arXiv subjects

Rodney Nillsen

Publications and source records attributed to Rodney Nillsen.

3 recordsLinked to original sources

arXiv2022

Sums Associated with Orbits in the Binary Dynamical System

In 1930, G. H. Hardy and J. E. Littlewood derived results concerning rates of divergence of certain series involving cosecants. In more recent terminology, one of their results can be interpreted in terms of the behaviour of orbits in a dynamical system that is a rotation on the unit circle. Now, the expansion of numbers in $[0,1)$ to the base $2$ can be associated with a different dynamical system -- the binary system. This article considers orbit behaviour in the binary system that corresponds to the behaviour that was, in effect, observed by Hardy and Littlewood in systems involving rotations. Given a typical number in $[0,1)$, the sequence of its binary digits may be arranged as an infinite sequence of consecutive, non-empty, finite blocks, each block consisting of all zeros or all ones. The relationships between the lengths of these blocks determine Hardy-Littlewood types of behaviour associated with the number. Amongst other results, upper and lower estimates are derived for the sums of powers of the reciprocals of points in the orbit of the number. These estimates are in terms of the lengths of the associated blocks. A necessary and sufficient condition is found for the essential `equivalence' of the upper and lower estimates. Almost all numbers in $[0,1)$ satisfy this condition.

math.DS↗

arXiv2016

Generalised differences and multiplier operators in $L^2({\mathbb R})$

Given two real numbers, the $L^2$ functions whose Fourier transforms vanish with a certain rapidity near the given numbers are characterised as those that are expressible as the sum of a certain number of generalised finite differences that is independent of the function. These generalised differences can be regarded as approximating the appropriate powers of first order ordinary differential operators. The upshot of this is that for operators in a certain class of ordinary differential operators that have polynomial multipliers, their ranges on the Sobolev spaces corresponding to the operators are those functions expressible as a finite sum of corresponding generalised differences, so that the latter form a weighted $L^2$ space under the Fourier transform. There is a connection with the continuity properties of invariant forms on $L^2$ spaces. The results presented here complement results previously obtained for the $L^2$ space of the circle group.

math.CA↗

arXiv2015

Generalised differences and a class of multiplier operators in Fourier analysis

The ranges of a certain type of second order differential operator, on a Sobolev subspace of the Lebesgue space $L^2$ of the circle group, can be characterised by the vanishing of the Fourier coefficients at (generally) two integers that are the zeros of the multiplier of the operator. It is proved here that the range of any such operator may be alternatively described as comprising those functions in $L^2$ that are the sum of five generalised second order differences, each such difference involving the zeros of the multiplier. In fact, higher order operators and differences are considered. There are applications to automatic continuity of linear forms on $L^2$. This work is related to earlier work of G. Meisters and W. Schmidt who derived, in effect, a description of the range of the ordinary differentiation operator D (whose multiplier vanishes at 0) in terms of first order differences.

math.CA↗