Error estimates for spaces arising from approximation by translates of a basic function

We look at aspects of error analysis for interpolation by translates of a basic function. In particular, we consider ideas of localisation and how they can be used to obtain improved error estimates. We shall consider certain seminorms and associated spaces of functions which arise in the study of such interpolation methods. These seminorms are naturally given in an indirect form, that is in terms of the Fourier Transform of the function rather than the function itself. Thus, they do not lend themselves to localisation. However, work by Levesley and Light [17] rewrites these seminorms in a direct form and thus gives a natural way of defining a local seminorm. Using this form of local seminorm we construct associated local spaces. We develop bounded, linear extension operators for these spaces and demonstrate how such extension operators can be used in developing improved error estimates. Specifically, we obtain improved L2 estimates for these spaces in terms of the spacing of the interpolation points. Finally, we begin a discussion of how this approach to localisation compares with alternatives.