The aim of this book is to propose a new algebraic approach to the study of norm stability of operator sequences which arise, for example, via discretization of singular integral equations on composed curves. A wide variety of discretization methods, including quadrature rules and spline or wavelet approximations, are covered and studied. The book is addressed to a wide audience, in particular to mathematicians working in operator theory and Banach algebras, as well as to applied mathematicians and engineers interested in theoretical foundations of various methods in general use, particularly splines and wavelets.