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 [332]. 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 in [504, Ch. 3, Pt. 2, pp. 82-84].^{5.5}