Representation theorems for semilocal and bounded linear shift-invariant operators on sequences
Paul Hughett · Signal Processing · 1998
Abstract Not every linear shift-invariant operator on two-sided sequences can be represented by a convolution. An operator defined on sequences that converge to zero in both directions is representable if and only if it is bounded in the sup norm. An operator defined on bounded sequences is representable if and only if it is semilocal in the sense that the error introduced by neglecting the tails of the input sequence can be made as small as desired. In either case, a representable operator has an absolutely summable impulse response.