For the ideal lowpass filter, we have

from which it follows that the maximum magnitude is

Note that this bound is attained at

Thus for the ideal lowpass filter , the pointwise error in the interpolated lookup of

The error bounds obtained for the ideal lowpass filter are typically
accurate also for lowpass filters used in practice. This is because the
error bound is a function of *M*_{2}, the maximum curvature of the impulse
response *h*(*t*), and most lowpass designs will have a value of *M*_{2} very
close to that of the ideal case. The maximum curvature is determined
primarily by the bandwidth of the filter since, generalizing equations
Eq.(10) and Eq.(11),

which is just the second moment of the lowpass-filter frequency response (which is real for symmetric FIR filters obtained by symmetrically windowing the ideal sinc function [#!RabinerAndGold!#]). A lowpass-filter design will move the cut-off frequency slightly below that of the ideal lowpass filter in order to provide a ``transition band'' which allows the filter response to give sufficient rejection at the ideal cut-off frequency which is where aliasing begins. Therefore, in a well designed practical lowpass filter, the error bound

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