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Unit sphere
About: Unit sphere is a research topic. Over the lifetime, 8818 publications have been published within this topic receiving 124102 citations.
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23 Jan 2012
TL;DR: In this paper, the boundary behavior of Cauchy integral functions is investigated and the results of the Schwarz Lemma are discussed, as well as the Zeros of Nevanlinna functions.
Abstract: Preliminaries.- The Automorphisms of B.- Integral Representations.- The Invariant Laplacian.- Boundary Behavior of Poisson Integrals.- Boundary Behavior of Cauchy Integrals.- Some Lp-Topics.- Consequences of the Schwarz Lemma.- Measures Related to the Ball Algebra.- Interpolation Sets for the Ball Algebra.- Boundary Behavior of H?-Functions.- Unitarily Invariant Function Spaces.- Moebius-Invariant Function Spaces.- Analytic Varieties.- Proper Holomorphic Maps.- The -Problem.- The Zeros of Nevanlinna Functions.- Tangential Cauchy-Riemann Operators.- Open Problems.
2,202 citations
01 Jul 1985
TL;DR: A new kind of spline curve is presented, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations, without quirks found in earlier methods.
Abstract: Solid bodies roll and tumble through space. In computer animation, so do cameras. The rotations of these objects are best described using a four coordinate system, quaternions, as is shown in this paper. Of all quaternions, those on the unit sphere are most suitable for animation, but the question of how to construct curves on spheres has not been much explored. This paper gives one answer by presenting a new kind of spline curve, created on a sphere, suitable for smoothly in-betweening (i.e. interpolating) sequences of arbitrary rotations. Both theory and experiment show that the motion generated is smooth and natural, without quirks found in earlier methods.
2,006 citations
Book•
28 Sep 1990TL;DR: In this paper, the basic fixed point theorems for non-pansive mappings are discussed and weak sequential approximations are proposed for linear mappings with normal structure and smoothness.
Abstract: Introduction 1. Preliminaries 2. Banach's contraction principle 3. Nonexpansive mappings: introduction 4. The basic fixed point theorems for nonexpansive mappings 5. Scaling the convexity of the unit ball 6. The modulus of convexity and normal structure 7. Normal structure and smoothness 8. Conditions involving compactness 9. Sequential approximation techniques 10. Weak sequential approximations 11. Properties of fixed point sets and minimal sets 12. Special properties of Hilbert space 13. Applications to accretivity 14. Nonstandard methods 15. Set-valued mappings 16. Uniformly Lipschitzian mappings 17. Rotative mappings 18. The theorems of Brouwer and Schauder 19. Lipschitzian mappings 20. Minimal displacement 21. The retraction problem References.
1,466 citations
TL;DR: In this paper, a measure on the closed subspaces of a Hilbert space is defined, which assigns to every closed subspace a non-negative real number such that if the subspace is a countable collection of mutually orthogonal sub-spaces having closed linear span B, then
Abstract: In his investigations of the mathematical foundations of quantum mechanics, Mackey1 has proposed the following problem: Determine all measures on the closed subspaces of a Hilbert space. A measure on the closed subspaces means a function μ which assigns to every closed subspace a non-negative real number such that if {Ai} is a countable collection of mutually orthogonal subspaces having closed linear span B, then
$$ \mu (B) = \sum {\mu \left( {{A_i}} \right)} $$
.
1,322 citations
TL;DR: In this article, the authors extend the duality between HI and BMO in terms of boundedness on L 2 of the commutator of a singular integral operator with a multiplication operator and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces.
Abstract: The purpose of this paper is to extend to Hardy spaces in several variables certain well known factorization theorems on the unit disk. The extensions will be carried out for various spaces of holomorphic functions on the unit ball of C" as well as for Hardy spaces defined by the Riesz systems on R". These results together with their proofs yield new characterizations of the space BMO (Bounded Mean Oscillation) and show a close relationship between BMO functions and certain linear operators on various LI and H2 spaces. The main tools are the result of Fefferman and Stein [8] on the duality between HI and BMO and a new characterization of BMO in terms of boundedness on L2 of the commutator of a singular integral operator with a multiplication operator. We begin by illustrating these ideas in the one dimensional case: Let F be holomorphic in {I z I < 1} and satisfy sup, 5 F(rete) I dO ? 1 (i.e., F is in H'(dO)). It is well known that F = GG2 with G1, G2 holomorphic and sup, I G,(rel0) 1' ! 1 (i.e., G, e H2(dO)). Write F = f + if, G, = gj + ig withf, g1, g, real. Thenf = Im(GG2) = sg1 1 + gi. Thusafunction f is an imaginary (or real) part of an HI function if and only if it can be represented as glg2 + g192 for L2 functions g, and g2. Furthermore,
1,304 citations