Although the Discrete Cosine Transform (DCT) is widely used for feature extraction in pattern recognition, it is shown that it converges slowly for most theoretically smooth functions. A modification of the DCT is described, based on a change of variable, which changes it to a new transform, called the Discrete Chebyshev Transform DChT, which converges very rapidly for the same smooth functions. Although this rapid convergence is largely destroyed by the noise in real experimental data, the Discrete Chebyshev Transform is still generally better than the DCT when the sampling of the data can be selected at non-equidistant points. The improvement over the DCT gives a theoretical explanation for improved speech recognition obtained using Mel Feature Cepstral Coefficients. These choose the sampling frequencies of a DCT to correspond to the human perception of pitch. It is shown that this sampling is similar to the sampling used in the Discrete Chebyshev Transform.