On Representations of Walsh Functions

A new recursive formula for defining Walsh functions on the real line is presented. It leads, in a natural way, to an explicit representation of Walsh futctions. The explicit representation can be identified with. the representation of Walsh functions by the products of Rademacher functions. It can also lead to other representations of Walsh functions in a straight-forward manner. Using the new-formulation, one can prove many known properties of Walsh functions easily and systematically and gain new insight.