Adaptive Fourier decomposition of functions in quaternionic Hardy spaces

We study decompositions of functions in the Hardy spaces into linear combinations of the basic functions in the orthogonal rational systems B_n, which are obtained in the respective contexts through Gram–Schmidt orthogonalization process on shifted Cauchy kernels. Those lead to adaptive decompositions of quaternionic‐valued signals of finite energy. This study is a generalization of the main results of the first author's recent research in relation to adaptive Takenaka–Malmquist systems in one complex variable. Copyright © 2011 John Wiley & Sons, Ltd.     

单位球面上具有非负M(o)bius截面曲率的超曲面

Let x : M → Sn 1 be a hypersurface in the (n 1)-dimensional unit sphere Sn 1 without umbilic point. The M(o)bius invariants of x under the M(o)bius transformation group of Sn 1 are M(o)bius metric, M(o)bius form, M(o)bius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem: Let x: M → Sn 1 (n ≥ 2) be an umbilic free hypersurface in Sn 1with nonnegative M(o)bius sectional curvature and with vanishing M(o)bius form. Then x is locally M(o)bius equivalent to one of the following hypersurfaces: (i) the torus Sk(a) × Sn-k(√1-a2) with 1 ≤ k ≤ n - 1; (ii) the pre-image of the stereographic projection of the standard cylinder Sk × Rn-k (∩) Rn 1 with 1 ≤ k ≤ n - 1; (iii) the pre-image of the stereographic projection of the cone in Rn 1: ～x(u, v, t) = (tu, tv),where (u,v,t) ∈ Sk(a) × Sn-k-1( √1- a2) × R .     

Sn中Moebius形式为零的曲面

本文证明了S^n中Moebius形式为零且法丛平坦的曲面的余维数约化定理，并且给出了这类曲面的分类．在此基础上，进一步给出了S^n中Moebius形式为零的曲面的分类．     

Asymptotics of orthogonal polynomials beyond the scope of Szegő’s theorem

First, we give a simple proof of a remarkable result due to Videnskii and Shirokov: let B be a Blaschke product with n zeros; then there exists an outer function φ, φ(0) = 1, such that ‖(Bφ)′‖ ? Cn, where C is an absolute constant. Then we apply this result to a certain problem of finding the asymptotics of orthogonal polynomials.     

Blaschke isoparametric hypersurfaces in the conformal space ℚ 1 n+1 , I

Let x : M → Q₁ⁿ⁺¹ be a regular hypersurface in the conformal space Q₁ⁿ⁺¹. We classify all the space-like Blaschke isoparametric hypersurfaces with two distinct Blaschke eigenvalues in the conformal space up to the conformal equivalence.     

The Hypersurfaces in a Unit Sphere with Nonnegative Mobius Sectional Curvature

Let x ： M→S^n＋1 be a hypersurface in the （n ＋ 1）-dimensional unit sphere S^n＋1 without umbilic point. The Mobius invariants of x under the Mobius transformation group of S^n＋1 are Mobius metric, Mobius form, Mobius second fundamental form and Blaschke tensor. In this paper, we prove the following theorem： Let x ： M→S^n＋1 （n≥2） be an umbilic free hypersurface in S^n＋1 with nonnegative Mobius sectional curvature and with vanishing Mobius form. Then x is locally Mobius equivalent to one of the following hypersurfaces： （i） the torus S^k（a） × S^n-k（√1- a^2） with 1 ≤ k ≤ n - 1; （ii） the pre-image of the stereographic projection of the standard cylinder S^k × R^n-k belong to R^n＋1 with 1 ≤ k ≤ n- 1; （iii） the pre-image of the stereographic projection of the Cone in R^n＋1 ： -↑x（u, v, t） = （tu, tv）, where （u,v, t）∈S^k（a） × S^n-k-1（ √1-a^2）× R^＋.     

On BMO-Type Characteristics for One Class of Holomorphic Functions in a Disk

Various BMO-type characteristics are given for some class of holomorphic functions with a mixed norm. Some criteria are established for Blaschke products to belong to analytic Besov spaces.     

Blaschke products and the rank of backward shift invariant subspaces over the bidisk

Let H²(D²) be the Hardy space over the bidisk. For sequences of Blaschke products {φn(z):−∞<n<∞} and {ψn(w):−∞<n<∞} satisfying some additional conditions, we may define a Rudin type invariant subspace M. We shall determine the rank of H²(D²)?M for the pair of operators and .