Let *h*(*t*) denote the lowpass filter impulse response, and assume it is
twice continuously differentiable for all *t*. By Taylor's theorem
[6, p. 119], we have

The linear interpolation error is

(7) |

where and . Thus

By definition,
*e*(*t*_{0})=*e*(*t*_{1})=0. That is, the interpolation error is
zero at the known samples. Let *t*_{e} denote any point at which
|e(*t*)| reaches a maximum over the interval (*t*_{0},*t*_{1}). Then we
have

Without loss of generality, assume . (Otherwise, replace

for some . Solving for

Defining

where the maximum is taken over , and noting that , we obtain the upper bound

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