In 1972 Gary Meisters and Wolfgang Schmidt showed that if the Fourier coefficient of a function is zero at the origin, then the function is a sum of three first order differences. They deduced from this that every translation invariant linear form on the space of square integrable functions on the circle group is automatically continuous. Their results were subsequently developed by the author for the case of n-dimensional Euclidean space (Lectures Notes in Mathematics, vol. 1586, Springer-Verlag,1994). The present paper gives an exposition of some of the ideas in this work, with the aim of minimising the technicalities. In particular, it is shown that if we differentiate s times the functions in the Sobolev space of order s on the real line, we obtain as the range of this operator a difference space of order s, where the form of the differences in this space can be explicitly described. This difference space is a Hilbert space, and the operation of differentiation s times is an isometry between these spaces. Whereas finite differences have long been used to approximate derivatives “locally”, these results reveal the precise “global” relationship. The main technique involves understanding how the operation of taking a derivative or a finite difference of a function affects the behaviour of the Fourier transform of the function near the origin on the real line.
Available at: http://works.bepress.com/rnillsen/9/