Harmonic functions can be constructed using two analytic functions acting as their analytic and coanalytic parts but the prediction of the behavior of convolution of harmonic functions, unlike the convolution of analytic functions, proved to be challenging. In this paper we use the shear construction of harmonic mappings and introduce dilatation conditions that guarantee the convolution of two harmonic functions to be harmonic and convex in the direction of imaginary axis.
AbstractIt is shown to be consistent with set theory that every set of reals of size ℵ1 is null yet there are ℵ1 planes in Euclidean 3-space whose union is not null. Similar results will be obtained for other geometric objects. The proof relies on results from harmonic analysis about the boundedness of certain harmonic functions and a measure theoretic pigeonhole principle.
Given a harmonic function u on a Riemann surface R, we define a period functionfor every one-dimensional cycle γ of the Riemann surface R. Γx(R) denote the totality of period functions Γu such that harmonic functions u satisfy a boundedness property X. As for X, we let B stand for boundedness, and D for the finiteness of the Dirichlet integral.
Abstract
The Hipparcos optical reference frame is
compared to the basic FK5 in order to determine
the orientation at
T0
= 1991.25 and the global spin between the two
frames. The components of the spin are significant
and suggest a correction the IAU76 value of the
precession constant and to a possible
non-precessional motion of the equinox of the FK5.
The regional errors are analysed with harmonic
functions and found to be as large as 150 mas in
position and 3 mas/yr in proper motion.
Directional Convexity of Convolutions of Harmonic Functions
