Abstract

This paper studies spaces of distributions on a dyadic half-line, which is the positive half-line equipped with bitwise binary addition and Lebesgue measure. We prove the nonexistence of a space of dyadic distributions which satisfies a number of natural requirements (for instance, the property of being invariant with respect to the Walsh–Fourier transform) and, in addition, is invariant with respect to multiplication by linear functions. This, in particular, is evidence that the space of dyadic distributions suggested by S. Volosivets in 2009 is optimal. We also show applications of dyadic distributions to the theory of refinement equations and wavelets on the dyadic half-line.

中文翻译：

---

二元分布的空间

摘要

本文研究了二进半线上的分布空间，二进半线上是带有按位二进制加法和勒贝格测度的正半线。我们证明了不存在满足许多自然要求的二元分布空间（例如，对于 Walsh-Fourier 变换的不变性），此外，对于线性函数的乘法也是不变的。这尤其证明了 S. Volosivets 在 2009 年建议的二元分布空间是最优的。我们还展示了二进分布在二进半线上的细化方程和小波理论中的应用。