The authors present a unique Banach algebra approach to the stability analysis of spline approximation methods for singular integral operators on the Euclidean space \(\mathbb{R}^ n\), which occur as local representatives of pseudodifferential operators on closed manifolds or on manifolds with a smooth boundary. Thus the paper may be regarded as a first step in establishing spline approximation methods for pseudodifferential equations on manifolds. After giving a treatise of the Fredholm theory in \(L_ p\) spaces, the authors study algebras of approximation sequences for large classes of convolution operators, leading to necessary and sufficient conditions for the stability of Galerkin, collocation and qualocation methods based on spline functions.