Multiresolution Analysis Based on Coalescence Hidden-variable FIF

0.21 MB·English

by  G. P. Kapoor

Multiresolution Analysis Based on
Coalescence Hidden-variable FIF
2
1
0
2
G. P. Kapoor1 and Srijanani Anurag Prasad2

n
a
J Department of Mathematics and Statistics

7
1 Indian Institute of Technology Kanpur

] Kanpur 208016, India
S
D
1 [email protected] 2 [email protected]
.
h
t
a
Abstract
m
[

In the present paper, multiresolution analysis arising from Coalescence Hidden-
1
v variable Fractal Interpolation Functions (CHFIFs) is accomplished. The availability
0
of a larger set of free variables and constrained variables with CHFIF in multiresolu-
0
5
tion analysis based on CHFIFs provides more control in reconstruction of functions
3
. in L (R) than that provided by multiresolution analysis based only on Affine Fractal
1 2
0 Interpolation Functions (AFIFs). In our approach, the vector space of CHFIFs is in-
2
1 troduced, its dimension is determined and Riesz bases of vector subspaces V ,k Z,
k
∈
:
v consisting of certain CHFIFs in L (R) C (R) are constructed. As a special case, for
2 0
i
X the vector space of CHFIFs of dimension 4, orthogonal bases for the vector subspaces
T
ar Vk,k Z, are explicitly constructed and, using these bases, compactly supported
∈
continuous orthonormal wavelets are generated.
Keywords: Fractal,Interpolation, Iteration,Affine, Coalescence, Attractor,Multiresolution
Analysis, Riesz Basis, Orthogonal Basis, Scaling Function, Wavelets
2010 Mathematics Subject Classification: Primary 42C40, 41A15; Secondary 65T60,
42C10, 28A80
1
1 Introduction
The theory of multiresolution analysis provides a powerful method to construct wavelets
having far reaching applications in analyzing signals and images [11, 14]. They permit effi-
cient representation of functions at multiple levels of detail, i.e. a function f L (R), the
2
∈
space of real valued functions g satisfying g = g(x) 2dx < , could be written as
L2
k k | | ∞
R
limit of successive approximations, each of which is Rsmoothed version of f. The multires-
olution analysis was first introduced by Mallat [10] and Meyer [13] using a single function.
The multiresolution analysis based upon several functions was developed in [6, 7, 9]. In [8],
multiresolution analysis of L2(R) were generated from certain classes of Affine Fractal In-
terpolation Functions (AFIFs). Such results were then generalized to several dimensions
in [3] and [4]. In [5], orthonormal basis for the vector space of AFIFs were explicitly con-
structed. A few years later, Donovan et al [2] constructed orthogonal compactly supported
continuous wavelets using multiresolution analysis arising from AFIFs. The interrelations
among AFIFs, Multiresolution Analysis and Wavelets are treated in detail by Massopust [12]
. However, multiresolution analysis of L (R) based on Coalescence Hidden-variable Fractal
2
Interpolation Functions (CHFIFs) which exhibits both self-affine and non-self-affine nature
has hitherto remained unexplored. In the present work, such a multiresolution analysis is
accomplished. The availability of a larger set of free variables and constrained variables in
multiresolution analysis based on CHFIFs provides more control in reconstruction of func-
tions in L (R) than that provided by multiresolution analysis based only on affine FIFs.
2
Further, orthogonal bases consisting of dilations and translations of scaling functions, for
the vector subspaces V ,k Z, are explicitly constructed and, using these bases, compactly
k
∈
supported continuous orthonormal wavelets are generated in the present work.
The organization of the paper is as follows: In Section 2, a brief introduction on the con-
struction of CHFIF is given, the vector space of CHFIFs is introduced and a few auxiliary
results, including a result on determination of dimension of this vector space, are found.
In Section 3, Riesz bases of vector subspaces V ,k Z, consisting of certain CHFIFs in
k
∈
L (R) C (R) are constructed. The multiresolution analysis of L (R) is then carried out in
2 0 2
terms of nested sequences of vector subspaces V ,k Z. As a special case, for the vector
T k ∈
space of CHFIFs of dimension 4, orthogonal bases for the vector subspaces V ,k Z, are
k
∈
explicitly constructed in Section 4 and, using these bases, compactly supported continuous
orthonormal wavelets are generated in the same section.
2
2 Preliminaries and Auxiliary Results
In this section, first a brief introduction on the construction of CHFIF is given. This is
followed by the development of some auxiliary results needed for the multiresolution analysis
generated by CHFIFs.
A Coalescence Hidden-variable Fractal Interpolation Function (CHFIF) is constructed such
that the graph of CHFIF is attractor of an IFS. Let the interpolation data be (x ,y )
i i
{ ∈
R2 : i = 0,1,...,N , where < x < x < ... < x < . By introducing a set of real
0 1 N
} −∞ ∞
parameters z for i = 0,1,...,N, consider the generalized interpolation data (x ,y ,z )
i i i i
{ ∈
R3 : i = 0,1,...,N . The contractive homeomorphisms L : I I for n = 1,...,N, are
n n
} →
defined by
L (x) = a x+b (2.1)
n n n
where, an = xxnN−xnx−01 and bn = xNxxnN−1−x0x0xn. For n = 1,...,N, define the maps ωn : I×R2 →
I R2 by − −
×
ω (x,y,z) = (L (x),F (x,y,z)) (2.2)
n n n
where, the functions F : I R2 R2 given by,
n
× →
F (x,y,z) = α y +β z +p (x),γ z +q (x) (2.3)
n n n n n n
(cid:0) (cid:1)
satisfy the join-up conditions
F (x ,y ,z ) = (y ,z ) and F (x ,y ,z ) = (y ,z ). (2.4)
n 0 0 0 n 1 n 1 n N N N n n
− −
Here, the variables α , γ are free variables and β are constrained variables such that
n n n
α < 1, γ < 1 and β + γ < 1 and the functions p (x) and q (x) are linear
n n n n n n
| | | | | | | |
polynomials given by
p (x) = c x+d and q (x) = e x+h . (2.5)
n n n n n n
It is proved in [1] that there exist a metric equivalent to Euclidean metric such that the
functions ω , defined by ω (x,y,z) = (L (x),F (x,y,z)), are contraction maps and, conse-
n n n n
quently,
I R2;ω ,n = 1,2,...,N (2.6)
n
{ × }
3
is the desired IFS for construction of CHFIF. Hence, there exists an attractor A in H(I R2)
×
N N
such that A = ω (A) = ω (x,y,z) : (x,y,z) A and is graph of a continuous
n n
{ ∈ }
n=1 n=1
function f : I SR2 such thaSt f(x ) = (y ,z ) for i = 0,1,...,N, i.e. A = (x,f(x)) :
i i i
→ {
x I, f(x) = (y(x),z(x)) . By expressing f component-wise as f = (f ,f ), Coalescence
1 2
∈ }
Hidden-variable Fractal Interpolation Function (CHFIF) [1] is defined as the continuous
function f for the given interpolation data (x ,y ) : i = 0,1,...,N .
1 i i
{ }
In order to develop the multiresolution analysis of L2(R), based on CHFIF, the space of
CHFIF needs to be introduced. For this purpose, let
t = (p ,q ) (2.7)
n n n
and t = (t ,t ,...,t ), where p and q are linear polynomials given by (2.5). Then, T =
1 2 N n n
t = (t ,...,t ) : t = (p ,q ),p ,q ,i = 1,2,...,N , with usual point-wise addition
1 N i i i i i 1
{ ∈ P }
and scalar multiplication, is a vector space, where is the class of linear polynomials. It
1
P
is easily seen that on B(I,R2), the set of bounded functions from I to R2 with respect to
maximum metric d (f,g) = max f (x) g (x) , f (x) g (x) , the function Φ defined by
∗ 1 1 2 2 t
x I {| − | | − |}
∈
(Φ f)(x) = F (L 1(x),f(L 1(x))) (2.8)
t n −n −n
for x I = [x ,x ], n = 1,2,...,N, is a contraction map. Therefore, by Banach
n n 1 n
∈ −
contraction mapping theorem, Φ has a unique fixed point f B(I,R2). By join-up
t t
∈
conditions (2.4), it follows that f C(I,R2), the set of continuous functions from I to R2.
t
∈
The following proposition gives the existence of a linear isomorphism between the vector
space T and the vector space C(I,R2).
Proposition 2.1. The mapping Θ : T C(I,R2) defined by Θ(t) = f is a linear isomor-
t
→
phism.
Proof. The assertion of the proposition is proved by establishing
(i) (af +f ) (x) = af (x)+f (x), i = 1,2, where f and af +f are written component-
t s i t,i s,i t t s
wise as f = (f ,f ) and af +f = ((af +f ) ,(af +f ) ), (ii) (af +f ) = f , (iii) Θ
t t,1 t,2 t s t s 1 t s 2 t s at+s
is onto and (iv) Θ is one-one.
The identity (i) follows by equating the components of left and right hand side in the identity
(af +f )(x) = af (x)+f (x).
t s t s
4
(ii) (af +f ) = f : By the definition of Φ ,
t s at+s t
(Φ (af +f ))(x)
at+s t s
= F (L 1(x),(af +f )(L 1(x)))
n −n t s −n
= α (af +f ) (L 1(x))+β (af +f ) (L 1(x))+(a p +pˆ )(L 1(x)),
n t s 1 −n n t s 2 −n n n −n
(cid:16)
γ (af +f ) (L 1(x))+(a q +qˆ )(L 1(x))
n t s 2 −n n n −n
(cid:17)
Using identity (i), it follows that,
(Φ (af +f ))(x)
at+s t s
= α af (L 1(x))+f (L 1(x)) +β af (L 1(x))+f (L 1(x))
n t,1 −n s,1 −n n t,2 −n s,2 −n
(cid:16)
+(cid:0)a p (L 1(x))+pˆ (L 1(x)) (cid:1), (cid:0) (cid:1)
n −n n −n
γ (cid:0)af (L 1(x))+f (L 1(x))(cid:1) + a q (L 1(x))+qˆ (L 1(x))
n t,2 −n s,2 −n n −n n −n
(cid:17)
(cid:0) (cid:1) (cid:0) (cid:1)
The above equation gives the following on simplification:
(Φ (af +f ))(x)
at+s t s
= a α f (L 1(x))+β f (L 1(x))+p (L 1(x)),γ f (L 1(x))+q (L 1(x))
n t,1 −n n t,2 −n n −n n t,2 −n n −n
(cid:16) (cid:17)
+ α f (L 1(x))+β f (L 1(x))+pˆ (L 1(x)),γ f (L 1(x))+qˆ (L 1(x))
n s,1 −n n s,2 −n n −n n s,2 −n n −n
(cid:16) (cid:17)
= af (x)+f (x)
t s
Therefore, af +f is a fixed point of Φ for all a R and t,s T. By uniqueness of fixed
t s at+s
∈ ∈
point of Φ , it follows that (af +f ) = f .
at+s t s at+s
(iii) Θ is onto : Let f = (f ,f ) C(I,R2). Define q (f) = f L γ f and p (f) =
1 2 i 2 i i 2 i
∈ ◦ −
f L α f β f for i = 1,...,N. Suppose t(f) = (t (f),t (f),...t (f)), where t (f) =
1 i i 1 i 2 1 2 N i
◦ − −
(p (f),q (f)). Then t(f) T whenever f C(I,R2). Also f = f.
i i t(f)
∈ ∈
(iv) Θ is one-one : Let f (x) = (0,0) for all values of x I. Then,
t
∈
f (x) = (0,0) Φ (f )(x) = (0,0)
t t t
⇔
F (L 1(x),f (L 1(x))) = (0,0)
⇔ n −n t −n
(p ,q ) = (0,0) for every n
n n
⇔
t = (0,...,0)
⇔
5
To introduce the space of CHFIFs, let the set consisting of functions f : I R2 be
0
S →
defined as = f : f = (f ,f ), f is a CHFIF passing through
0 1 2 1
S {
(x ,y ) R2 : i = 0,1,...,N and f is an AFIF passing through (x ,z ) R2 : i =
i i 2 i i
{ ∈ } { ∈
0,1,...,N . Then, is a vector space, with usual point-wise addition and scalar multi-
0
}} S
plication. The space of CHFIFs is now defined as follows:
Definition 2.1. Let 1 be the set of functions f : I R that are first components of
S0 1 →
functions f . The space of CHFIFs is the set 1 together with the maximum metric
∈ S0 S0
d (f,g) = max f(x) g(x) .
∗
x I | − |
∈
It is easily seen that the space of CHFIFs 1 is also a vector space with point-wise addition
S0
and scalar multiplication. The following proposition gives the dimension of 1:
S0
Proposition 2.2. The dimension of space of CHFIFs is 2N.
Proof. Consider the operator Φ f = (Φ f ,Φ f ). The operators Φ : B(I,R) B(I,R),
t t,1 1 t,2 2 t,i
→
i = 1,2, where B(I,R) is the set of bounded functions, satisfy
Φ f (x) = α f (L 1(x))+β f (L 1(x))+p (L 1(x)) (2.9)
t,1 1 n 1 −n n 2 −n n −n
and
Φ f (x) = γ f (L 1(x))+q (L 1(x)) (2.10)
t,2 2 n 2 −n n −n
for x [x ,x ]. By Proposition 2.1 and (2.10), it follows that f is completely determined
n 1 n 2
∈ −
by f ( i ) for i = 0,1,...N. Further, it follows by (2.9) that f depends on f . Then, for
2 N 1 2
f = (f ,f ) , the function f is the unique CHFIF passing through ( i ,y ), while the
1 2 ∈ S0 1 N i
function f is the unique AFIF passing through ( i ,z ). Hence,
2 N i
dimension of = 2(N +1). (2.11)
0
S
ConsidertheprojectionmapP : S S1. Then, KernelofP = f such thatP(f) = 0
0 → 0 { ∈ S0 }
isaproper subset ofS andconsists ofelements intheform(0,0)and(0,f ). Fortheelement
0 2
(0,f ) KerP, it is observed that β f (L 1(x)) + p (L 1(x)) = 0 for x I . Hence, for
2 ∈ n 2 −n n −n ∈ n
all x I, it is seen that f (x) = 1p (x). With x = x , it follows that c = βic and
∈ 2 −βn n 0 i β1 1
d = βid ,i = 2,...,N. Consequently, if (0,f ) KerP then f is a linear polynomial.
i β1 1 2 ∈ 2
So, dimension of KerP = 2. Therefore, by Rank-Nullity Theorem [56], dimension of 1 =
S0
dimension of dimension of Ker P = 2(N +1) 2 = 2N.
0
S − −
6
Remark 2.2. By Proposition 2.1 and (2.11), it follows that the map θ : RN+1 RN+1
0
× → S
defined by θ(y,z) = f is a linear isomorphism, where f = (f ,f ) , f is the unique
1 2 0 1
∈ S
CHFIF passing through the points (x ,y ) and f is the unique AFIF passing through the
i i 2
points (x ,z ), y = (y ,y ,...,y ) and z = (z ,z ,...,z ). Thus, is linearly isomorphic
i i 0 1 N 0 1 N 0
S
to RN+1 RN+1. Consider the metric space (RN+1 RN+1,d ), where d is given
R2(N+1) R2(N+1)
× ×
by d (y z,y¯ z¯) = max ( y y¯ , z z¯ ), y = (y ,y ,...,y ), z = (z ,z ,...,z ),
R2(N+1) i i i i 0 1 N 0 1 N
× × 0 i N | − | | − |
≤≤
y¯ = (y¯ ,y¯ ,...,y¯ ) and z¯ = (z¯ ,z¯ ,...,z¯ ). Then, with the metric d on the set , it is
0 1 N 0 1 N ∗ 0
S
observed by (2.8) that the maps θ and θ 1 are continuous. Hence is closed and complete
− 0
S
subspace of L (R).
2
Remark 2.3. Let f be a sequence in 1 such that lim f = f and f = (f ,f )
{ n,1} S0 n n,1 1∗ { n n,1 n,2 }
be a convergent sequence in , where f are AFIFs. →S∞ince is closed, lim f = f
0 n,2 0 n ∗
S S n ≡
(f ,f ) . Thus, f 1 and consequently, 1 is closed subspace of L (R→).∞
1∗ 2∗ ∈ S0 1∗ ∈ S0 S0 2
3 Multiresolution Analysis Based on CHFIF
In this section, the multiresolution analysis of L (R) is generated by using a finite set of
2
CHFIFs. For this purpose, the sets V ,k Z, consisting of collection of CHFIFs are defined.
k
∈
It is first shown that the sets V form a nested sequence. The multiresolution analysis of
k
L (R) is then generated by constructing Riesz bases of vector subspaces V consisting of
2 k
orthogonal functions in L (R).
2
To introduce certain sets of CHFIFs needed for multiresolution of L (R), let L (R,R2) be a
2 2
collection of functions f : R R2 such that f = (f ,f ) and f , f L (R) and C (R,R2)
1 2 1 2 2 0
→ ∈
be a collection of functions f : R R2 such that f = (f ,f ) and f , f C (R), the set of
1 2 1 2 0
→ ∈
all real valued continuous functions defined on R which vanish at infinity. Define the set V˜
0
as
V˜ = S˜ L (R,R2) C (R,R2)
0 0 2 0
where, S˜ = f : f = (f ,f ), f \is a CHFIF\andf is an AFIF, i Z .
0 1 2 1 [i 1,i) 2 [i 1,i)
{ | − | − ∈ }
That the set V˜ is not empty is easily seen by considering a function f = (f ,f ) , with
0 1 2 0
∈ S
f(x ) = (0,0) = f(x ) and f(x) = (0,0) for x I, which obviously belongs to V˜ . Let, for
0 N 0
6∈
k Z,
∈
V˜ = f : f(N k ) V˜ .
k − 0
{ · ∈ }
7
The sets V˜ and V˜ are easily seen to be closed sets as follows: Let f be a sequence
0 k n
{ }
in V˜ such that lim f = f = (f ,f ). Now, lim f [i 1,i) = f [i 1,i) = (f [i
0 n n ∗ 1∗ 2∗ n n| − ∗| − 1∗| −
1,i),f [i 1,i)). →B∞y Remark 2.2 and Remark 2.3,→it∞is observed that f [i 1,i) is a CHFIF
2∗| − 1∗| −
and f [i 1,i) is an AFIF, i Z. Thus, f S˜ , which implies that S˜ is a closed set.
2∗| − ∈ ∗ ∈ 0 0
Consequently, V˜ and V˜ ,,k Z are closed sets. Now, consider sets V and V ,k Z 0,
0 k 0 k
∈ ∈ \
of CHFIFs defined as follows:
V = f : f is the first component of some f = (f ,f ) V˜ (3.1)
0 1 1 1 2 0
{ ∈ }
and
V = f : f (N k ) V . (3.2)
k 1 1 − 0
{ · ∈ }
The sets V ,k Z, with L -norm, are subspaces of L (R). The following proposition shows
k 2 2
∈
that the subspaces V ,k Z, of L (R), form a nested sequence:
k 2
∈
Proposition 3.1. The subspaces V ,k Z are vector subspaces of L (R) and form a nested
k 2
∈
sequence ... V V V ....
1 0 1
⊇ − ⊇ ⊇ ⊇
Proof. It follows from Proposition 2.1 that the sets V ,k Z are vector subspaces of L (R).
k 2
∈
Now, to show V V for all k Z, it suffices to prove the inclusion relation for k = 0.
k k+1
⊇ ∈
Let f V . Then, f = g for some g = (g ,g ) V˜ . If G = graph(g )
1 [0,N) 1 [0,N) 1 2 1 [0,N]
∈ | | ∈ |
N N
then, G = w (G) implies, for j 1,...,N , w (G) = w w w 1(w (G)),
i ∈ { } j j ◦ i ◦ j− j
i=1 i=1
where wi(G) =S(Li(x),Fi(x,y,z)) for all (x,y,z) G , i = 1,.S..N. Expressing wi and
∈
w w w 1 in matrix form as w (x,y,z) = A (x,y,z) + B and w w w 1(x,y,z) =
j ◦ i ◦ j− i i i j ◦ i ◦ j−
A (x,y,z)+B , it is observed that non-zero entries in matrices A and A occur at the
i,j i,j i i,j
same places. Consequently, w (G) is graph of g , so that g V˜ . It therefore follows
j [j 1,j) 0
| − ∈
that g is a CHFIF on the interval [j 1,j). Thus, the function f = g is
1 [j 1,j) [j 1,j) 1 [j 1,j)
| − − | − | −
a CHFIF on the interval [j 1,j) and consequently, f V .
0
− ∈
In order to generate a multiresolution analysis of L (R) using CHFIFs, the inner product on
2
the space V ,k Z is defined by f ,fˆ = f (x)fˆ(x) dx. Using the following recurrence
k 1 1 1 1
∈ h i
R
relations, R
f (L (x)) = α f (x)+β f (x)+p (x)
1 n n 1 n 2 n
and
ˆ ˆ ˆ ˆ
f (L (x)) = αˆ f (x)+β f (x)+pˆ (x),
1 n n 1 n 2 n
8
it is observed that, for f ,fˆ V ,
1 1 0
∈
N
ˆ ˆ ˆ ˆ ˆ
a α β f ,f +β αˆ f ,f +β β f ,f
n n n 1 2 n n 2 1 n n 2 2
h i h i h i
n=1
P (cid:16) +α f ,pˆ +αˆ fˆ,p +β f ,pˆ
n 1 n n 1 n n 2 n
h i h i h i
ˆ ˆ
+β f ,p + p ,pˆ
n 2 n n n
ˆ h i h i
f ,f = (3.3)
h 1 1i N (cid:17)
1 a α αˆ
n n n
−
n=1
P
ˆ
where, a , α and β , p ; αˆ and β , pˆ , are given by (2.1), (2.3), (2.5) respectively
n n n n n n n
for the interpolation data (x ,y ,z ) : i = 0,1,...,N and (x ,yˆ,zˆ) : i = 0,1,...,N .
i i i i i i
{ } { }
Using (3.3), the set of orthogonalfunctions that formsthe Riesz basis of set V isconstructed
0
as follows:
Let, the free variables α , γ and constrained variables β , j = 1,...,N, N > 1, in the
j j j
construction of CHFIF be chosen such that α + β = γ for atleast one j. Consider, the
j j j
6
points y and z Rn+1, i = 0,...,N, given by
i i
∈
y = (1,r ,...,r ,0), y = (0,s ,...,s ,1),
0 1 N 1 N 1 N 1
− −
y = (0,...,1,...,0), i = 1,...,N 1, (3.4)
i 
−

y = (0,u ,...,u ,0), i = 0,...,N; 
N+1+i i,1 i,N 1
−


z = (0,...,0), z = (0,...,1,...,0), i = 0,...,N (3.5)
i N+1+i
˜ ˜ ˜ ˜
and a set of 2(N+1) functions f = (f ,f ) , i = 0,...,2N+1, where the CHFIF f
i i,1 i,2 0 i,1
∈ S
passes through the points (x ,y ),k = 0,...,N +1, y being the kth component of y and
k ik ik i
AFIF f˜ passes through the points (x ,z ),k = 0,...,N +1, z being the kth component
i,2 k ik ik
of z . Let the function f˜ : R R2, i = 0,1,...2N + 1, be the extension of the function
i i∗ →
f˜ : I R2 such that f˜ (x) = f˜(x) for x I and f˜ (x) = (0,0) for x I.
i → i∗ i ∈ i∗ 6∈
˜
For ensuring the orthogonality of the functions f with respect to the inner product in
i∗,1
L (R), let the values of r , s and u , i,j = 1,...,N 1, in (3.4) be chosen such that
2 i i i,j
−
˜ ˜ ˜ ˜ ˜ ˜
f ,f = 0, f ,f = 0, f ,f = 0. (3.6)
i,1 0,1 i,1 N,1 N+1+j,1 i,1
h i h i h i
Let, for i = 1,2,...,N 1,
−
˜ ˜ ˜ ˜
ζ =< f ,f > and η =< f ,f > . (3.7)
i N+1+i,1 0,1 i N+1+i,1 N,1
9
The free variables α ,γ and constrained variables β , j = 1,2,...,N, in (2.3) are 3N
j j j
variables and ζ = η = 0, i = 1,...,N 1 is a system of 2N 2 equations. Suppose there
i i
− −
exist no α ,γ and β , j = 1,...,N, in ( 1,1) such that ζ = η = 0, i = 1,...,N 1,
j j j i i
− −
then dimension of S1 < 2N, which is a contradiction. Hence, there exists atleast one set of
0
α ,γ and β , j = 1,...,N, in ( 1,1) such that ζ = η = 0, i = 1,...,N 1. The free
j j j i i
− −
variables α ,γ and constrained variables β , j = 1,2,...,N, in (2.3) are chosen such that,
j j j
for i = 1,2,...,N 1, ζ = 0 and η = 0.
i i
−
˜ ˜
It is easily seen that the functions f , i = 0,...,2N + 1, f , i = 0,...,N and the
i∗ i∗,1
˜ ˜
functions f , j = N + 1,...,2N + 1, are linearly independent. Now, by (2.8), f (x) =
j∗,2 j∗,1
α f˜ (L 1(x)) + β f˜ (L 1(x)) + p (L 1(x)) and f˜ (x) = γ f˜ (L 1(x)) + q (L 1(x)),
n j∗,1 −n n j∗,2 −n n,j −n j∗,2 n j∗,2 −n n,j −n
j = 0,1,...,2N + 1, where p and q are linear polynomials. By (3.5), the func-
n,j n,j
N 1
tions f˜ , j = N + 2,...,2N, are not linear polynomials. Hence, − a f˜ (x)
j∗,2 k N∗+1+k,1 −
k=1
N 1 P
α − a f˜ (L 1(x)) = 0if andonlyifa = 0, which implies f˜ , k = 1,...,N 1,
n k N∗+1+k,1 −n k N∗+1+k,1 −
k=1
arePlinearlyindependent. Thelinearindependenceof f˜ , f˜ ,k = 1,...,N 1together
k∗,1 N∗+1+k,1 −
with (3.6) will yield same number of orthogonal functions in Gram-Schmidt process.
Let φ 2N 1 V ,i = N,beasequenceoforthogonalfunctionsobtainedfrom f˜ 2N ,i =
{ i,1}i=1− ⊂ 0 6 { i∗,1}i=1 6
N,N +1 by the Gram-Schmidt process. Set,
˜
f (x) x [0,1)
N∗,1 ∈
φ = f˜ (x 1) x [1,2)
N,1  0∗,1 − ∈

 0 otherwise


It is easily seen by Proposition 2.2 that φ , i = 1,2,...,2N 1, are non-zero functions.
i,1
−
Further, by (3.6) and (3.7), it follows that φ : i = 1,2,...,2N 1 is an orthogonal
i,1
{ − }
set. This is the set required for the generation of multiresolution analysis of L (R) in the
2
following theorem :
Theorem 3.1. Let free variables α , γ and constrained variables β , j = 1,...,N, N > 1,
j j j
in the construction of CHFIF be chosen such that α +β = γ for atleast one j and ζ , η
j j j i i
6
given by (3.7) be such that ζ = 0 , η = 0, i = 1,...,N 1. Then,
i i
−
V = clos span φ ( l) : i = 1,...,2N 1, l Z , (3.8)
0 L2 i,1
{ ·− − ∈ }
where, φ V . Also, the set φ 2N 1 generates a continuous, compactly supported
i,1 ∈ 0 { i,1}i=1−
multiresolution analysis of L (R).
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10