ξ) =
{
f̂(ξ ⊕k) ∈ l2(Z+) : f ∈ S,k ∈ Z+

}
.

3. p-FRAME MULTIRESOLUTION ANALYSIS ON A POSITIVE HALF-LINE

We first introduce the notion of a p-frame multiresolution analysis (p-FMRA) of
L2(R+).

Definition 3.1. A p-frame multiresolution analysis of L2(R+) is a sequence of
closed subspaces {Vj}j∈Z such that

(i) Vj ⊂ Vj+1 for all j ∈ Z;

(ii)
⋃
j∈ZVj is dense in L

2(R+) and
⋂
j∈ZVj = {0};

(iii) f(·) ∈ Vj if and only if f(p·) ∈ Vj+1 for all j ∈ Z;

(iv) The function f lies in V0 implies that the collection f(·	k) lies in V0, for all
k ∈ Z+.

(v) The sequence {τkϕ := ϕ(·	k) : k ∈ Z+} is a frame for the subspace V0.

The function ϕ is known as the scaling function while the subspaces Vj’s are
known as approximation spaces or multiresolution subspaces. An p-FMRA is said
to be non-exact and respectively exact if the frame for the subspace V0 is non-exact
and respectively exact. In p-MRA’s studied in [4], the frame condition is replaced
by that of an orthonormal basis or an exact frame.

Next, we establish several properties of multiresolution subspaces that will
help in the construction of p-FMRA’s. The following proposition shows that for
every j ∈ Z, the sequence {ϕj,k}k∈Z+ , where

ϕj,k(x) = p
j/2ϕ(pjx	k), (3.1)

is a frame for Vj.

Proposition 3.2. Let {τkϕ} be a frame for V0 = span{τkϕ : k ∈ Z+} and

Vj =
{
f ∈ L2(R+) : f(p−j.) ∈ V0

}
, j ∈ Z. (3.2)

Then, the sequence {ϕj,k : k ∈ Z+} defined in (3.1) is a frame for Vj with the same
bounds as those for V0.



6 SHAH

Proof. For any f ∈ Vj, we have∑
k∈Z+

∣∣〈D−jf,τkϕ〉∣∣2 = ∑
k∈Z+

∣∣∣∣
∫
R+
p−jf(p−jx) pj/2ϕ(x	k) dx

∣∣∣∣2

=
∑
k∈Z+

∣∣∣∣
∫
R+
f(x) pj/2ϕ(pjx	k) dx

∣∣∣∣2

=
∑
k∈Z+

|〈f,ϕj,k〉|2.

Since {τkϕ}k∈Z+ be a frame for V0, therefore, we have

A‖f‖22 = A‖D
−jf‖22 ≤

∑
k∈Z+

|〈f,ϕj,k〉|2 ≤ B‖D−jf‖22 = B‖f‖
2
2.

This completes the proof of the Proposition.

Next, we characterize all functions of FSI space in terms of its Walsh-Fourier trans-
form.

Proposition 3.3. Let {τkϕ : k ∈ Z+,ϕ ∈ Ω} be a frame for its closed linear span
V , where Ω = {ϕ1,ϕ2, ...,ϕL} ⊂ L2(R+). Then, f ∈ L2(R+) lies in V if and only
if there exist periodic functions h` ∈ L2[0, 1],` = 1, ...,L such that

f̂(ξ) =

L∑
`=1

h`(ξ)ϕ̂`(ξ). (3.3)

Proof. Since the system {τkϕ : k ∈ Z+,ϕ ∈ Ω} is a frame for V , then by Theorem
2.2, there exist a sequence

{
a`k
}
∈ l2(Z+), for ` = 1, ...,L such that

f(x) =

L∑
`=1

∑
k∈Z+

a`kϕ`(x	k). (3.4)

Taking Walsh-Fourier transform on both sides of (3.4), we obtain

f̂(ξ) =

L∑
`=1

h`(ξ)ϕ̂`(ξ),

where h`(ξ) =
∑
k∈Z+ a

`
kχk(ξ) are the periodic functions in L

2[0, 1]. The converse is
established by taking h` as above and applying the inverse Walsh-Fourier transform
on both sides of (3.3).

We now study some properties of the multiresolution subspaces Vj of the form
(3.2) by means of the Walsh-Fourier transform.

Proposition 3.4. Let {τkϕ} be a frame for V0 = span{τkϕ : k ∈ Z+} and for
j ∈ Z, define Vj by (3.2). Then for any function ψ ∈ V1, there exist periodic



p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE WALSH FUNCTIONS 7

functions G ∈ L2[0, 1] such that

ψ̂(pξ) = p−1/2G(ξ)ϕ̂(ξ). (3.5)

Proof. By the definition of Vj, it follows that ψ(p
−1x) ∈ V0. By Proposition 3.3,

there exists a periodic function G ∈ L2[0, 1] such that
(
ψ(p−1x)

)∧
= ψ̂(pξ) =

p−1/2G(ξ)ϕ̂(ξ) lies in L2(R+).

The following theorem establishes a sufficient condition to ensure that the
nesting property holds for the subspaces Vj’s.

Theorem 3.5. Let {τkϕ} be a frame for V0 = span{τkϕ : k ∈ Z+} and for j ∈ Z,
define Vj by (3.2). Assume that there exists a periodic function H ∈ L∞[0, 1] such
that

ϕ̂(ξ) = p−1/2H(p−1ξ)ϕ̂(p−1ξ). (3.6)

Then, Vj ⊆ Vj+1, for every j ∈ Z.

Proof. Given any f ∈ Vj, there exist as sequence {ak}k∈Z+ ∈ l
2(Z+) such that

f(x) =
∑
k∈Z+

pj/2akϕ(p
jx	k). (3.7)

Let m0(ξ) =
∑
k∈Z+ akχk(ξ) ∈ L

2[0, 1], m1(p
−1ξ) = m0(ξ)H(p

−1ξ). Then, clearly

m1 lies in L
2[0, 1] as H lies in L∞[0, 1]. Therefore, by Parsevals identity, there exist

a sequence {bk}k∈Z+ ∈ l
2(Z+) such that m1(ξ) =

∑
k∈Z+ bkχk(ξ) lies in L

2(R+).

Applying the Walsh-Fourier transform to (3.7), we obtain

f̂(ξ) = p
−j
2 m0(p

−jξ) ϕ̂(p−jξ)

= p
−j−1

2 m0(p
−jξ)H(p−j−1ξ) ϕ̂(p−j−1ξ)

= p
−j−1

2 m1(p
−j−1ξ) ϕ̂(p−j−1ξ).(3.8)

Implementing inverse Walsh-Fourier transform to (3.8), we obtain

f(x) = p
j+1
2

∑
k∈Z+

bk ϕ(p
j+1x	k). (3.9)

Thus the function f lies in Vj+1 by Proposition 3.2. Moreover, it is easy to verify
that the function H in (3.6) is not unique.

The following theorem is a converse to Theorem 3.5.

Theorem 3.6. Let {τkϕ} be a frame for V0 = span{τkϕ : k ∈ Z+} and for j ∈ Z,
define Vj by (3.2). Assume that V0 ⊆ V1 and Φ(ξ) = ‖ϕ̂(ξ ⊕k)‖

2
l2(Z+) . Then there

exists periodic function H ∈ L∞[0, 1] such that (3.6) holds.



8 SHAH

Proof. Since {τkϕ}k∈Z+ is a frame for V0, therefore, there exist positive constants
A and B such that

A ≤ Φ(ξ) ≤ B a.e on σ(V0).

Since V0 ⊆ V1, we have ϕ ∈ V1. By Proposition 3.4, there exists a periodic function
H0 ∈ L2[0, 1] such that

ϕ̂(pξ) = p−1/2H0(ξ)ϕ̂(ξ).

Therefore, we have

|ϕ̂(ξ)|2 = p−1
∣∣H0(p−1ξ)∣∣2 ∣∣ϕ̂(p−1ξ)∣∣2 a.e. (3.10)

Let S = [0, 1)\σ(V0) and H ∈ L2[0, 1] be a periodic function such that H = H0, a.e
on σ(V0) and H is bounded on S by a positive constant C. Then, it follows from
the above fact that H is not unique so that (3.10) also holds for H, i.e.,

|ϕ̂(ξ)|2 = p−1|H(p−1ξ)|2|ϕ̂(p−1ξ)|2 a.e.

Taking n = kp + r, where k ∈ Z+ and r = 0, 1, . . . ,p− 1, we have

|ϕ̂(ξ ⊕n)|2 = p−1|H(p−1ξ ⊕p−1r)|2|ϕ̂(p−1ξ ⊕rp−1 ⊕k|2 a.e. (3.11)

Summing up (3.11) for all k ∈ Z+ and r = 0, 1, ...,p− 1, we have

∑
n∈Z+

|ϕ̂(ξ ⊕n)|2 = p−1
p−1∑
r=0

|H(p−1ξ ⊕p−1r)|2
∑
k∈Z+

|ϕ̂(p−1ξ ⊕rp−1 ⊕k|2 a.e,

which is equivalent to

Φ(ξ) = p−1
p−1∑
r=0

|H(p−1ξ ⊕p−1r)|2 Φ(p−1ξ ⊕p−1r) a.e,

or

Φ(pξ) = p−1
p−1∑
r=0

|H(ξ ⊕p−1r)|2 Φ(ξ ⊕p−1r) a.e. (3.12)

Note that Φ(pξ) ≤ B a.e and hence, (3.12) becomes

p−1∑
r=0

|H(ξ ⊕p−1r)|2 Φ(ξ ⊕p−1r) ≤ pB a.e.

This implies that for almost every ξ ∈ [0, 1
p
) and r = 0, 1, ...,p− 1, we have

|H(ξ ⊕p−1r)|2 Φ(ξ ⊕p−1r) ≤ pB.



p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE WALSH FUNCTIONS 9

Further, if Φ(ξ ⊕p−1r) = 0, then |H(ξ ⊕p−1r)| ≤ C and if Φ(ξ ⊕p−1r) > 0, then
we may assume that A ≤ Φ(ξ ⊕p−1r) ≤ B. Thus, for almost every ξ ∈ [0, 1

p
) and

r = 0, 1, . . . ,p− 1, we have

|H(ξ ⊕p−1r)|2 ≤ max
{
C2,pBA−1

}
.

Hence H is essentially bounded on the interval [0, 1). This proves the theorem
completely.

The following two propositions are proved in [4]:

Proposition 3.7. Suppose V0 = span{τkϕ : k ∈ Z+} and for each j ∈ Z, define
Vj by (3.2) such that V0 ⊆ V1. Assume that |ϕ̂| > 0, a.e on a neighborhood of zero.
Then, the union

⋃
j∈ZVj is dense in L

2(R+).

Proposition 3.8. Let ϕ ∈ L2(R+) and define V0 = span{τkϕ : k ∈ Z+}. For each
j ∈ Z, define Vj by (3.2). Then, we have

⋂
j∈ZVj = {0}.

Lemma 3.9. Let Vj be the family of subspaces defined by (3.2) with Vj ⊆ Vj+1, for
each j ∈ Z. Suppose ϕ ∈ L2(R+) be a non-zero function with V0 = span{τkϕ : k ∈ Z+}.
Then, for every j ∈ Z, Vj is a proper subspace of Vj+1.

Proof. Suppose that V` = V`+1 for some ` ∈ Z. Let f ∈ Vj+1, then for any given
j ∈ Z, we have f(p−j−1+`+1x) ∈ Vj+1. Since f(p−j+`x) ∈ V`, therefore f lies in
Vj and Vj = Vj+1. Hence,

⋂
j∈ZVj = V0. Therefore, it follows from Proposition 3.8

that Vj = {0}, which is a contradiction.

Combining all our results so far, we have the following theorem.

Theorem 3.10. Let ϕ ∈ L2(R+) and define V0 = span{τkϕ : k ∈ Z+}. For each
j ∈ Z, define Vj by (3.2) and Φ(ξ) = ‖ϕ̂(ξ ⊕k)‖

2
l2(Z+) . Suppose that the following

hold:

(i) A ≤ Φ(ξ) ≤ B a.e on σ(V0)

(ii) There exists a periodic function H ∈ L∞[0, 1] such that

ϕ̂(ξ) = p−1/2H(p−1ξ)ϕ̂(p−1ξ), a.e.

(iii) |ϕ̂| > 0, a.e on a neighborhood of zero.

Then, {Vj}j∈Z defines a p-frame multiresolution analysis of L
2(R+).

Proof. Since V0 is a shift-invariant subspace of L
2(R+). Therefore, the system {τkϕ}

forms a frame for V0 with frame bounds A and B. Then, it follows from Theorem
3.5 and Lemma 3.9 that Vj ⊂ Vj+1, for every j ∈ Z. Now, by the definition of
Vj,f lies in Vj if and only if f(p

−j.) lies in V0, while f(p.) lies in Vj+1 if and only if
f(p−j−1(p.)) lies in V0. Thus, f lies in Vj if and only if f(p.) lies in Vj+1. Further, by



10 SHAH

our assumption (iii) and Proposition 3.8, it follows that
⋃
j∈ZVj is dense in L

2(R+)
and

⋂
j∈ZVj = {0}. Therefore, the sequence {Vj}j∈Z satisfies all the conditions to

be an p-FMRA of L2(R+).

In order to construct p-wavelet frames associated with p-FMRA on a positive
half-line R+, we introduce the orthogonal complement subspaces {Wj}j∈Z of Vj in
Vj+1. Further, it is easy verify that the sequence of subspaces {Wj}j∈Z also satisfies
the scaling property, i.e.,

Wj =
{
f ∈ L2(R+) : f(p−j·) ∈ W0

}
, j ∈ Z. (3.13)

Theorem 3.11. Let {Vj}j∈Z be an increasing sequence of closed subspaces of
L2(R+) such that

⋃
j∈ZVj is dense in L

2(R+) and
⋂
j∈ZVj = {0}. Let Wj be the

orthogonal complement of Vj in Vj+1, for each j ∈ Z. Then, the subspaces Wj are
pairwise orthogonal and

L2(R+) =
⊕
j∈Z

Wj.

Proof. Assume that i < j, then 〈fi,fj〉 = 0, for any fi ∈ Wj as Wi ⊂ Vi+1 ⊂ Vj. Let
Pj be the orthogonal projection operators from L

2(R+) onto Vj, then limj→∞Pjf =
f, limj→−∞Pjf = 0 and Wj = {f −Pjf : f ∈ Vj+1}. Therefore, for any f ∈
L2(R+), we have

f =
∑
j∈Z

(Pj+1f −Pjf).

Thus, the result of the direct sum follows since Pj+1−Pj is the orthogonal projector
from L2(R+) onto Wj.

4. CHARACTERIZATION OF p-WAVELET FRAMES

In this section, we give the characterization all p-wavelet frames associated with
p-FMRA on a half-line R+. First, we shall characterize the existence of a function
ψ in W0, where W0 is the orthogonal complement of V0 in V1, by virtue of the
analysis filters G and H.

Theorem 4.1. Let H be a periodic function associated with an p-FMRA {Vj : j ∈ Z}
such that (3.6) holds. Define W0 as the orthogonal complement of V0 in V1. Let
ψ ∈ V1 such that

ϕ̂(ξ) = p−1/2G(ξ/p)ϕ̂(ξ/p). (4.1)

where G is a periodic function in L2[0, 1]. Then ψ lies in W0 if and only if

p−1∑
r=0

H
(
p−1ξ ⊕p−1r

)
Φ
(
p−1ξ ⊕p−1r

)
G (p−1ξ ⊕p−1r) = 0 a.e. (4.2)

Proof. We note that ψ lies in W0 if and only if



p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE WALSH FUNCTIONS 11

〈ψ,τkψ〉 = 〈ψ,ψ(.	k)〉 = 0, for all k ∈ Z+. (4.3)
Define

F(ξ) =
∑
k∈Z+

ϕ̂(ξ ⊕k) ψ̂(ξ ⊕k).

Then, it is easy to verify that F lies in L1[0, 1] by using Monotonic Convergence
Theorem and the Plancherel Theorem as

∫ 1
0

|F(ξ)|dξ ≤
∫ 1

0

∑
k∈Z+

∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ
=

∑
k∈Z+

∫ 1
0

∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ
=

∫
R+

∣∣∣ϕ̂(ξ)ψ̂(ξ)∣∣∣dξ
≤

∥∥ϕ̂∥∥
2

∥∥∥ψ̂∥∥∥
2

=
∥∥ϕ∥∥

2

∥∥ψ∥∥
2
.

Now, for a fixed n ∈ Z+, let FM be the function defined by

FM (ξ) =

M∑
k=0

ϕ̂(ξ ⊕k) ψ̂(ξ ⊕k) χn(ξ).

Then, in view of (3.6) and (4.1), we have

FM (ξ) =

p−1∑
r=0

∑
|pk+r|≤M

H
(
p−1ξ ⊕p−1r

)
|ϕ̂
(
p−1ξ ⊕p−1r ⊕k

)
|2 G (p−1ξ ⊕p−1r) χn(ξ).

(3.4)
Implementation of Monotonic Convergence Theorem and the Cauchy-Schwartz in-
equality yields

∥∥FM −Fχn∥∥L2[0,1] ≤
∫ 1

0

∑
|k|≥M+1

∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ
=

∑
|k|≥M+1

∫ 1
0

∣∣∣ϕ̂(ξ ⊕k)ψ̂(ξ ⊕k)∣∣∣dξ
=

∑
|k|≥M+1

∫ k+1
k

∣∣∣ϕ̂(ξ)ψ̂(ξ)∣∣∣dξ
≤
∫
|ξ|>M

∣∣∣ϕ̂(ξ)ψ̂(ξ)∣∣∣dξ



12 SHAH

≤

{∫
|ξ|>M

|ϕ̂(ξ)|2 dξ

}1/2 {∫
|ξ|>M

∣∣∣ψ̂(ξ)∣∣∣ |2dξ
}1/2

→ 0 as M →∞.

Thus,

lim
M→∞

∥∥FM −Fχn∥∥L2[0,1] = 0. (4.5)
Therefore, there exists a subsequence

{
FMj

}
such that

lim
j→∞

∥∥FMj −Fχn∥∥L2[0,1] = 0, a.e.
Hence

F(ξ) =

p−1∑
r=0

p−1H
(
p−1ξ ⊕p−1r

)
Φ
(
p−1ξ ⊕p−1r

)
G (p−1ξ ⊕p−1r) a.e.

Using (4.5) and the Dominated Convergence Theorem, we have for all n ∈ Z+,

〈ψ,τ−nϕ〉 =
∫
R+
ψ̂(ξ)ϕ̂(ξ)χn(ξ)dξ

=
∑
k∈Z+

∫ k+1
k

ψ̂(ξ)ϕ̂(ξ)χn(ξ)dξ

= lim
M→∞

M∑
k=0

∫ 1
0

ψ̂(ξ ⊕k)ϕ̂(ξ ⊕k)χn(ξ)χk(ξ)dξ

= lim
M→∞

∫ 1
0

FM (ξ)dξ

=

∫ 1
0

F(ξ)χn(ξ)dξ.

Consequently, F = 0 a.e, is the necessary and sufficient condition for (4.3) to hold
for all n ∈ Z+.

Lemma 4.2. Let {Wj : j ∈ Z} be a sequence of pairwise orthogonal closed subspaces
of L2(R+) such that L2(R+) =

⊕
j∈Z Wj. Then, for every f ∈ L

2(R+), there exist
fj ∈ Wj,j ∈ Z such that f(x) =

∑
j∈Z fj(x). Furthermore,∥∥f∥∥2

2
=
∑
j∈Z

∥∥fj∥∥22.
Proof. For any arbitrary function f ∈ L2(R+), we have

lim
n→∞

∥∥∥f − n∑
j=−n

fj

∥∥∥
2

= 0,



p-FRAME MULTIRESOLUTION ANALYSIS RELATED TO THE WALSH FUNCTIONS 13

where fj ∈ Wj, for each j ∈ Z. Moreover, for a fixed n ∈ N, we have∥∥∥ n∑
j=−n

fj

∥∥∥2
2

=

n∑
j=−n

‖fj‖22.

Since the norm ‖.‖2 is continuous, hence the desired result is obtained by taking
n →∞ on both sides of the above equality.

Theorem 4.3. Let ϕ be the scaling function for an p-FMRA {Vj : j ∈ Z} and sup-
pose that Wj be the orthogonal complement of Vj in Vj+1. Let Ψ = {ψ1,ψ2, . . . ,ψL}⊂
W0. Then, the collection

FΨ =
{
ψ`j,k(x) := p

j/2ψ`(pjx	k),j ∈ Z,k ∈ Z+,` = 1, . . . ,L
}

(4.6)

forms a p-wavelet frame for L2(R+) with frame bounds A and B if and only if

{
τkψ

` : k ∈ Z+,` = 1, . . . ,L
}

is a frame for W0 with frame bounds A and B.

Proof. Suppose that the system FΨ given by (4.6) is a p-wavelet frame for L2(R+)
with bounds A and B. Therefore, it follows from (3.13) that the family of functions
ψ`j,k lies in Wj, for ` = 1, . . . ,L,j ∈ Z and k ∈ Z

+.

By applying Theorem 3.11 to an arbitrary function f ∈ W0, we have∑
j∈Z

∑
k∈Z+

|〈f,ψ`j,k〉|
2 =

∑
k∈Z+

|〈f,τkψ`〉|2.

Consequently, using the p-wavelet frame condition of the system FΨ, we have

A‖f‖22 ≤
L∑
`=1

∑
k∈Z+

|〈f,τkψ`〉|2 ≤ B‖f‖22,

and it follows that the collection
{
τkψ

` : k ∈ Z+,` = 1, . . . ,L
}

is a frame for W0.

Conversely, suppose that the collection
{
τkψ

` : k ∈ Z+,` = 1, . . . ,L
}

is a frame
for W0 with bounds A and B. For any fixed j ∈ Z and f ∈ Wj, we have from e-
quation (3.13) that f(p−j·) ∈ W0. Further, by making use of the fact that

〈
f,ψ`j,k

〉
=

∫
R+
f(x) pj/2ψ`(pjx	k) dx

and

∥∥∥p−j/2f(p−j.)∥∥∥2
2

= p−j
∫
R+

∣∣f(p−jx)∣∣2dx = ∥∥f∥∥2
2
,



14 SHAH

we have

A
∥∥∥p−j/2f(p−j·)∥∥∥2

2
≤

L∑
`=1

∑
k∈Z+

∣∣〈f,ψ`j,k〉∣∣2 ≤ B ∥∥∥p−j/2f(p−j.)∥∥∥2
2
. (4.7)

Therefore, for a given j ∈ Z, the collection
{
ψ`j,k : k ∈ Z

+,` = 1, . . . ,L
}

is a frame

for Wj with frame bounds A and B.

Let f be an arbitrary function in L2(R+), then by Theorem 3.11 and Lemma
4.2, there exist fj ∈ Wj such that

f =
∑
j∈Z

fj, and
〈
fi,ψ

`
j,k

〉
= 0, i 6= j.

Thus, we have

L∑
`=1

∑
j∈Z

∑
k∈Z+

∣∣〈f,ψ`j,k〉∣∣2 = L∑
`=1

∑
j∈Z

∑
k∈Z+

∣∣∣∑
i∈Z

〈
fi,ψ

`
j,k

〉∣∣∣2

=

L∑
`=1

∑
j∈Z

∑
k∈Z+

∣∣〈fj,ψ`j,k〉∣∣2.(4.8)
It follows from (4.7) that

A
∑
j∈Z

∥∥fj∥∥22 ≤ L∑
`=1

∑
j∈Z

∑
k∈Z+

∣∣〈fj,ψ`j,k〉∣∣2 ≤ B ∑
j∈Z

∥∥fj∥∥22. (4.9)
By combining (4.8), (4.9) and Lemma 4.2, we obtain

A
∥∥fj∥∥22 ≤ L∑

`=1

∑
j∈Z

∑
k∈Z+

∣∣〈fj,ψ`j,k〉∣∣2 ≤ B∥∥fj∥∥22.
This completes the proof of the theorem.

Acknowledgement

The author was partially supported by the University of Kashmir, under seed money
grant scheme, letter no. F(Seed Money Grant)Res/KU; dated: March 26, 2013.

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Department of Mathematics, University of Kashmir, South Campus, Anantnag-192101,

Jammu and Kashmir, India