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本文对于在实线上定义 Walsh 函数提出了一种新的递归公式,并由此导得了 Walsh函数的一种显式表示法。这种显式表示法可与 Rademacher 函数之积的 Walsh 函数表示法等同。由此公式用直接的方法也可导得其它的 Walsh 函数表示法。采用这种新的公式,可容易地、系统地证明许多熟知的 Walsh 函数的性质,且增进新的理解。
This paper presents a new recursive formula for defining Walsh functions on a solid line, and derives an explicit representation of Walsh functions. This explicit notation is equivalent to the Walsh function notation of the product of the Rademacher functions. From this formula, direct methods can be used to derive other Walsh function notations. With this new formula, the properties of many well-known Walsh functions can be easily and systematically proven, and new understandings enhanced.