# Iterative Matrix Techniques Based on Averages
## 7 Solution of a Fredholm Integral Equation of the First Kind with Separable Kernel  
We consider now the case where \(K\) has a separable kernel \(k\), that is to say, \(k\) can be expressed as  
\[k(x,y)=\sum_{j=1}^{r}\alpha_{j}(x)\beta_{j}(y),\]  
for some \(r\in\mathbb{N}\). **Spectrum of \(K\)**:  
As suggested by the equality (13), we can guess that \(K\) has eigenfunctions of type  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]  
In order to find \(\varphi\), we consider that  
\[K\varphi(x)=\int_{a}^{b}(\sum_{j=1}^{r}\alpha_{j}(x)\beta_{j}(y))(\sum_{j=1}^ {r}c_{j}\alpha_{j}(y))dy=\sum_{i,j=1}^{r}c_{j}\alpha_{j}(x)\bigg{(}\int_{a}^{ b}\beta_{i}(y)\alpha_{j}(y)dy\bigg{)},\]  
\[K\varphi(x)=\sum_{i,j=1}^{r}A_{ij}c_{j}\alpha_{i}(x)=\lambda\varphi(x),\]  
where  
\[A_{ij}=\int_{a}^{b}\beta_{i}(y)\alpha_{j}(y)dy, \tag{14}\]  
and the problem is reduced to solving the eigenvalue issue:  
\[A.c=\lambda c,\]where \(c=(c_{1},c_{2},\ldots c_{r})^{T}\) is the vector of coefficients of \(\varphi\) with respect to \((\alpha_{j})\). **Null space of \(K\)**:  
Let us consider now the null space of the operator \(K\). If \(h\in Ker(K)\),  
\[Kh(x)=\int_{a}^{b}k(x,y)h(y)dy=\sum_{j=1}^{r}\alpha_{j}(x)\int_{a}^{b}\beta_{j} (y)h(y)dy.=\sum_{j=1}^{r}<\beta_{j},h>\alpha_{j}(x)=0.\]  
If the maps \(\alpha_{j}\) are linearly independent, a characterization of the functions belonging to the null space is as follows:  
\[<h_{t},\beta_{j}>=0,\]  
for \(j=1,2,\ldots,r\). Let us consider an orthogonal system of functions \((p_{j})\) for \(j=1,2,\ldots,r\) defined following a Gram-Schmidt process from \((\beta_{j})\). If \(w\in\mathcal{L}^{2}([a,b])\) is any function,  
\[\bar{w}=w-\sum_{j=1}^{r}\frac{<w,p_{j}>}{<p_{j},p_{j}>}p_{j}\]  
belongs to the null space of \(K\) since  
\[<\bar{w},p_{i}>=0\]  
for \(i=1,2,\ldots,r\). In order to define an orthonormal basis of the kernel, let us consider a system of functions \((h_{i})\) such that \(\{p_{1},p_{2},\ldots p_{r},h_{1},h_{2},...\}\) is linearly independent and complete. Then, we define  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]0  
A Gram-Schmidt process on \((\bar{h}_{i})\) provides an orthonormal basis \((\psi_{j})\) of the null space of \(K\). **General solution of \(Kf=g\)**:  
The general solution of the equation \(Kf=g\) has the following form:  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]1  
where \(\varphi_{j}\) are the eigenfunctions, the coefficients \(a_{j}\) are to be computed and the \(b_{j}\) are arbitrary. To find \(a_{j}\) let us consider that  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]2  
and we have a linear system of \(r\) equations with unknown \(a_{j}\)'s, \(j=1,2,\ldots,r\). _Example_  
Consider the following Fredholm integral equation of the first kind:  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]3  
Let us study the characteristics of the operator \(K\). **Spectrum of \(K\)**:  
The first eigenvalue of the operator was computed in Example 5, obtaining a value of \(\lambda_{1}=1.709201\). The second eigenvalue was found by applying the averaged power method,but subtracting at every step the component of the approximation with respect to the first eigenfunction (\(\varphi_{1}\)). The value obtained is \(\lambda_{2}=-0.292534\). The maps of the kernel \(k\) are as follows:  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]4  
\[\varphi=\sum_{j=1}^{r}c_{j}\alpha_{j}.\]5  
Let us define the matrix \(A\) as in (14), \(A_{ij}=\int_{0}^{1}\beta_{i}(y)\alpha_{j}(y)dy:\)  
\[A= \begin{pmatrix}2/3&2\\ 1/2&3/4\end{pmatrix} \]  
The spectrum of the matrix is \(\sigma(A)=\{1.709201,-0.292534\}\). Two eigenfunctions associated with these eigenvalues were computed as follows: \(\varphi_{1}(x)=0.575913+1.104832x^{2}\) and \(\varphi_{2}(x)=1-2.08508x^{2}\). Since the map \(g\in span(\varphi_{1},\varphi_{2})\), the equation has a solution. **Null space of \(K\)**:  
As explained previously, the maps \(h\) belonging to the null space of \(K\) are characterized by the following equations:  
\[<h,\beta_{i}>=0,\]  
for \(i=1,2\).