The isometric extension of the into mapping from the unit sphere S 1( E ) to S 1( l ∞(Γ))

This paper considers the isometric extension problem concerning the mapping from the unit sphere S ₁(E) of the normed space E into the unit sphere S ₁(l ^∞(Γ)). We find a condition under which an isometry from S ₁(E) into S ₁(l ^∞(Γ)) can be linearly and isometrically extended to the whole space. Since l ^∞(Γ) is universal with respect to isometry for normed spaces, isometric extension problems on a class of normed spaces are solved. More precisely, if E and F are two normed spaces, and if V ₀: S ₁(E) → S ₁(F) is a surjective isometry, where c ₀₀(Γ) ⊆ F ⊆ l ^∞(Γ), then V ₀ can be extended to be an isometric operator defined on the whole space. This work is supported by Natural Science Foundation of Guangdong Province, China (Grant No. 7300614)     

de Sitter空间S1m+1中具有平行Blaschke张量的正则类空超曲面

本文引入两个以de Sitter空间为模型的非齐性坐标来覆盖共形空间Q₁^m+1.利用球面S^m+1中超曲面的Möbius几何的方法,本文研究了Q₁^m+1中正则类空超曲面的共形几何.作为其结果,本文对所有具有平行Blaschke张量的正则类空超曲面进行了完全分类.     

The uniform norm of hyperinterpolation on the unit sphere in an arbitrary number of dimensions

In this paper we study the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere S ^r−1 ⊂ R^r. The hyperinterpolation approximation L _n ƒ, where ƒ ∈ C(S ^r −1), is derived from the exact L ² orthogonal projection Π ƒ onto the space P _n ^r (S ^r −1) of spherical polynomials of degree n or less, with the Fourier coefficients approximated by a positive weight quadrature rule that integrates exactly all polynomials of degree ≤ 2n. We extend to arbitrary r the recent r = 3 result of Sloan and Womersley [9], by proving that under an additional “quadrature regularity” assumption on the quadrature rule, the order of growth of the uniform norm of the hyperinterpolation operator on the unit sphere is O(n ^r /2−1), which is the same as that of the orthogonal projection Π_n, and best possible among all linear projections onto P _n ^r (S ^r −1).     

The structure of some classes of <Emphasis Type="Italic">K</Emphasis>-contact manifolds

We study projective curvature tensor in K-contact and Sasakian manifolds. We prove that (1) if a K-contact manifold is quasi projectively flat then it is Einstein and (2) a K-contact manifold is ξ-projectively flat if and only if it is Einstein Sasakian. Necessary and sufficient conditions for a K-contact manifold to be quasi projectively flat and φ-projectively flat are obtained. We also prove that for a (2n + 1)-dimensional Sasakian manifold the conditions of being quasi projectively flat, φ-projectively flat and locally isometric to the unit sphere S ²ⁿ⁺¹ (1) are equivalent. Finally, we prove that a compact φ-projectively flat K-contact manifold with regular contact vector field is a principal S ¹-bundle over an almost Kaehler space of constant holomorphic sectional curvature 4.     

The Ruziewicz problem and distributing points on homogeneous spaces of a compact Lie group

LetG be any compact non-commutative simple Lie group not locally isomorphic to SO(3). We present a generalization of a theorem of Lubotzky, Phillips and Sarnak on distributing points on the sphere S² (or S³) to any homogeneous space ofG, in particular, to all higher dimensional spheres. Our results can also be viewed as a quantitative solution to the generalized Ruziewicz problem for any homogeneous space ofG. Partially supported by DMS-0070544 and DMS-0333397.     

Uniform algebras on the sphere invariant under group actions

It is shown under certain conditions that a uniform algebra on the unit sphere S in C ² that is invariant under the action of the 2-torus must be C(S). Contrasting with this, an example is presented showing that the statement becomes false when 2 is replaced by n > 2. It is also shown that C(M) is the only uniform algebra on a smooth manifold M that is invariant under a transitive Lie group action on its maximal ideal space. The results presented answer a question raised by Ronald Douglas in connection with a conjecture of William Arveson.     

A lower bound for the worst-case cubature error on spheres of arbitrary dimension

This paper is concerned with numerical integration on the unit sphere S^r of dimension r≥2 in the Euclidean space ℝ^r+1. We consider the worst-case cubature error, denoted by E(Q_m;H^s(S^r)), of an arbitrary m-point cubature rule Q_m for functions in the unit ball of the Sobolev space H^s(S^r), where s>, and show that The positive constant c_s,r in the estimate depends only on the sphere dimension r≥2 and the index s of the Sobolev space H^s(S^r). This result was previously only known for r=2, in which case the estimate is order optimal. The method of proof is constructive: we construct for each Q_m a `bad' function f_m, that is, a function which vanishes in all nodes of the cubature rule and for which Our proof uses a packing of the sphere S^r with spherical caps, as well as an interpolation result between Sobolev spaces of different indices.     

Laplacians on quotients of Cauchy-Riemann complexes and Szegö map for L-harmonic forms

We compute the spectra of the Tanaka type Laplacians on the Rumin complex Q, a quotient of the tangential Cauchy-Riemann complex on the unit sphere S²ⁿ⁻¹ in ⁿ. We prove that Szegö map is a unitary operator from a subspace of (p, q −1)-forms on the sphere defined by the operators and the normal vector field onto the space of L²-harmonic (p, q)-forms on the unit ball. Our results generalize earlier result of Folland.     

Wavelets on the 2-Sphere: A Group-Theoretical Approach

We present a purely group-theoretical derivation of the continuous wavelet transform (CWT) on the 2-sphere S², based on the construction of general coherent states associated to square integrable group representations. The parameter space X of our CWT is the product of SO(3) for motions and ⁺_* for dilations on S², which are embedded into the Lorentz group SO₀(3, 1) via the Iwasawa decomposition, so that X SO₀(3, 1) M Y S O L N, where N . We select an appropriate unitary representation of SO₀(3, 1) acting in the space L²(S², d μ) of finite energy signals on S². This representation is square integrable over X; thus it yields immediately the wavelets on S² and the associated CWT. We find a necessary condition for the admissibility of a wavelet, in the form of a zero mean condition. Finally, the Euclidean limit of this CWT on S² is obtained by redoing the construction on a sphere of radius R and performing a group contraction for R → ∞. Then the parameter space goes into the similitude group of ² and one recovers exactly the CWT on the plane, including the usual zero mean necessary condition for admissibility.     

A geometric inequality for a finite set of points on a sphere and its applications

In this paper we present a geometric inequality for a finite number of points on an (n–1)-dimensional sphere S _n–1(R). As an application, we obtain a geometric inequality for finitely many points in the n-dimensional Euclidean space E ⁿ.     

Approximation power of RBFs and their associated SBFs: a connection

Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long time. Regardless of the underlying manifold, for example ℝⁿ or S ⁿ, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝⁿ⁺¹, to the unit sphere S ⁿ. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent to a Sobolev space H _s(ℝⁿ⁺¹), then the restriction to S ⁿ has a native space equivalent to H _s−1/2(S ⁿ). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of these were known earlier. Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation.     

Isometric Stochastic Flows on Spheres

We consider isometric stochastic flows on the sphere S ^n–1 with the same one point motion. In particular, we will show that when n>3, the set of such flows with Brownian motion as one point motion can be represented by a cube in some Euclidean space.     

Preconditioners for pseudodifferential equations on the sphere with radial basis functions

In a previous paper a preconditioning strategy based on overlapping domain decomposition was applied to the Galerkin approximation of elliptic partial differential equations on the sphere. In this paper the methods are extended to more general pseudodifferential equations on the sphere, using as before spherical radial basis functions for the approximation space, and again preconditioning the ill-conditioned linear systems of the Galerkin approximation by the additive Schwarz method. Numerical results are presented for the case of hypersingular and weakly singular integral operators on the sphere \mathbbS²{\mathbb{S}^2} .     

On the number of triple points of an immersed surface with boundary

Given a generic immersionf:S ¹→S ² of a circle into the sphere, we find the best possible lower estimation for the number of triple points of a generic immersionF: (M, S ¹)→(B ³,S ²) extendingf, whereM is an oriented surface with boundary ∂M=S ¹,B ³ is the 3-dimensional ball with boundaryS ². Supported by the Hungarian National Science and Research Foundation OTKA 2505 Supported by the Hungarian National Science and Research Foundation OTKA T4232 This article was processed by the author using the Springer-Verlag T_EX P Jour1g macro package 1991.     

Homogeneous surfaces in Lie sphere geometry

In this paper, we study surfaces of S ³ in the context of Lie sphere geometry. We construct invariants with respect to Lie sphere transformations on the surfaces, which determine the surfaces up to a Lie sphere transformation. Finally we classify completely the homogeneous surfaces in S ³ with respect to the Lie sphere transformation group of S ³.     

Associative Cones and Integrable Systems

Abstract We identify ℝ⁷ as the pure imaginary part of octonions. Then the multiplication in octonions gives a natural almost complex structure for the unit sphere S⁶. It is known that a cone over a surface M in S⁶ is an associative submanifold of ℝ⁷ if and only if M is almost complex in S⁶. In this paper, we show that the Gauss-Codazzi equation for almost complex curves in S⁶ are the equation for primitive maps associated to the 6-symmetric space G₂=T², and use this to explain some of the known results. Moreover, the equation for S¹-symmetric almost complex curves in S⁶ is the periodic Toda lattice, and a discussion of periodic solutions is given. (Dedicated to the memory of Shiing-Shen Chern) * Partially supported by NSF grant DMS-0529756.     

Summation of Fourier-Laplace Series in the Space <Emphasis Type="Italic">L</Emphasis>(<Emphasis Type="Italic">S</Emphasis><Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>)

We establish estimates for the rate of convergence of a group of deviations on a sphere in the space L(S ^m ), m ≥ 3. __________ Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 4, pp. 496–504, April, 2005.     

Extremal Systems of Points and Numerical Integration on the Sphere

This paper considers extremal systems of points on the unit sphere S ^r⊂R ^r+1, related problems of numerical integration and geometrical properties of extremal systems. Extremal systems are systems of d _n =dim P _n points, where P _n is the space of spherical polynomials of degree at most n, which maximize the determinant of an interpolation matrix. Extremal systems for S ² of degrees up to 191 (36,864 points) provide well distributed points, and are found to yield interpolatory cubature rules with positive weights. We consider the worst case cubature error in a certain Hilbert space and its relation to a generalized discrepancy. We also consider geometrical properties such as the minimal geodesic distance between points and the mesh norm. The known theoretical properties fall well short of those suggested by the numerical experiments.

     

On the commutant of certain multiplication operators on spaces of analytic functions

Let ℬ be a Banach space of analytic functions defined on the open unit disk. We characterize the commutant ofM _Z ² (the operator of multiplication by the square of independent variable defined on ℬ) and show that for an operatorS in the commutantM _Z ² ifSM _Z ^2k+1−M _Z ^2k+1 S is compact for some nonnegative integerk, thenS=M _ϕ whereϕ is a multiplier of ℬ. Letn be a positive integer andS be an operator in the commutant ofM _Z ⁿ defined on a functional Hilbert spaces of analytic functions. We show that under certain conditionsS has the formM _ϕ. Research supported by the Shiraz University Grant 78-SC-1188-657.     

On the weak non-defectivity of veronese embeddings of projective spaces

Fix integers n, x, k such that n≥3, k>0, x≥4, (n, x)≠(3, 4) and k(n+1)<( _n ^n+x ). Here we prove that the order x Veronese embedding ofP ⁿ is not weakly (k−1)-defective, i.e. for a general S⊃P ⁿ such that #(S) = k+1 the projective space | I _2S (x)| of all degree t hypersurfaces ofP ⁿ singular at each point of S has dimension ( _n ^/n+x )−1− k(n+1) (proved by Alexander and Hirschowitz) and a general F∈| I _2S (x)| has an ordinary double point at each P∈ S and Sing (F)=S. The author was partially supported by MIUR and GNSAGA of INdAM (Italy).