Analog frequency sampling filters interpolate a frequency response through the filter’s Fourier series coefficients, which are evenly spaced samples from the filter’s frequency response. Although the frequency response of a frequency sampling filter passes through the filter’s frequency samples, or Fourier series coefficients, the frequency response may not behave well between samples. This paper develops a method that controls the interpolation errors between frequency samples by minimizing a weighted power of the error between the desired and actual frequency responses over the pass-band and stop-band frequencies subject to constraints on the frequency response. This design method describes the frequency sampling filter design problem as a constrained optimization problem which is solved using the Lagrange multiplier optimization method. This method is used to design Type 1 and Type 2 frequency sampling filters.