The Inversion of Sampling Solved Algebraically

We show that Shannon’s reconstruction formula can be written as a ∗ (b · c) = c = (a ∗ b) · c with tempered distributions a, b, c ∈ S′(􏰓n) where ∗ is convolution, · is multiplication, c is the function being sampled and restored after sampling, b· is sampling and a∗ its inverse. The requirement a ∗ b = 1 which describes a smooth partition of unity where b = III is the Dirac comb implies that a is satisfied by unitary functions introduced by Lighthill (1958). They form convolution inverses of the Dirac comb. Choosing a = sinc yields Shannon’s reconstruction formula where the requirement a ∗ b = 1 is met approximately and cannot be exact because sinc is not integrable. In contrast, unitary functions satisfy this requirement exactly and stand for the set of functions which solve the problem of inverse sampling algebraically.