A note on Unser-Zeruhia generalized sampling theory applied to the linear interpolator

In this correspondence, we calculate the condition number of the linear operator that maps sequences of samples f(2k), f(2k+a), k∈Z of an unknown continuous f∈L² (R) consistently (in the sense of the Unser-Zeruhia generalized sampling theory) onto the set of continuous, piecewise linear functions in L²(R) with nodes at the integers as a function of a∈(0,2). It turns out that the minimum condition numbers occur at a=√2/3 and a=2-√2/3 and not at a=1 as we might have expected. The theory is verified using the example of video deinterlacing