### Proof of Maximum Flatness at DC

The maximumally flat fractional-delay FIR filter is obtained by equating to zero all leading terms in the Taylor (Maclaurin) expansion of the frequency-response error at dc:

This is a linear system of equations of the form , where is a Vandermonde matrix. The solution can be written as a ratio of Vandermonde determinants using Cramer's rule [329]. As shown by Cauchy (1812), the determinant of a Vandermonde matrix , can be expressed in closed form as

Making this substitution in the solution obtained by Cramer's rule yields that the impulse response of the order , maximally flat, fractional-delay FIR filter may be written in closed form as

which is the formula for Lagrange-interpolation coefficients (Eq.(4.6)) adapted to this problem (in which abscissae are equally spaced on the integers from 0 to ). Further details regarding the theory of Lagrange interpolation can be found (online) in [502, Ch. 3, Pt. 2, pp. 82-84].
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