This work addresses the problem of the implementation of fractional sample delays necessary for accurate digital beamforming with small steering delay quantization. Past work in this area has employed techniques which upsample by an integer factor, providing an intermediate higher sampling rate at which more accurate delays can be implemented. The signal is then downsampled to the original rate. This results in a minimum delay resolution equal to the original sampling interval divided by the integer valued upsampling factor. The approach presented herein uses existing signal processing techniques to perform resampling without requiring a conversion to an intermediate higher sampling rate [J. O. Smith and P. Gossett, Proc. Intl. Conf. on ASSP (March 1984), Vol. 2, pp. 19.4.1--19.4.2]. This method allows for continuously variable delay values. This technique is extended by the inclusion of interpolation filters which are optimal in a Chebychev or min--max sense. It is shown that the design of fractional delay filters is a complex-valued optimization problem. Specifically, the design of Chebychev optimal filters can be posed as a quadratically constrained quadratic program. Techniques for designing Chebychev optimal fractional delay filters, as well as several examples, will be presented.