Some approaches to the numerical evaluation of Fourier coefficients are outlined. Numerical quadrature formulas of Filon type are developed using higher-order polynomial approximations, in particular third- and fifth-order Hermite interpolants. Adaptive methods of implementation are described. The resulting formulas and the original Filon formulas are compared in numerical examples. Higher-order Lagrange interpolants were also used, the fifth-order being less effective than the corresponding Hermite approximation.