Two new digital filter design techniques are proposed for obtaining the transfer function, in the z-domain, of a recursive digital filter which is to simulate the performance of a given continuous filter. The first one which we refer to as "frequency method" is obtained by minimizing the expected value of the squared error at the sampling instants when the input signal is band-limited. The resultant digital filters will have the best frequency responses in the Nyquist frequency range. The second one which we call the "spline method" relies on the approximation of the convolution integrand associated with the continuous filter by a simple cubic interpolating spline. The best approximation properties of the spline interpolation and the approximation error are studied in the context of the problem under consideration. Some numerical results obtained on a Burroughs 5500 digital computer are presented which illustrate the frequency responses of the resultant digital filters.