Lagrange Frequency Response Magnitude Bound

The amplitude response of fractional delay filters based on Lagrange interpolation is observed to be bounded by 1 when the desired delay $\Delta$ lies within half a sample of the midpoint of the coefficient span [506, p. 92], as was the case in all preceeding examples above. Moreover, even-order interpolators are observed to have this boundedness property over a two-sample range centered on the coefficient-span midpoint [506, §3.3.6]. These assertions are easily proved for orders 1 and 2. For higher orders, a general proof appears not to be known, and the conjecture is based on numerical examples. Unfortunately, it has been observed that the gain of some odd-order Lagrange interpolators do exceed 1 at some frequencies when used outside of their central one-sample range [506, §3.3.6].