Projection method is a traditional method to numerically handle problem of linear integral equation. The routine way is to do it in $ L^2 $ setting. For example:

Let $ A:L^2(a,b) \to L^2(a,b) $ be a compact injective operator, introduce a basis subspace $ X_n := \{ \xi_k\}^n_{k=1} $, which is increasing and eventually dense in $ L^2(a,b) $, that is, \begin{equation*} X_n \subseteq X_{n+1}, \overline{\cap_{n \in \mathbb{N}} X_n} =L^2(a,b) \end{equation*} then define a sequence of orthogonal projection operator $ \{ P_n \} $, which project $ L^2(a,b) $ onto $ X_n $. Now for $ y \in \mathcal{R}(A) $, $ A^\dagger_n y_n $ could be a natural approximate scheme to $ A^{-1} y $. where \begin{equation*} A_n := P_n A P_n : X_n \longrightarrow X_n \ \text{and} \ y_n := P_n y \in X_n \end{equation*} (Of course we could describe above system in a inner product form.) The convergence result could be seen in [Theorem 13.6] of

R. Kress: Linear Integral Equations, Springer-Verlag, Berlin, 1989.

However, we notice that the convergence analysis permits the compact operator defined in Banach space ($ L^p $), but we have not seen a numerical example with projection procedure in $ L^p $ setting. So we want to know if this result in $ L^p $ setting is only of theoretical importance? If not, please indicate some references with numerical example which handle linear integral equation with projection method in $ L^p $ setting.

Thank you in advance!

Projection method is a traditional method to numerically handle problem of linear integral equation. The routine way is to do it in $ L^2 $ setting. For example:

Let $ A:L^2(a,b) \to L^2(a,b) $ be a compact injective operator. We introduce a sequence of basis subspace $ X_n := \{ \xi_k\}^n_{k=1} $, which is increasing and eventually dense in $ L^2(a,b) $, that is, \begin{equation*} X_n \subseteq X_{n+1}, \overline{\bigcup_{n \in \mathbb{N}} X_n} =L^2(a,b). \end{equation*} Then define a sequence of orthogonal projection operator $ \{ P_n \} $, which project $ L^2(a,b) $ onto $ X_n $. Now for $ y \in \mathcal{R}(A) $, $ A^\dagger_n y_n $ could be a natural approximate scheme to $ A^{-1} y $, where \begin{equation*} A_n := P_n A P_n : X_n \longrightarrow X_n \ \text{and} \ y_n := P_n y \in X_n \end{equation*} (Of course we could describe above system in a inner product form). The convergence result could be seen in [Theorem 13.6] of

R. Kress: Linear Integral Equations, Springer-Verlag, Berlin, 1989.

However, we notice that the convergence analysis permits the compact operator $ A $ to be defined in Banach space ($ L^p $), but we have not seen a numerical example with projection procedure in $ L^p $ setting. So we want to know if this convergence result in $ L^p $ setting is of practical value? or only of theoretical importance? 

If the former, please indicate some references with numerical example which handle linear integral equation with projection method in $ L^p $ setting.

Thank you in advance!