AN APPROXIMATE SOLUTION OF FREDHOLM INTEGRAL EQUATION OF THE SECOND KIND BY BAND-LIMITED SCALING FUNCTION

Fredholm integral equation of the second kind appears naturally in different areas of mathematics, physics and engineering. In this paper we consider Fredholm integral equation of the second kind with convolution type kernel. We assume f in the approximation space and prove the existence and uniqueness of approximate solution by band-limited scaling function. Since these functions are infinitely differentiable and possess decay property, methods based on these functions would be very accurate. Convergence analysis has been discussed to validate our approximate solution. AMS Subject Classification: 41B05, 45L05, 65G99


Introduction
Integral equations arise naturally in various fields of science and engineering [1]. These equations are encountered in a variety of applications from many fields including potential theory, mathematical physics, quantum mechanics, acoustics and fluid mechanics [2,3]. Because of its applicability in various fields, these equations have been studied extensively both at theoretical and compuational level; see [1,2,3]. They are closely related to differential equations. A large class of initial and boundary value problems can be converted to Volterra or Fredholm integral equations.
In this paper we study existence and uniqueness of the approximate solution of Fredholm integral equation of the second kind with convolution type kernel which is given by Here u(x) is the unknown function while f (x) is a known right-hand side. The function of two variable k(x, y) is called kernel. The subject area of wavelets, developed mainly in the last two decades, has attracted people from various branches of science and technology. Due to time-frequency localization property, these functions are widely used in image processing, signal processing and data compression [6,7]. Haar wavelet is the simplest example of wavelet but it is not smooth. Smoother class of wavelets were constructed by Meyer in [17], Lemarie in [18], Daubechies in [8], Battle in [19] and Stromberg in [16]. We use infinitely differentiable band-limited scaling function to get the approximate solution of Fredholm integral equation of the second kind.
In the past few years many authors have studied approximate solution of Fredholm integral equation of the second kind using wavelets; see [10,11,12,13]. Authors have used Haar wavelets, multiwavelets and B-spline wavelets. Haar wavelets and multiwavelets do not have sufficient regularity while B-spline wavelets does not form orthonormal basis. Regularity in basis functions and orthonormality properties, both are essential to get accurate solution of integral equations. In 2005, Shim et al. used Meyer wavelet to prove the existence of solution of Fredholm integral equation of the first kind. We provide an alternate way to find approximate solution of Fredholm integral equation of the second kind. We apply infinitely differentiable band-limited scaling function which is generated by a class of band-limited wavelets. Since these band-limited scaling functions are infinitely differentiable and possess sufficient decay property, methods based on these functions would provide accurate results for both partial differential equations and integral equations. Also convergence analysis in the setting of band-limited wavelets can be done easily comparing to other wavelets.
The content of this paper is organized as follows. In Section 2, we summarize some basics of orthonormal wavelets and multiresolution analysis. In Section 3, we consider a class of band-limited wavelets and prove a lemma. In Section 4, we prove the existence of approximate solution by band-limited scaling function generated by a class of band-limited wavelets. Convergence analysis is studied in Section 5. Section 6 concludes the paper and gives a brief idea of the future work.

Orthonormal Wavelets and Multiresolution Analysis
Good references for orthonormal wavelets and multiresolution analysis are [4,5,6,7,8,17]. Wavelets are square integrable functions generated from one scaling function by dilations and translations. These functions possess several properties such as arbitrary regularity, high order vanishing moments and an ability to represent functions at different resolutions.
Definition 1 (see [5]). An orthonormal wavelet on R is a function Definition 2 (see [5]). A multiresolution analysis (MRA) of L 2 (R) consists of a sequence of closed subspaces V j (j ∈ Z) of L 2 (R) satisfying for all j ∈ Z; (v) There exists a function φ ∈ V 0 (known as scaling function) such that {φ(· − k) : k ∈ Z} is an orthonormal basis for V 0 .

Remark 3. Most of the orthonormal wavelets are constructed using multiresolution analysis theory.
Approximation space V j is defined as In a similar way, wavelet space W j is defined as Some properties of scaling function in the frequency domain: The orthogonal projection P j : where . denotes the standard inner product in L 2 (R). In a similar way, a function can be represented in wavelet space. The orthogonal projection Q j : L 2 (R) → W j is defined as Since the family {ψ [j,k] (x) : j, k ∈ Z } forms an orthonormal basis for L 2 (R), every function f ∈ L 2 (R) admits the L 2 (R) convergent wavelet expansion Definition 4 (see [5]). A scaling function φ ∈ L 2 (R) is said to be bandlimited scaling function if support ofφ is contained in a finite interval.
These functions are used to get the approximate solution of Fredholm integral equation of the second kind.

A Class of Band-Limited
Wavelets, see [5], [9] In this section, we consider all orthonormal wavelets ψ with supp (ψ) con- All these wavelets are characterized by some results which we present below. Then we prove a lemma that provide us the relationship between scaling function and wavelet in the frequency domain.
Definition 5 (see [5]). An orthonormal wavelet ψ ∈ L 2 (R) is said to be band-limited wavelet if support ofψ is contained in a finite interval.
Example 6. A function ψ ∈ L 2 (R) such that its Fourier transform is given byψ then ψ is said to be Shannon wavelet.
The following two theorems completely characterize all orthonormal wavelets ψ for whichψ has support contained in [ −8π Then ψ is an orthonormal wavelet if and only if: . Theorem 8 (see [5] 3 }, from remark 10, we can choose α(ξ) to be any measurable function; in particular we can take α(ξ) = ξ 2 .
make ψ an orthonormal wavelet.
Now fix a ξ such that −2π 3 < ξ < 0. It is clear that there exists a unique n ∈ N s.t.

Convergence Analysis
Theorem 15 (see [14]). Let f ∈ (L 2 ∩ C k )(R), k > 0, φ and ψ be scaling function and wavelet respectively. Assume that φ, ψ ∈ C β for some 0 < β < k. Also, assume that φ, ψ satisfy the following decay condition: for some constants c, ǫ > 0. For J > 0, define the projection operator P J by Then P J (f ) converges to f in the . ∞ norm. Moreover, P J (f ) − f ∞ ≤ c2 −Jk for some constant c depending only on f .
Remark 16. We note that theorem 15 is written in our paper's convention.
Proof. Since u is the exact solution, it satisfies Similarly, the approximate solution u J+1 satifies From (5) and (6), we have Taking Fourier transform on both sides, we get It follows that This implies (10), Plancherel theorem [5] and assumption of the theorem imply that From remark 17, f J → f in L 2 -norm as J → ∞. So we can conclude that u J+1 → u in L 2 -norm.

Conclusion and Future Work
We have established the existence and uniqueness of approximate solution of Fredholm integral equation of the second kind with convolution type kernel using band-limited scaling function. Convergence analysis has been discussed to validate our method. In future, we will use these scaling functions to get the local solution of Fredholm integral equations. Since these band-limited scaling functions are infinitely differentiable and possess sufficient decay property, these functions would be very effective for the numerical solution of integral equations.