47 research outputs found
The Mahler measure of the Rudin-Shapiro polynomials
Littlewood polynomials are polynomials with each of their coefficients in
{-1,1}. A sequence of Littlewood polynomials that satisfies a remarkable
flatness property on the unit circle of the complex plane is given by the
Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure
and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of
the complex plane have the same size. It is also shown that the Mahler measure
and the maximum norm of the Rudin-Shapiro polynomials have the same size even
on not too small subarcs of the unit circle of the complex plane. Not even
nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials
have been known before
On the Rudin-Shapiro transform
AbstractThe Rudin–Shapiro transform (RST) is a linear transform derived from the remarkable Rudin–Shapiro polynomials discovered in 1951. The transform has the notable property of forming a spread spectrum basis for RN, i.e. the basis vectors are sequences with a nearly flat power spectrum. It is also orthogonal and Hadamard, and it can be made symmetric. This presentation is partly a tutorial on the RST, partly some new results on the symmetric RST that makes the transform interesting from an applicational point-of-view. In particular, it is shown how to make a very simple O(NlogN) implementation, which is quite similar to the Haar wavelet packet transform
On moments of a polytope
We show that the multivariate generating function of appropriately normalized
moments of a measure with homogeneous polynomial density supported on a compact
polytope P in R^d is a rational function. Its denominator is the product of
linear forms dual to the vertices of P raised to the power equal to the degree
of the density function. Using this, we solve the inverse moment problem for
the set of, not necessarily convex, polytopes having a given set S of vertices.
Under a weak non-degeneracy assumption we also show that the uniform measure
supported on any such polytope is a linear combination of uniform measures
supported on simplices with vertices in S.Comment: 28 pages, 3 figure