Function Approximation by Walsh Series

Function approximation by a finite Walsh series is considered. There are two methods for selecting the terms of a series. The process of threshold sampling gives a least-square error approximation, but no error analysis technique is available. However, error analysis is possible if terms are selected according to degrees and subdegrees. It is shown that truncation is equivalent to dropping all terms with degrees greater than some amount. The error caused is a weighted integral of the first derivative, and an upper bound on the expression can be derived. It is also shown that a truncated Walsh series corresponds to a simple function table. Data compression is equivalent to dropping terms with large enough subdegrees, with an estimable error. After a Walsh series has been selected, it is possible to modify the coefficients using Lawson's algorithm and reduce the maximum error.