The purpose of this project was to investigate the method of discrete least squares polynomial approximation and to provide an extension of the method which would allow for a reasonable data-fit while lowering the number of undetermined coefficients. Also, computer programs were to be provided so that one may use the extended method. This project is an outgrowth of work done by Dr. R. De an Riess, Mathematics Department, V.P.I.S.U. There exist several areas of investigation of the theory and some extensions of the computer programs which appear to merit future effort. Possible investigations of the theory include the interpolation result that was mentioned in the discussion of the extended problem, and error bound results. Possible extensions of the programs include the introduction of decision making capabilities so that the user need only specify the data and a tolerance level to achieve a good approximation, and the implementation of the extended programs on the central system. / Master of Science