# Iterative Matrix Techniques Based on Averages
## 5 Solution of a Fredholm Integral Equation of the First Kind by Means of Orthogonal Polynomials
: _In order to solve the following Fredholm integral equation of the first kind_  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]1  
_we used the Legendre polynomials defined in the interval \([0,1]\):_  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]2  
_We have considered an expansion with \(q=4\) and obtained a system of \(5\times 5\) linear equations with matrix \(C\in\mathcal{M}(5\times 5)\),_  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]3  
_The unknowns \(V=(v_{0},\dots,v_{4})\) are the coefficients of \(v\) with respect to (\(p_{m}\)) and \(U=(0.5,0.288675,0,0,0)\) is the vector of coefficients of \(u\) with respect to the same basis. The system \(U=CV\) has been transformed into \(V=(I-C)V+U\)._  
_Since \(\rho(I-C)\simeq 0.91\) (close to \(1\)), we have applied the N-algorithm with all the parameters equal to \(1/2\). Starting at \(V_{0}=(0.5,0.5,0.5,0.5,0.5)\), we obtain at the \(18\)-th iteration the approximate solution:_  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]4  
_with an error of_  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]5  
_and stop criterion \(E_{n}<10^{-3}\). Figure 3 displays the \(6 th\), \(12 th\) and \(18 th\) approximation function to the solution of the equation. For the \(18 th\) Picard iteration, \(\bar{v}(x)\), the error is as follows:_  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]6  
_Consequently, the N-iteration may be better even if the Picard method can be applied._  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]7 Sixth (blue), twelve (yellow) and eighteen (green) approximations of the solution of a Fredholm equation. Spectrum of a Compact Operator and General Solution of Fredholm Integral Equations of the First Kind  
In this section we explore the relationships between spectral theory and the resolution of a Fredholm integral equation of the first kind given by following the expression:  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]8  
where \(f,g\in\mathcal{L}^{2}([a,b])\), and \(K:\mathcal{L}^{2}([a,b])\to\mathcal{L}^{2}([a,b])\) is defined as  
\[p_{n}(x):=P_{n}(x)/||P_{n}(x)||_{2}=\sqrt{\frac{2n+1}{2}}P_{n}(x),\]9  
where \(k\in\mathcal{L}^{2}([a,b]\times[a,b])\). It is well known that \(K\) is a linear and compact operator. A linear map \(K:E\to E\), where \(E\) is a normed space, is self-adjoint if \(K=K^{*}\) where \(K^{*}\) is the adjoint of \(K\). The set of eigenvalues of a linear operator is called the point spectrum of \(K\), \(\sigma(K)\), and \(\lambda\in\sigma(K)\) if there exists \(v\
eq 0\) such that \(Kv=\lambda v\). In this case \(v\) is an eigenvector of \(K\) (or an eigenfunction, in the case where \(E\) is a functional space). The set \(E_{\lambda}=\{v\in E:Kv=\lambda v\}\) is the eigenspace associated with \(\lambda\). The next theorems describe the spectrum of compact operators (see the references [26; 27], for instance), and provide a kind of \"diagonalization\" of these maps. Theorem 6.1: _Let \(K:E\to E\) be a linear, compact and self-adjoint operator. Then \(K\) has an orthonormal basis of eigenvectors and_  
* _If the cardinal of_ \(\sigma(K)\) _is not finite then the spectrum is countably infinite and the only cluster point of_ \(\sigma(K)\) _is_ \(0\)_, that is to say,_ \(\lim_{m\to\infty}\lambda_{m}=0\) _where_ \(\lambda_{m}\in\sigma(K)\)_._
* _The eigenspaces_ \(E_{\lambda}\) _corresponding to different eigenvalues are mutually orthogonal._
* _For_ \(\lambda\
eq 0\)_,_ \(E_{\lambda}\) _has finite dimension._  
According to this theorem, the eigenvalues of a compact self-adjoint operator can be ordered as follows:  
\[f(x)=\sum_{m=0}^{\infty}c_{m}p_{m}(x),\]0  
and the action of \(K\) on every element \(u\in E\) can be expressed as follows:  
\[f(x)=\sum_{m=0}^{\infty}c_{m}p_{m}(x),\]1  
where \((e_{m})\) is an orthonormal basis of eigenvectors of \(K\). If \(\sigma(K)\) is finite, then \(K\) has a finite range.