A Fourier method of filtering digital data called Lanczos filtering is described. Its principal feature is the use of “sigma factors” which significantly reduce the amplitude of the Gibbs oscillation. A pair of graphs is developed that can be used to determine filter response quality given the number of weights and the value of the cutoff frequency, the only two inputs required by the method. Examples of response functions in one and two dimensions are given and comparisons are made with response functions from other filters. The simplicity of calculating the weights and the adequate response make Lanczos filtering an attractive filtering method.