Search results for "Unit sphere"
showing 10 items of 54 documents
Interpolating sequences on uniform algebras
2009
Abstract We consider the problem of whether a given interpolating sequence for a uniform algebra yields linear interpolation. A positive answer is obtained when we deal with dual uniform algebras. Further we prove that if the Carleson generalized condition is sufficient for a sequence to be interpolating on the algebra of bounded analytic functions on the unit ball of c 0 , then it is sufficient for any dual uniform algebra.
An optimal extension of Marstrand?s plane-packing theorem
2003
We prove that if F is a subset of the 2-dimensional unit sphere in $\mathbb{R}^3$, with Hausdorff dimension strictly greater than 1, and E is a subset of $\mathbb{R}^3$ such that for each $e \in F$, E contains a plane perpendicular to the vector e, then E must have positive 3-dimensional Lebesgue measure.
Uniform continuity of quasiconformal mappings and conformal deformations
2008
We prove that quasiconformal maps onto domains satisfying a suitable growth condition on the quasihyperbolic metric are uniformly continuous even when both domains are equipped with internal metric. The improvement over previous results is that the internal metric can be used also in the image domain. We also extend this result for conformal deformations of the euclidean metric on the unit ball of R n \mathbb {R}^n .
Radial growth of solutions to the poisson equation
2001
We establish a radial growth estimate of the type of the iterated law of the logarithm for solutions to the Poisson equation in the unit ball.
The average over a sphere
1980
Abstract The N points ri and the N segments ΔΩi of the unit sphere used in the numerical approximation of the average over the sphere are optimized to approximate the average of the set of spherical harmonics {;Yl,m;l = 0, 1, 2, …, L}; up to L = 18. The symmetry of f( r ) can be taken into acount by using only a distinct subquantity of the N point {; r i , ΔΩ i }; . Sets for N = 48n (n = 1, 2, …, 6) are tabulated. The advantage of the method is shown by the calculation of a powder Mossbauer spectrum including electric and magnetic hyperfine interactions.
Grover’s Search with Faults on Some Marked Elements
2016
Grover's algorithm is a quantum query algorithm solving the unstructured search problem of size N using $$O\sqrt{N}$$ queries. It provides a significant speed-up over any classical algorithm [2]. The running time of the algorithm, however, is very sensitive to errors in queries. Multiple authors have analysed the algorithm using different models of query errors and showed the loss of quantum speed-up [1, 4]. We study the behavior of Grover's algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of marked elements in $$O\sqrt{N}$$ queries.
The Daugavet equation for polynomials
2007
In this paper we study when the Daugavet equation is satisfied for weakly compact polynomials on a Banach space X, i.e. when the equality ‖Id + P‖ = 1 + ‖P‖ is satisfied for all weakly compact polynomials P : X −→ X. We show that this is the case when X = C(K), the real or complex space of continuous functions on a compact space K without isolated points. We also study the alternative Daugavet equation max |ω|=1 ‖Id + ω P‖ = 1 + ‖P‖ for polynomials P : X −→ X. We show that this equation holds for every polynomial on the complex space X = C(K) (K arbitrary) with values in X. The result is not true in the real case. Finally, we study the Daugavet and the alternative Daugavet equations for k-h…
Bloch functions on the unit ball of an infinite dimensional Hilbert space
2015
The Bloch space has been studied on the open unit disk of C and some ho- mogeneous domains of C n . We dene Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(BE) is invariant under composition with the automorphisms of the ball, leading to a norm that- modulo the constant functions - is automorphism invariant as well. All bounded analytic functions on BE are also Bloch functions. ones, resulting the fact that if for a given n; the restrictions of the function to the n-dimensional subspaces have their Bloch norms uniformly bounded, then the function is a Bloch one and conversely. We also introduce an equivalent norm forB(BE) obtained by repla…
Weak chord-arc curves and double-dome quasisymmetric spheres
2014
Let $\Omega$ be a planar Jordan domain and $\alpha>0$. We consider double-dome-like surfaces $\Sigma(\Omega,t^{\alpha})$ over $\overline{\Omega}$ where the height of the surface over any point $x\in\overline{\Omega}$ equals $\text{dist}(x,\partial\Omega)^{\alpha}$. We identify the necessary and sufficient conditions in terms of $\Omega$ and $\alpha$ so that these surfaces are quasisymmetric to $\mathbb{S}^2$ and we show that $\Sigma(\Omega,t^{\alpha})$ is quasisymmetric to the unit sphere $\mathbb{S}^2$ if and only if it is linearly locally connected and Ahlfors $2$-regular.
Relatively weakly open convex combinations of slices
2018
We show that c 0 c_0 and, in fact, C ( K ) C(K) for any scattered compact Hausdorff space K K have the property that finite convex combinations of slices of the unit ball are relatively weakly open.