# Is projection method really applicable for numerical solution of linear integral equations in $L^p \ (p \neq 2)$ setting?

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!