Difference Spaces and Invariant Linear Forms

Difference spaces arise by taking sums of finite or fractional differences. Linear forms which vanish identically on such a space are invariant in a corresponding sense. The difference spaces of L2 (Rn) are Hilbert spaces whose functions are characterized by the behaviour of their Fourier transforms near, for example the origin. The aim of this text is to establish connections between these spaces and differential operators, singular integral operators and wavelets. It discusses aspects of these ideas which emphasize invariant linear forms on locally compact groups. The work primarily presents new results, but does so from a unified viewpoint, which emphasises connections with related work.