A generalized approach to local regularization of linear Volterra problems in Lp spaces

Abstract

A generalized version of local regularization is developed. The convergence criteria set forth in the generalization are shown to be realized naturally by a particular local regularization scheme when applied to finitely-smoothing linear Volterra convolution equations in the Banach spaces Lp(0, 1), 1 ⩽ p ⩽ ∞. The method leads to convergence with a priori parameter selection for 1 < p < ∞ and under assumptions of increased regularity of the true solution for 1 ⩽ p ⩽ ∞. Rates of convergence are established beyond those previously known for local regularization of this problem in C[0, 1] and under more general source conditions. Numerical examples are included to illustrate implementation and effectiveness of the method.