Brad Osgood

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This note is a sequel to our paper [OS] where we generalized the Schwarzian derivative to conformal mappings of Riemannian manifolds. There we found that many of the phenomena familiar from the classical theory have counterparts in the more general setting. Here we advance this another step by giving a generalization of the well known univalence criterion(More)
It is shown that an analytic function taking circles to ellipses must be a Möbius transformation. It then follows that a harmonic mapping taking circles to ellipses is a harmonic Möbius transformation. Analytic Möbius transformations take circles to circles. This is their most basic, most celebrated geometric property. We add the adjective 'analytic'(More)
In this note we study the zeros of solutions of differential equations of the form u ′′ + pu = 0. A criterion for oscillation is found, and some sharper forms of the Sturm comparison theorem are given. §1. Number of zeros. Consider the linear differential equation u ′′ (x) + p(x) u(x) = 0 , where p(x) = 1 (1 − x 2) 2 , (1) on the interval −1 < x < 1. Two(More)
—We study the problem of interpolating all values of a discrete signal f of length N when d < N values are known, especially in the case when the Fourier transform of the signal is zero outside some prescribed index set J ; these comprise the (generalized) bandlimited spaces B J. The sampling pattern for f is specified by an index set I, and is said to be a(More)
For analytic functions in the unit disk, general bounds on the Schwarzian derivative in terms of Nehari functions are shown to imply uniform local univalence and in some cases finite and bounded valence. Similar results are obtained for the Weierstrass–Enneper lifts of planar harmonic mappings to their associated minimal surfaces. Finally certain classes of(More)
—We study the problem of finding unitary submatrices of the discrete Fourier transform matrix. This problem is related to a diverse set of questions on idempotents on Z N , tiling Z N , difference graphs and maximal cliques. Each of these is related to the problem of interpolating a discrete bandlimited signal using an orthogonal basis.