Ratio Mathematica Volume 39, 2020, pp. 187-212

CAS wavelet approximation of functions of
Hölder’s class Hα[0,1) and Solution of

Fredholm Integral Equations

Shyam Lal*
Satish Kumar†

Abstract

In this paper, cosine and sine wavelet is considered. Two new CAS
wavelet estimators E(1)

2k,2M+1
(f) and E(2)

2k,2M+1
(f) for the

approximation of a function f whose first derivative f
′

and second
derivative f

′′
belong to Hölder’s class Hα[0, 1) of order 0 < α 6 1,

have been obtained. These estimators are sharper and best in wavelet
analysis. Using CAS wavelet, a computational method has been
developed to solve Fredholm integral equation of second kind. In
this process, Fredholm integral equations are reduced into a system of
linear equations. Approximation of functions by CAS wavelet method
is applied in obtaining the solution of Fredholm integral equation of
second kind. CAS wavelet coefficient matrices are prepared using the
properties of CAS wavelets. Two examples are illustrated to show the
validity and efficiency of the technique discussed in this paper.

Keywords: CAS wavelet, CAS Wavelet Approximation, Function
of Hölder’s class, Orthonormal basis, Fredholm integral equation.
Mathematics Subject Classification:42C40, 65T60, 45G10,45B05.
1

*Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-
221005, India; shyam lal@rediffmail.com.

†Department of Mathematics, Institute of Science, Banaras Hindu University, Varanasi-
221005, India); satishkumar3102@gmail.com.

1Received on June 21st, 2020. Accepted on December 15th, 2020. Published on December
31st, 2020. doi: 10.23755/rm.v39i0.549. ISSN: 1592-7415. eISSN: 2282-8214. ©Lal and Satish
Kumar. This paper is published under the CC-BY licence agreement.

187



Shyam Lal and Satish Kumar

1 Introduction
Wavelet is a very recent and powerful tool in pure as well as

applied mathematical research area. It has wide range applications in engineering,
science and technology, signal analysis, time-frequency analysis, fast numerical
algorithm. Several problems of Physics, Engineering, science and Technology
are found in the form of integral equations. In some cases, integral equations are
reformulated into ordinary differential equations and partial differential equations.
In many cases, it is very difficult to solve integral equations analytically and
hence there is a need of approximate solution of integral equations. In recents
years, the approximate solutions of integral equations have been obtained by
orthogonal basis functions as well as orthogonal wavelets. The main advantage
of using orthonormal basis is that it converts the mathematical problems to a
system of algebraic equations. Working in same direction, several researchers
like [2], Sahu [3] etc. have been solved integral equations. It is known that
wavelets are considerably useful in the solution of integral equations. In science
and Technology, some problems are available in the form of Fredholm integral
equations of second kind:

u(x) = f(x) +

∫ 1
0

K(x,y)u(y)dy (1)

where f ∈ L2[0, 1) and K ∈ L2[0, 1) × L2[0, 1) are known functions and u is
unknown function to be determined (Ray and Sahu [3]).

In best of our knowledge, there is no work associated with the solution of
Fredholm integral eqn (1) by CAS wavelet method. The main objectives of the
research paper are as follows:

1. To estimate the approximation of functions belonging to Hölder’s class
Hα[0, 1) of order 0 < α 6 1 by CAS wavelet method.

2. To develop a procedure to solve Fredholm integral equation of second kind
by using CAS wavelet approximation.

3. To compare the solutions of Fredholm integral eqn (1) obtained by CAS
wavelet, Legendre wavelet and Haar wavelet method with their exact
solutions.

It is remarkable to note that the solution of Fredholm integral eqn (1)
obtained by CAS wavelet method and its exact solution are almost same. The
solution of Fredholm integral eqn (1) obtained by CAS wavelet method is
better and more closed to its exact solution than the solutions obtained by
Legendre wavelet and Haar wavelet method. It is observed in numerical

188



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

comparison of these solutions. It is a significant achievement of the proposed
method.

2 Definitions and Preliminaries

2.1 Basic Wavelets And CAS Wavelets
Let ψ ∈ L2(IR). ψ is called a basic wavelet if it satisfies the admissibility

condition:

Cψ =

∫ ∞
−∞

| ψ̂ |2

| w |
dw < ∞ (Chui [1]) (2)

The integral wavelet transform, relative to a basic wavelet ψ, is defined by

(Wψf)(b,a) = |a|−1/2
∫ ∞
−∞

f(t)ψ(
b−a
a

)dt ,f ∈ L2(IR) (3)

where a,b ∈ IR,a 6= 0 . Set

ψb,a(t) = |a|−1/2ψ(
b−a
a

). (4)

This is a family of wavelets. If we restrict the parameters a and b to discrete values

a = a−k0 ,b = nb0a
−k
0 ,a0 > 1,b0 > 0

where n and k are positive integers, then

ψb,a(t) = ψn,k(t) = |a0|k/2ψ(ak0t−nb0). (5)

Taking a0 = 2,b0 = 1 in eqn (5),

ψn,k(t) = 2
k/2ψ(2kt−n). (6)

If

ψ(2kt−n) = cos(2mπ(2kt−n + 1)) + sin(2mπ(2kt−n + 1)) (7)
= CASm(2

kt−n + 1). (8)

Using eqn(7), eqn (6) becomes

ψn,m(t) =

{
2
k
2{cos(2mπ(2kt−n + 1)) + sin(2mπ(2kt−n + 1))}, if n−1

2k
6 t < n

2k
,

0, otherwise.

{ψn,m}n,m∈Z are orthonormal CAS wavelets defined on [0,1) .

189



Shyam Lal and Satish Kumar

3 Function belonging to Hölder’s class Hα[0,1)
A function f is said to belong to Hölder’s class Hα[0, 1) of order 0 < α 6 1

if f satifies the following condition :

|f(x) −f(y)| 6 A|x−y|α, ∀x,y ∈ IR (9)

for some positive constant A (Zheng, Wei [4]).

3.1 Proposition
Let f be a function such that its second derivative f

′′
is in Hα[0, 1), then its

first derivative f
′

is in Hα[0, 1).
Proof : Let φ

′′ ∈ Hα[0, 1) .

f(x) =

∫ xα
0

φ
′
(t) dt

f
′
(x) =

∫ xα
0

φ
′′
(t) dt and f

′
(y) =

∫ yα
0

φ
′′
(t) dt

|f
′
(x) −f

′
(y)| = |

∫ xα
0

φ
′′
(t) dt−

∫ yα
0

φ
′′
(t) dt| = |

∫ xα
yα

φ
′′
(t) dt|

≤ M|xα −yα| ≤ M|x−y|α, M = sup
t∈[0,1)

{φ
′′
(t)}

Converse is not true. Consider the example f(x) = x
α+1

α+1
0 < α < 1.Then,

f
′
(x) = xα and f

′′
(x) = αxα−1. For x = 1

N
1

1−α
, y = 1

(1+N)
1

1−α
, we have

|x−y| ≤ 1
N

1
1−α
− 1

(1+N)
1

1−α
≤ 1

N
1

1−α
= δ.

And |f ′′(x) −f ′′(y)| = α(1 + N −N) = α
If 0 < � < α, then |f ′′(x) − f ′′(y)| � � whenever |x− y| ≤ δ = 1

N
1

1−α
. Hence,

f
′ ∈ Hα[0, 1) but f ′′ 6∈ Hα[0, 1).

3.2 Difference between Hölder’s class and Lipschitz class
1. Consider the function f(x) =

√
x2 + 5 ∀x ∈ [0, 1]. Then

|f(x) −f(y)| ≤ |
√
x2 + 5 −

√
y2 + 5| ≤ |

√
x2 −y2| ≤

√
2|x−y|

1
2 (10)

Eqn(10) shows that f ∈ H
1
2 [0, 1). And also, we have

|f
′
(x)| ≤ |

x

x2 + 5
| ≤ 1, ∀ x ∈ [0, 1] (11)

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CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

Eqn(10) and Eqn(11) shows that f ∈ Lip1
2
[0, 1).

2. Define the function f(x) =
√
x ∀x ∈ [0, 1], then we have

|f(x) −f(y)| ≤ |
√
x−
√
y| ≤ |x−y|

1
2 =⇒ f ∈ H

1
2 [0, 1).

And since, f
′
(x) = 1

2
√
x
→∞ as x → 0+. Hence, f is not bounded.

∴ f 6∈ Lip1
2
[0, 1). Hence, we conclude that Lipα[0, 1] ⊂ Hα[0, 1].

4 Approximation of function
Since {ψn,m}n,m∈Z forms an orthonormal basis for L2[0, 1] , therefore a func-

tion f ∈ L2[0, 1) can be expressed into CAS wavelet series as:

f(t) =
∞∑
n=1

∞∑
m=−∞

cn,mψn,m(t) (12)

where the coefficients cn,m are given by

cn,m =< f,ψn,m > (13)

(2k, 2M + 1)th partial sum S2k,2M+1(f)(t) of (12) is given by

S2k,2M+1(f)(t) =
2k∑
n=1

M∑
m=−M

cn,mψn,m(t) = C
T Ψ(t) (14)

where C and Ψ(t) are given by
C = [c1,(−M),c1,(−M+1), ...,c1,M,c2,(−M), ...,c2,M, ...,c2k,(−M), ...,c2k,M ]

T

and

Ψ(t) = [ψ1,(−M)(t),ψ1,(−M+1)(t), ...,ψ1,M (t),ψ2,(−M)(t), ...,ψ2,M (t), ...,

ψ2k,(−M)(t), ...,ψ2k,M (t)]
T .

Extended Legendre Wavelet expansion of function f ∈ L2[0, 1) is

f(x) =
∞∑
n=1

∞∑
m=0

cn,mψ
(µ)
n,m(x),

and its (µk,M)th partial sum is

Sµk,M (f)(x) =

µk∑
n=1

M∑
m=0

cn,mψ
(µ)
n,m(x).

191



Shyam Lal and Satish Kumar

The extended Legendre wavelet approximation Eµk,M (f) of f by (µ
k,M)thpartial

sum Sµk,M (f) is defined by

Eµk,M (f) = min
S
µk,M

(f)
||f −Sµk,M (f)||2 .

In our case, the CAS wavelet approximation E2k,2M+1(f) of f by (2
k, 2M + 1)th

partial sum S2k,2M+1(f) of series (12) is defined by

E2k,2M+1(f) = min
S
2k,2M+1

(f)
||f −S2k,2M+1(f)||2 . (15)

5 Theorems
In this paper, we prove the following theorems:

Theorem 5.1. If f ∈ L2[0, 1) is a function such that f ′ ∈ Hα[0, 1) and its CAS
wavelet expansion is

f(t) =
∞∑
n=1

∞∑
m=−∞

cn,mψn,m(t) (16)

then the approximation error E(1)
2k,2M+1

(f) of f by (2k, 2M + 1)th partial sum

S2k,2M+1(f)(t) =
2k∑
n=1

M∑
m=−M

cn,mψn,m(t) (17)

of expansion 16 is given by

E
(1)

2k,2M+1
(f) = min

S
2k,2M+1

(f)
||f − (S2k,2M+1f)||2 = O(

1
√
M + 1 2k(α+1)

) (18)

Theorem 5.2. If f ∈ L2[0, 1) is a function such that f ′′ ∈ Hα[0, 1) and its
CAS wavelet expansion is given by the series (16) , then the approximation error
E

(2)

2k,2M+1
(f) of f by (2k, 2M + 1)th partial sum S2k,2M+1(f)(t) of series (16) is

given by

E
(2)

2k,2M+1
(f) = min

S
2k,2M+1

(f)
||f − (S2k,2M+1f)||2 = O(

1

(M + 1)
3
2 2k(α+2)

) (19)

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CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

Proof of theorem (5.1) Since

f(t) =
∞∑
n=1

∞∑
m=−∞

cn,mψn,m(t)

and

S2k,2M+1(f)(t) =
2k∑
n=1

M∑
m=−M

cn,mψn,m(t)

∴ f(t) −S2k,2M+1(f)(t) =
∞∑
n=1

∞∑
m=−∞

cn,mψn,m(t) −
2k∑
n=1

M∑
m=−M

cn,mψn,m(t)

= (
2k∑
n=1

+
∞∑

n=2k+1

)(
−M−1∑
m=−∞

+
M∑

m=−M

+
∞∑

m=M+1

)cn,mψn,m(t)

−
2k∑
n=1

M∑
m=−M

cn,mψn,m(t)

=
2k∑
n=1

−M−1∑
m=−∞

cn,mψn,m(t) +
2k∑
n=1

∞∑
m=M+1

cn,mψn,m(t)

(f(t) −S2k,2M+1(f)(t))2 =
2k∑
n=1

−M−1∑
m=−∞

c2n,mψ
2
n,m(t) +

2k∑
n=1

∞∑
m=M+1

c2n,mψ
2
n,m(t)

+2
∑∑

16 n6=n′≤ 2k

∑∑
−∞≤m6=m′≤−M−1

cn,mcn′,m′ψ
T
n,m(t)ψn′,m′(t)

+2
∑∑

16 n6=n′≤ 2k

∑∑
M+1≤m 6=m′≤∞

cn,mcn′,m′ψ
T
n,m(t)ψn′,m′(t)

||f −S2k,2M+1(f)||22 =
∫ 1
0

|f(t) −S2k,2M+1(f)(t)|2dt

6
2k∑
n=1

−M−1∑
m=−∞

|cn,m|2
∫ 1
0

|ψn,m(t)|2dt

+
2k∑
n=1

∞∑
m=M+1

|cn,m|2
∫ 1
0

|ψn,m(t)|2dt

+ 2
∑∑

16 n6=n′≤ 2k

∑∑
−∞≤m 6=m′≤−M−1

|cn,m||cn′,m′|

∫ 1
0

|ψTn,m(t)ψn′,m′(t)|dt

193



Shyam Lal and Satish Kumar

+2
∑∑

16 n6=n′≤ 2k

∑∑
M+1≤m6=m′≤∞

|cn,m||cn′,m′|
∫ 1
0

|ψTn,m(t)ψn′,m′(t)|dt

=
2k∑
n=1

−M−1∑
m=−∞

|cn,m|2 +
2k∑
n=1

∞∑
m=M+1

|cn,m|2 , by orthonormality of {ψn,m}n,m∈Z

||f −S2k,2M+1(f)||22 ≤
2k∑
n=1

(
−M−1∑
m=−∞

+
∞∑

m=M+1

)|cn,m|2 (20)

cn,m = < f,ψn,m >

=

∫ n
2k

n−1
2k

f(t) 2
k
2{cos(2mπ(2kt−n + 1)) + sin(2mπ(2kt−n + 1))} dt

=
1

2
k
2

∫ 1
0

f(
x + n− 1

2k
) (cos(2mπx) + sin(2mπx)) dx, 2kt−n + 1 = x

=
1

(2mπ) 2
3k
2

∫ 1
0

f
′
(
x + n− 1

2k
)(cos(2mπx) − sin(2mπx))dx, integrating by part

=
1

(2mπ) 2
3k
2

[

∫ 1
0

{f
′
(
x + n− 1

2k
) −f

′
(
n− 1

2k
)}(cos(2mπx) − sin(2mπx))dx

−f
′
(
n− 1

2k
)

∫ 1
0

(cos(2mπx) − sin(2mπx))dx]

=
1

(2mπ) 2
3k
2

∫ 1
0

{f
′
(
x + n− 1

2k
) −f

′
(
n− 1

2k
)}(cos(2mπx) − sin(2mπx))dx

|cn,m| 6
1

(2mπ) 2
3k
2

∫ 1
0

|f
′
(
x + n− 1

2k
) −f

′
(
n− 1

2k
)| |cos(2mπx) − sin(2mπx)|dx

6
A

(2mπ) 2
3k
2

∫ 1
0

|
x

2k
|α |cos(2mπx) − sin(2mπx)| dx, since f

′
∈ Hα[0, 1)

Now by Cauchy Schwarz inequality, we have

|cn,m| 6
A

(2mπ) 2
3k
2

{
∫ 1
0

|
x

2k
|2α dx}

1
2 {
∫ 1
0

|cos(2mπx) − sin(2mπx)|2dx}
1
2

=
A

(2mπ) 2(
3k
2
+kα)
{
∫ 1
0

|x|2α dx}
1
2

=
A

(2mπ) 2(
3
2
+α)k

1
√

2α + 1

|cn,m| 6
A

2mπ
√

2α + 1 2(
3
2
+α)k

194



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

By eqn (20) and (21) , we have

||f −S2k,2M+1(f)||22 ≤
2k∑
n=1

(
−M−1∑
m=−∞

+
∞∑

m=M+1

)
A2

4m2π2(2α + 1) 2(3+2α)k
,

=
A2

4π2(2α + 1)
(
−M−1∑
m=−∞

+
∞∑

m=M+1

)
2k

2(3+2α)k m2

=
A2

4π2(2α + 1)

1

2(2+2α)k
(
−M−1∑
m=−∞

1

m2
+

∞∑
m=M+1

1

m2
)

=
A2

4π2(2α + 1)

1

2(1+α)2k
(

1

M + 1
+

1

M + 1
)

=
A2

2π2(2α + 1)

1

2(1+α)2k
1

M + 1

∴ min
S
2k,2M+1

(f)
||f −S2k,2M+1(f)||2 6

A

π
√

2(2α + 1)

1

2k(α+1)
1

√
M + 1

∴ E
(1)

2k,2M+1
(f) = min

S
2k,2M+1

(f)
||f −S2k,2M+1(f)||2 = O(

1
√
M + 1 2k(α+1)

)

Thus, theorem (5.1) is completely established.

Proof of theorem (5.2) Following the steps of the proof of theorem ( 5.1)

cn,m =
1

(2mπ) 2
3k
2

∫ 1
0

f
′
(
x + n− 1

2k
)(cos(2mπx) − sin(2mπx))dx

=
−1

(4m2π2) 2
5k
2

∫ 1
0

f
′′
(
x + n− 1

2k
)(cos(2mπx) + sin(2mπx))dx,

=
−1

(4m2π2) 2
5k
2

[

∫ 1
0

{f
′′
(
x + n− 1

2k
) −f

′′
(
n− 1

2k
)}(cos(2mπx) + sin(2mπx))dx

−f
′′
(
n− 1

2k
)

∫ 1
0

(cos(2mπx) − sin(2mπx))dx]

|cn,m| 6
1

(4m2π2) 2
5k
2

∫ 1
0

|f
′′
(
x + n− 1

2k
) −f

′′
(
n− 1

2k
)| |cos(2mπx) + sin(2mπx)|dx

6
B

(4m2π2) 2
5k
2

∫ 1
0

|
x

2k
|α |cos(2mπx) + sin(2mπx)|dx, sincef

′′
∈ Hα[0, 1)

Now by Cauchy Schwarz inequality, we have

|cn,m| 6
B

(4m2π2) 2(
5k
2
+kα)
{
∫ 1
0

|
x

2k
|2αdx}

1
2{
∫ 1
0

|cos(2mπx) + sin(2mπx)|2dx}
1
2

195



Shyam Lal and Satish Kumar

|cn,m| 6
B

(4m2π2) 2(
5
2
+α)k

(
1

√
2α + 1

)

|cn,m| 6
B

4m2π2
√

2α + 1 2(
5
2
+α)k

(21)

From eqn (20) and (21), we have

||f −S2k,2M+1(f)||22 =
2k∑
n=1

(
−M−1∑
m=−∞

+
∞∑

m=M+1

)|cn,m|2

6
2k∑
n=1

(
−M−1∑
m=−∞

+
∞∑

m=M+1

)
B2

16m4π4 (2α + 1) 2(5+2α)k
,

=
B2

16π4 (2α + 1) 2(5+2α)k
(
−M−1∑
m=−∞

+
∞∑

m=M+1

)
2k

m4

=
B2

16π4 (2α + 1) 2(4+2α)k
(
−M−1∑
m=−∞

1

m4
+

∞∑
m=M+1

1

m4
)

=
B2

16π4 (2α + 1) 2(4+2α)k
(

1

3(M + 1)3
+

1

3(M + 1)3
)

=
B2

24π4 (2α + 1) 22k(α+2)
1

(M + 1)3

∴ min
S
2k,M

(f)
||f −S2k,2M+1(f)||2 6

B

2
√

6π2
√

(2α + 1) 2k(α+2)
1

(M + 1)
3
2

∴ E
(2)

2k,2M+1
(f) = min

S
2k,2M+1

(f)
||f −S2k,2M+1(f)||2 = O(

1

(M + 1)
3
2 2k(α+2)

)

Hence, theorem (5.2) has been proved.

6 Solution of the Fredholm integral equation of sec-
ond kind

Consider the Fredholm integral equation of second kind given by eqn (1).
Using CAS wavelet approximations,

u(x) = UT Ψ(x) = ΨT (x)U , (22)
f(x) = FT Ψ(x) = ΨT (x)F ,

and K(x,y) = ΨT (x)KΨ(y) ,

196



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

where K is a square matrix of order 2k(2M + 1), which is calculated as follows∫ 1
0

∫ 1
0

ψn,m(x)ψn′,m′(y)K(x,y)dxdy , (23)

where 1 6 n,n
′
6 2k and −M 6 m,m′ 6 M , equation (1) becomes

Ψt(x)U = Ψt(x)F + Ψt(x)K
∫ 1
0

Ψ(y)Ψt(y)Udy (24)

By orthonormality of CAS wavelets, equation (24) reduces to

U = (I − K)−1F (25)

where I is identity matrix of order 2k(2M + 1) . Subtituting the value of U from
eqn (25) in eqn (22) , the solution u(x) of Fredholm integral equation of second
kind (1) can be obtained.

6.1 Solution of integral eqn (1) by Haar wavelet method
Let Haar wavelet solution of intgral eqn (1) be of the form

u(x) =
2M∑
i=1

aihi(x) (26)

Subtituting the eqn (26) in eqn (1) , we have

2M∑
i=1

ai(hi(x) −gi(x)) = f(x) (27)

where

gi(x) =

∫ 1
0

k(x,y)hi(y)dy (28)

Taking the collocation points xk =
k−1

2

2M
, k = 1, 2, ..., 2M, in eqns (27) and (26),

we obtain

2M∑
i=1

ai(hi(xk) −gi(xk)) = f(xk) (29)

197



Shyam Lal and Satish Kumar

and u(xk) =
2M∑
i=1

aihi(xk) (30)

The wavelet coefficients ai, i = 1, 2, ..., 2M are obtained by solving 2M
system of equations in (29). Subtituting these coefficients in the eqn( 30) we
can obtain the Haar wavelet solution of the integral eqn (1).

7 Illustrated Numerical Examples
Two Fredholm integral equations have been solved by proposed method ie.

CAS wavelet method discussed in this paper. Exact solutions of
considered integral eqn are compared with their approximate solutions obtained
by CAS wavelet, Legendre wavelet and Haar wavelet method. The graphs of
these solutions are plotted. It is observed that exact solution and approximate
solutions of Fredholm integral equations obtained by CAS wavelet method are
almost equal. The solutions of Fredholm integral equation derived by the help of
CAS wavelet method are more closed than the solutions of this integral equation
obtained by Legendre wavelet and Haar wavelet method. This comparison shows
the advantages of proposed method of this paper. This is illustrated in following
two examples.

Example 1
Subtituting f(x) = sin(8πx) and K(x,y) = y2 , in the Fredholm integral

equation (1), it reduces to

u(x) = sin(8πx) +
∫ 1
0

y2u(y)dy (31)

The exact solution of integral eqn (31) is given by

u(x) = sin(8πx) −
3

16π
(32)

CAS wavelet solution
For CAS wavelet solution, take k = 2,M = 1 in the eqn (14) . In this case,

Ψ(x) = [ψ1,−1(x),ψ1,0(x),ψ1,1(x),ψ2,−1(x),ψ2,0(x),ψ2,1(x),

ψ3,−1(x),ψ3,0(x),ψ3,1(x),ψ4,−1(x),ψ4,0(x),ψ4,1(x)]
T (33)

198



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

where

ψ1,−1(x) = 2(cos(8πx) − sin(8πx))
ψ1,0(x) = 2

ψ1,1(x) = 2(cos(8πx) + sin(8πx))


 0 6 x < 14 ,

ψ2,−1(x) = 2(cos(8πx) − sin(8πx))
ψ2,0(x) = 2

ψ2,1(x) = 2(cos(8πx) + sin(8πx))


 14 6 x < 12 ,

ψ3,−1(x) = 2(cos(8πx) − sin(8πx))
ψ3,0(x) = 2

ψ3,1(x) = 2(cos(8πx) + sin(8πx))


 12 6 x < 34 ,

and

ψ4,−1(x) = 2(cos(8πx) − sin(8πx))
ψ4,0(x) = 2

ψ4,1(x) = 2(cos(8πx) + sin(8πx))


 34 6 x < 1 .

F = [
−1
4
, 0,

1

4
,
−1
4
, 0,

1

4
,
−1
4
, 0,

1

4
,
−1
4
, 0,

1

4
]T ,

The matrix K is calculated as follows:

199



Shyam Lal and Satish Kumar

Ki,j =

∫ 1
0

∫ 1
0

ψi(x)K(x,y)ψj(y)dydx

=

∫ 1
0

ψi(x) (

∫ 1
0

y2ψj(y)dy) dx

= (

∫ 1
0

ψi(x)dx) (

∫ 1
0

y2ψj(y)dy)

K =




π+1
64π2

1
96

−π+1
64π2

3π+1
64π2

7
96

−3π+1
64π2

5π+1
64π2

19
96

−5π+1
64π2

7π+1
64π2

37
96

−7π+1
96π2




[
0 1

2
0 0 1

2
0 0 1

2
0 0 1

2
0
]

200



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

K =




0 π+1
128π2

0 0 π+1
128π2

0 0 π+1
128π2

0 0 π+1
128π2

0

0 1
192

0 0 1
192

0 0 1
192

0 0 1
192

0

0 −π+1
128π2

0 0 −π+1
128π2

0 0 −π+1
128π2

0 0 −π+1
128π2

0

0 3π+1
128π2

0 0 3π+1
128π2

0 0 3π+1
128π2

0 0 3π+1
128π2

0

0 7
192

0 0 7
192

0 0 7
192

0 0 7
192

0

0 −3π+1
128π2

0 0 −3π+1
128π2

0 0 −3π+1
128π2

0 0 −3π+1
128π2

0

0 5π+1
128π2

0 0 5π+1
128π2

0 0 5π+1
128π2

0 0 5π+1
128π2

0

0 19
192

0 0 19
192

0 0 19
192

0 0
19

192
0

0 −5π+1
128π2

0 0 −5π+1
128π2

0 0 −5π+1
128π2

0 0 −5π+1
128π2

0

0 7π+1
128π2

0 0 7π+1
128π2

0 0 7π+1
128π2

0 0 7π+1
128π2

0

0 37
192

0 0 37
192

0 0 37
192

0 0 37
192

0

0 −7π+1
128π2

0 0 −7π+1
128π2

0 0 −7π+1
128π2

0 0 −7π+1
128π2

0




∴ U = (I − K)−1F

= [
−1
4
, 0,

1

4
,
−1
4
, 0,

1

4
,
−1
4
, 0,

1

4
,
−1
4
, 0,

1

4
]T (34)

Putting the values of Ψ(x) and U from eqns (33) and (34) in eqn (22), we have

u(x) = sin(8πx) (35)

which is the CAS wavelet solution of the integral equation (31) .

201



Shyam Lal and Satish Kumar

Legendre wavelet solution

Legendre wavelets ψ(L)n,m(t) = ψ(L)(k,n,m,t) having four arguments; k =
2, 3, ...,
2n − 1 , n = 1, 2, 3, ..., 2k−1, m is the order of the Legendre polynomial and t
is the normalised time, are defined by :

ψ(L)n,m(t) =

{
(m + 1

2
)
1
2 2

k
2 Pm(2

kt− 2n + 1), if n−1
2k−1

6 t < n
2k−1

,

0, otherwise.
(36)

where Pm(t) are Legendre ploynomials of order m (Rehman and Khan [7]). The
set {ψ(L)n,m}n,m∈Z of Legendre wavelets forms an orthonormal set. A function f ∈
L2[0, 1) may be expanded into Legendre wavelet series as:

f(t) =
∞∑
n=1

∞∑
m=0

cn,mψ
(L)
n,m(t), (37)

where cn,m =< f,ψ
(L)
n,m > .The series (37) may be truncated as:

(f)(t) ≈
2k−1∑
n=1

M−1∑
m=0

cn,mψ
(L)
n,m(t) = C

T Ψ(L)(t) (38)

where C and Ψ(L)(t) are 2k−1M × 1 matrices given by:

C = [c1,0,c1,1, ...,c1,M−1,c2,0, ...,c2,M−1, ...,

c2k−1,0, ...,c2k−1,M−1]
T

and

Ψ(L)(t) = [ψ
(L)
1,0 (t),ψ

(L)
1,1 (t), ...,ψ

(L)
1,M−1(t),ψ

(L)
2,0 (t), ...,ψ

(L)
2,M−1(t), ...,

ψ
(L)

2k−1,0
(t), ...,ψ

(L)

2k−1,M−1(t)]
T

Similarly, a function K ∈ L2[0, 1) ×L2[0, 1) may be approximated as:

K(x,y) ≈ (Ψ(L))T (x)K(L)Ψ(L)(y),

where K(L) is 2k−1M × 2k−1M matrix, whose entries are given by

K(L)i,j =< ψ
(L)
i (x),< K(x,y),ψ

(L)
j (y) >> . (39)

For Legendre wavelet solution, take M = 3,k = 3 in eqn (38), then twelve basis
functions are given by

202



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

Ψ(L)(x) = [ψ
(L)
1,0 (x),ψ

(L)
1,1 (x),ψ

(L)
1,2 (x),ψ

(L)
2,0 (x),ψ

(L)
2,1 (x),ψ

(L)
2,2 (x),

ψ
(L)
3,0 (x),ψ

(L)
3,1 (x),ψ

(L)
3,2 (x),ψ

(L)
4,0 (x),ψ

(L)
4,1 (x),ψ

(L)
4,2 (x)]

T (40)

where

ψ
(L)
1,0 (x) = 2

ψ
(L)
1,1 (x) = 2

√
3(8x− 1)

ψ
(L)
1,2 (x) =

√
5(3(8x− 1)2 − 1)


 0 6 x <

1

4
,

ψ
(L)
2,0 (x) = 2

ψ
(L)
2,1 (x) = 2

√
3(8x− 3)

ψ
(L)
2,2 (x) =

√
5(3(8x− 3)2 − 1)




1

4
6 x <

1

2
,

ψ
(L)
3,0 (x) = 2

ψ
(L)
3,1 (x) = 2

√
3(8x− 5)

ψ
(L)
3,2 (x) =

√
5(3(8x− 5)2 − 1)




1

2
6 x <

3

4
,

and

ψ
(L)
4,0 (x) = 2

ψ
(L)
4,1 (x) = 2

√
3(8x− 7)

ψ
(L)
4,2 (x) =

√
5(3(8x− 7)2 − 1)




3

4
6 x < 1 .

203



Shyam Lal and Satish Kumar

K(L) =




1
192

0 0 1
192

0 0 1
192

0 0 1
192

0 0

√
3

384
0 0

√
3

384
0 0

√
3

384
0 0

√
3

384
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

7
192

0 0 7
192

0 0 7
192

0 0 7
192

0 0

√
3

128
0 0

√
3

128
0 0

√
3

128
0 0

√
3

128
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

19
192

0 0 19
192

0 0 19
192

0 0 19
192

0 0

5
√
3

384
0 0 5

√
3

384
0 0 5

√
3

384
0 0 5

√
3

384
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

37
92

0 0 37
92

0 0 37
92

0 0 37
92

0 0

7
√
3

384
0 0 7

√
3

384
0 0 7

√
3

384
0 0 7

√
3

384
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0

√
5

1920
0 0




F(L) = [0,
−
√

3

2π
, 0, 0,

−
√

3

2π
, 0, 0,

−
√

3

2π
, 0, 0,

−
√

3

2π
, 0]T ,

U(L) = (I − K(L))−1F(L)

= [0,
−
√

3

2π
, 0, 0,

−
√

3

2π
, 0, 0,

−
√

3

2π
, 0, 0,

−
√

3

2π
, 0]T . (41)

Putting the values of Ψ(L)(x) and U(L) from eqns (40) and (41) in eqn (22), we get
the Legendre wavelet solution of the integral equation (31) as:

u(x) = −
√

3

2π
ψ

(L)
1,1 (x) −

√
3

2π
ψ

(L)
2,1 (x) −

√
3

2π
ψ

(L)
3,1 (x) −

√
3

2π
ψ

(L)
4,1 (x) (42)

204



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

Haar wavelet solution
The Haar wavelet family for x ∈ [0, 1] is defined as follows:

hi(x) =




1 if x ∈ [ k
m
,
k+ 1

2

m
),

−1 ifx ∈ [k+
1
2

m
, k+1
m

),

0, otherwise

(43)

where m = 2b, b = 0, 1, ...,J is the level of wavelet; k = 0, 1, ...,m − 1 is
the translation parameter. J is the maximum level of resulution. i is calculated by
i = m + k + 1. The minimum value of i for m = 1,k = 0 is 2. The maximum
value of i is i = 2M = 2J+1 (Arbabi and Darvishi [6]).

For i = 1, h1(x) is taken to be scaling function which is defined as follows:

h1(x) =

{
1 if x ∈ [0, 1),
0, otherwise

Any function f(x) can be expressed in terms of Haar wavelets as follows:

f(x) =
2M∑
i=1

aihi(x), (44)

where the wavelet coefficients ai, i = 1, 2, ..., 2M are to be determined. For Haar
wavelet solution take J = 3 in eqn (43), b = 0, 1, 2, 3 , then m = 2b = 1, 2, 4, 8.
By eqns (28) the Haar wavelet coefficients ai, i = 1, 2, ..., 16 are given by

[−0.008071, 0.001459, 0.002497, 0.001447, 0.000485, 0.006380,
0.000488,−0.000476, 1.000010, 1, 1, 0.988178, 1, 1, 0.999039, 1] (45)

Putting these values of ai in the eqn (26), we get the solution of integral equation
(31) by Haar wavelet method. The Haar wavelet solutions of integral eqn 31 are
shown in the Table (1).

The exact solution and approximate solutions of Fredholm integral equation
(31) obtained by CAS wavelet, Legendre wavelet and Haar wavelet method for
different values of x are given in the Table (1).

205



Shyam Lal and Satish Kumar

Table (1)
x Exact soln CAS wavelet soln Legendre wavelet soln Haar wavelet soln

by eqn 32 by eqn (35) by eqn (42) by eqn (26)
0 -0.059680 0 0.954930 0.996370

0.1 0.528105 0.587785 0.190986 -1.003630
0.2 -1.010736 -0.951056 -0.572958 0.995399
0.3 0.891376 0.951056 0.572958 0.995399
0.4 -0.647465 -0.587785 -0.190986 0.972689
0.5 -0.059680 0 -0.954930 0.992404
0.6 0.528105 0.587785 0.190986 -1.008083
0.7 -1.010736 -0.951056 -0.572958 -1.007595
0.8 0.891376 0.951056 0.572958 0.987585
0.9 -0.647465 -0.587785 -0.190986 0.989498

The graphs of the exact solution and approximate solutions of integral equation
(31) obtained by CAS wavelet, Legendre wavelet and Haar wavelet method are
shown in the Fig.(1).

Fig.(1)

By numerical comparison in Table(1) and graphs shown in Fig.(1), it is clear
that the solution of Fredholm integral equation (31) by CAS wavelet method is
better than solutions obtained by Legendre wavelet and Haar wavelet methods.

206



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

Example 2
Consider the Fredholm integral equation:

u(x) = sin(4πx) +
∫ 1
0

xyu(y)dy . (46)

It is obtained by subtituting f(x) = sin(4πx) and K(x,y) = xy , in the Fredholm
integral equation (1). The exact solution of Fredholm integral equation (46) is
given by

u(x) = sin(4πx) −
3x

8π
(47)

CAS wavelet solution
For CAS wavelet solution, take k = 1,M = 1 in eqn (14), then following the

procedure of example (31), we have

F∗ = [
−1

2
√

2
, 0,

1

2
√

2
,
−1

2
√

2
, 0,

1

2
√

2
]T ,

The matrix K∗ is calculated as follows:

K∗i,j =

∫ 1
0

∫ 1
0

ψi(x)K(x,y)ψj(y)dydx

=

∫ 1
0

ψi(x) (

∫ 1
0

xyψj(y)dy) dx

= (

∫ 1
0

xψi(x)dx) (

∫ 1
0

yψj(y)dy)

K =




√
2

8π

√
2
8

−
√
2

8π

√
2

8π

√
2
8

−
√
2

8π




[ √
2

8π

√
2
8
−
√
2

8π

√
2

8π

√
2
8
−
√
2

8π

]

207



Shyam Lal and Satish Kumar

K∗ =




1
32π2

1
32π

−1
32π2

1
32π2

1
32π

−1
32π2

1
32π

1
32

−1
32π

1
32π

1
32

−1
32π

−1
32π2

−1
32π

1
32π2

−1
32π2

−1
32π

1
32π2

1
32π2

1
32π

−1
32π2

1
32π2

1
32π

−1
32π2

3
32π

3
32

−3
32π

3
32π

3
32

−3
32π

−1
32π2

−1
32π

1
32π2

−1
32π2

−1
32π

1
32π2




and

U∗ = [
−1

2
√

2
, 0,

1

2
√

2
,
−1

2
√

2
, 0,

1

2
√

2
]T .

u(x) = 1.0188 sin(4πx) − 0.0294 (48)

This is the approximate solution of the integral equation (46) by CAS wavelet
method.

Legendre wavelet solution

For Legendre wavelet solution, take M = 3,k = 2 in eqn (38), then we have

Ψ(L)(x) = [ψ
(L)
1,0 (x),ψ

(L)
1,1 (x),ψ

(L)
1,2 (x),ψ

(L)
2,0 (x),ψ

(L)
2,1 (x),ψ

(L)
2,2 (x)]. (49)

Following the procedure of the example (1), we have

(F∗)(L) = [0,
−
√

6

2π
, 0, 0,

−
√

6

2π
, 0]T ,

(U∗)(L) = [−0.0211,−0.4020, 0,−0.0633,−0.4020, 0]T (50)

Putting the values of Ψ(L)(x) and (U∗)(L) from eqns (49) and (50) in eqn (22), we
get the solution of the integral equation (46) by Legendre wavelet method as

u(x) = −0.0211ψ(L)1,0 (x)−0.4020ψ
(L)
1,1 (x)−0.0633ψ

(L)
2,0 (x)−0.4020ψ

(L)
2,1 (x) (51)

208



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

(K∗)(L) =




1
32

√
3

96
0 3

32

√
3

96
0

√
3

96
1
96

0
√
3

32
1
96

0

0 0 0 0 0 0

3
32

√
3

32
0 9

32

√
3

32
0

√
3

96
1
96

0
√
3

32
1
96

0

0 0 0 0 0 0




Haar wavelet solution

For Haar wavelet solution, take J = 2 in eqn (43),b = 0, 1, 2 then m = 2b =
1, 2, 4. The Haar wavelet coefficients ai, i = 1, 2, ..., 8 are given by

[0.061361, 0.027885, 0.616015, 0.670955, 0.000922, 1.465270, 0.000264, 0.004906]

Putting these values of ai in the eqn (26), we get the solution of integral equation
(46) by Haar wavelet method. The Haar wavelet solutions of integral eqn 46 are
given in the Table (2).

The exact solution and approximate solutions of Fredholm integral equation
(46) obtained by CAS wavelet, Legendre wavelet and Haar wavelet method for
different values of x are given in the Table (2).

Table (2)
x Exact soln CAS wavelet soln Legendre wavelet soln Haar wavelet soln

by eqn (47) by eqn (48) by eqn (51) by eqn (26)
0 0 -0.0294 0.9549 0.6955

0.1 0.9391 0.9395 0.5610 0.6955
0.2 0.5639 0.5694 0.1671 0.6722
0.3 -0.6236 -0.6282 -0.2268 -0.7701
0.4 -0.9988 -0.9983 -0.6207 -1.7239
0.5 -0.0597 -0.0294 0.8952 0.6194
0.6 0.8794 0.9395 0.5013 0.6194
0.7 0.5042 0.5694 0.1074 -0.3672
0.8 -0.6833 -0.6282 -0.2865 -0.8337
0.9 -1.0585 -0.9983 -0.6803 -0.8532

209



Shyam Lal and Satish Kumar

The graphs of the exact solution and approximate solutions of integral equation
(46) obtained by CAS wavelet, Legendre wavelet and Haar wavelet method are
shown in the Fig.(2).

Fig.(2)

By numerical comparison in Table(2) and graphs shown in Fig.(2), it is observed
that the solution of Fredholm integral equation (46) by CAS wavelet method is
more accurate than solutions obtained by Legendre wavelet and Haar wavelet
methods.
Note: The solutions of Fredholm integral equations in examples (1) and (2) by
CAS wavelet method propoesd in this research paper and their numerical
comparison with Legendre wavelet and Haar wavelet methods show the
advantages of CAS wavelet method than Legendre wavelet and Haar wavelet
methods.

8 Remarks

1. CAS wavelet approximation of Theorem (5.1) is given by
E

(1)

2k,2M+1
(f) = O( 1√

M+1 2k(α+1)
) . E(1)

2k,2M+1
(f) → 0 as M →∞, k →∞ .

CAS wavelet approximation of Theorem (5.2) is given by
E

(2)

2k,2M+1
(f) = O( 1

(M+1)
3
2 2k(α+2)

) . E(2)
2k,2M+1

(f) → 0 as M →∞, k →∞ .

Therefore, estimators E(1)
2k,2M+1

(f) and E(2)
2k,2M+1

(f) are best possible in wavelet

210



CAS wavelet approximation of functions of Hölder’s class Hα[0, 1)...

analysis (Zygmund [5]).

2. ∵ (M + 1)
3
2 2k(α+2) > (M + 1)

1
2 2k(α+1), M > 1,k > 1

∴
1

(M + 1)
3
2 2k(α+2)

6
1

(M + 1)
1
2 2k(α+1)

ie. E
(2)

2k,2M+1
(f) 6 E

(1)

2k,2M+1
(f).

Hence, estimator E(2)
2k,2M+1

(f) is sharper than estimator E(1)
2k,2M+1

(f) . This shows
that the estimator of a function f having f ′′ ∈ Hα[0, 1) is sharper than the estima-
tor of f having f ′ ∈ Hα[0, 1).

3. CAS wavelet method is more effective than Legendre wavelet and Haar wavelet
method in finding the solution of Fredholm integral equations (31) and (46).

4. Fredholm integral equation of first kind,∫ 1
0

K(x,t)y(t)dt = f(x)

can be solved by CAS wavelet method as follows:∫ 1
0

Ψ(x)KΨT (t)Ψ(t)Y = Ψ(x)F

Using orthonormality of CAS wavelet, we get KY = F . By finding the matrix K
and F as in the case of Fredholm integral of second kind, we can find Y and hence
the solution y(x).

9 Acknowledgments
Shyam Lal, one of the authors, is thankful to DST - CIMS for encouragement

to this work.

Satish Kumar, one of the authors, is grateful to C.S.I.R. (India) for provid-
ing financial assistance in the form of Junior Research Fellowship vide Ref. No.
17/12/2017 (ii) EU-V Dated:13-02-2019 for his research work.

Authors are greatful to the referee for his valuable comments and suggestions,
to improve the quality of the research paper.

211



Shyam Lal and Satish Kumar

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[3] S. Saha Ray and P.K. Sahu :Numerical Methods for Solving Fredholm
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