Spaces of Dyadic Distributions

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.