Direct form seminorms arising in the theory of interpolation by translates of a basis function

In the error analysis of the process of interpolation by translates of a single basis function, certain spaces of functions arise naturally. These spaces are de ned with respect to a seminorm which is given in terms of the Fourier transform of the function. We call this an indirect seminorm. In certain well-understood cases, the seminorm can be rewritten trivially in terms of the function, rather than its Fourier transform. We call this a direct seminorm. The direct form allows better error estimates to be obtained. In this paper, we show how to rewrite most of the commonly arising indirect form seminorms in direct form, and begin a little of the analysis required to obtain the improved error estimates.