Dyadic generalized functions and applications: a distribution of a power series on a dyadic half-line

The spaces of distributions on a dyadic half-line, which is the positive half-line equipped with the digitwise binary addition and Lebesgue measure, are studied. We prove the non-existence of such 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, allows the space of dyadic distributions suggested by S. Volosivets in 2009 to be optimal. We also show the applications of dyadic distributions to the theory of refinement equations as well as wavelets on a dyadic half-line. As an interesting application we are exploring the dyadic analogue of one of the Paul Erdos problem, namely, the existence of a probability density of a random variable (which is a power series), extended to a dyadic half-line. We consider the power series with coefficients being either zeroes or ones at the fixed point $x$ of the $(0, 1)$ interval. The question is whether there is a density from $\mathbb{L}_1$? In classic case it is still an opened problem for $x$ greater than one half (P. Erdos proved the non-existence of the density for lambdas equal to $\frac{1}{p}$, where $p$ is the Pisot number). Moreover, we study the so called "dual problem". The same random variable, but the point $x$ is fixed now ($x = \frac{1}{2}$) and the coefficients are integer and belong to $[0; N]$ segment for some natural $N$. Here we answer the same question and provide criteria of the existence of a density in terms of the solution of the refinement equation as well as in terms of the coefficients of a random variable.