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The ranges of a certain type of second order differential operator, on aSobolev subspace of the Lebesgue space $L^2$ of the circle group, can becharacterised by the vanishing of the Fourier coefficients at (generally) twointegers that are the zeros of the multiplier of the operator. It is provedhere that the range of any such operator may be alternatively described ascomprising those functions in $L^2$ that are the sum of five generalised secondorder differences, each such difference involving the zeros of the multiplier.In fact, higher order operators and differences are considered. There areapplications to automatic continuity of linear forms on $L^2$. This work isrelated to earlier work of G. Meisters and W. Schmidt who derived, in effect, adescription of the range of the ordinary differentiation operator D (whosemultiplier vanishes at 0) in terms of first order differences.

Generalised differences and a class of multiplier operators in Fourier analysis