Riesz sequences of translates and generalized duals with support on [0, 1]

If the integer translates of a function ø with compact support generate a frame for a subspace W of L ²(?),then it is automatically a Riesz basis for W, and there exists a unique dual Riesz basis belonging to W. Considerable freedom can be obtained by considering oblique duals, i.e., duals not necessarily belonging to W. Extending work by Ben-Artzi and Ron, we characterize the existence of oblique duals generated by a function with support on an interval of length one. If such a generator exists, we show that it can be chosen with desired smoothness. Regardless whether ø is polynomial or not, the same condition implies that a polynomial dual supported on an interval of length one exists.     

Asymptotics of Studentized <Emphasis Type="Italic">U</Emphasis>-type processes for changepoint problems

This paper investigates weighted approximations for Studentized U-statistics type processes, both with symmetric and antisymmetric kernels, only under the assumption that the distribution of the projection variate is in the domain of attraction of the normal law. The classical second moment condition E|h(X ₁, X ₂)|² < ∞ is also relaxed in both cases. The results can be used for testing the null assumption of having a random sample versus the alternative that there is a change in distribution in the sequence.     

On the weighted Kneser-Poulsen conjecture

Suppose that p = (p1, p2, …, pN) and q = (q1, q2, …, qN) are two configurations in , which are centers of balls B ^d (p _i , r _i ) and B ^d (q _i , r _i ) of radius r _i , for i = 1, …, N. In [9] it was conjectured that if the pairwise distances between ball centers p are contracted in going to the centers q, then the volume of the union of the balls does not increase. For d = 2 this was proved in [1], and for the case when the centers are contracted continuously for all d in [2]. One extension of the Kneser-Poulsen conjecture, suggested in [6], was to consider various Boolean expressions in the unions and intersections of the balls, called flowers, where appropriate pairs of centers are only permitted to increase, and others are only permitted to decrease. Again under these distance constraints, the volume of the flower was conjectured to change in a monotone way. Here we show that these generalized Kneser-Poulsen flower conjectures are equivalent to an inequality between certain integrals of functions (called flower weight functions) over , where the functions in question are constructed from maximum and minimum operations applied to functions each being radially symmetric monotone decreasing and integrable. Research supported in part by NSF Grant No. DMS-0209595.     

On weighted L p -spaces of vector-valued entire analytic functions

The weighted L _p -spaces of entire analytic functions are generalized to the vector-valued setting. In particular, it is shown that the dual of the space is isomorphic to when the function χ _K is an L _p,ρ (E)-Fourier multiplier. This result allows us to give some new characterizations of the so-called UMD-property and to represent several ultradistribution spaces by means of spaces of vector sequences. J. Motos is partially supported by DGI (Spain), Grant BFM 2002-04013 and Grant MTEM 2005-08350-C03-03.     

The empirical distribution function and partial sum process of residuals from a stationary arch with drift process

The weak convergence of the empirical process and partial sum process of the residuals from a stationary ARCH-M model is studied. It is obtained for and consistent estimate of the ARCH-M parameters. We find that the limiting Gaussian processes are no longer distribution free and hence residuals cannot be treated as i.i.d. In fact the limiting Gaussian process for the empirical process is a standard Brownian bridge plus an additional term, while the one for partial sum process is a standard Brownian motion plus an additional term. In the special case of a standard ARCH process, that is an ARCH process with no drift, the additional term disappears. We also study a sub-sampling technique which yields the limiting Gaussian processes for the empirical process and partial sum process as a standard Brownian bridge and a standard Brownian motion respectively.     

Fourier series of functions whose Hankel transform is supported on [0, 1]

LetJ denote the Bessel function of order . For >–1, the system x^–/2–1/2J_+2n+1(x^1/2, n=0, 1, 2,..., is orthogonal onL ²((0, ),x dx). In this paper we study the mean convergence of Fourier series with respect to this system for functions whose Hankel transform is supported on [0, 1].Communicated by Mourad Ismail.     

Derivation of mixture distributions and weighted likelihood function as minimizers of KL-divergence subject to constraints

In this article, mixture distributions and weighted likelihoods are derived within an information-theoretic framework and shown to be closely related. This surprising relationship obtains in spite of the arithmetic form of the former and the geometric form of the latter. Mixture distributions are shown to be optima that minimize the entropy loss under certain constraints. The same framework implies the weighted likelihood when the distributions in the mixture are unknown and information from independent samples generated by them have to be used instead. Thus the likelihood weights trade bias for precision and yield inferential procedures such as estimates that can be more reliable than their classical counterparts.     

On the complexity of a family related to the Legendre symbol

Ahlswede, Khachatrian, Mauduit and A. Sárközy introduced the notion of family-complexity of families of binary sequences. They estimated the family-complexity of a large family related to Legendre symbol introduced by Goubin, Mauduit and Sárközy. Here their result is improved, and apart from the constant factor the best lower bound is given for the family-complexity.     

On SU(1,D)/[U(1,D),U(1,D)] for a quaternion division algebra D

Let D be a quaternion division algebra with an involution of the second kind. We show that the quotient group SU(1,D)/[U(1,D),U(1,D)] is nontrivial in general. For global fields, we completely determine this group. Received: 11 May 2007, Revised: 15 December 2007     

Construction of almost periodic solutions to one quasilinear system with two delays

In this paper we construct an almost periodic solution to a quasilinear system with constant and linear delays. We prove the asymptotic stability of this solution.     

Normal <Emphasis Type="Italic">π</Emphasis>-complements for finite groups

Let G be a finite group. For a finite p-group P the subgroup generated by all elements of order p is denoted by Ω₁(p). Zhang [5] proved that if P is a Sylow p-subgroup of G, Ω₁(P) ≦ Z(P) and N _G (Z(P)) has a normal p-complement, then G has a normal p-complement. The object of this paper is to generalize this result. This paper was partly supported by Hungarian National Foundation for Scientific Research Grant # T049841 and T038059.     

Operator Space Embedding of Schatten p-Classes Into Von Neumann Algebra Preduals

Let X₁ and X₂ be subspaces of quotients of R OH and C OH respectively. We use new free probability techniques to construct a completely isomorphic embedding of the Haagerup tensor product into the predual of a sufficiently large QWEP von Neumann algebra. As an immediate application, given any 1 < q ≤ 2, our result produces a completely isomorphic embedding of (equipped with its natural operator space structure) into with a QWEP von Neumann algebra. Received: June 2006, Revision: June 2007, Accepted: September 2007     

Abelian groups small with respect to reduced groups

In this paperwe study

Open image in new window

-small abelian groups, where

Open image in new window

is the class of all reduced groups.

     

A generalization of the monotonicity condition and applications

We introduce a new class of sequences called NBVS to generalize GBVS, essentially extending monotonicity from “one sided” to “two sided”, while some important classical results keep true. Supported in part by Natural Science Foundation of China under grant number 10471130.     

Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator

We give a complete classification of -invariant real hypersurfaces in complex two-plane Grassmannians G ₂(C ^m+2) with commuting normal Jacobi operator . The first author was supported by MCYT-FEDER grant BFM 2001-2871-C04-01, the second author by grant Proj. No. KRF-2006-351-C00004 from Korea Research Foundation and the third author by grant Proj. No. R14-2002-003-01001-0 from Korea Research Foundation, Korea 2006 and Proj. No. R17-2007-006-01000-0 from KOSEF.     

Uniform boundedness of (<Emphasis Type="Italic">C</Emphasis>, 1) means of Jacobi expansions in weighted sup norms. II (Some necessary estimations)

The paper is concerned with bounds for integrals of the type

A characterization of a class of [Z] groups via Korovkin theory

We characterize all the central topological groupsG for which the centreZ(L ¹(G)) of the group algebra admits a finite universal Korovkin set. It is proved thatZ(L ¹(G)) has a finite universal Korovkin set iffĜ is a finite dimensional, separable metric space. This is equivalent to the fact thatG is separable, metrizable andG/K has finite torsion free rank, whereK is a compact open normal subgroup of certain direct summand ofG.     

Equivalence of curves with respect to the action of the symplectic group

We establish an equivalence criterion for finite systems of curves with respect to the action of the symplectic group.     

Holonomic -modules and positive characteristic

We discuss a hypothetical correspondence between holonomic -modules on an algebraic variety X defined over a field of zero characteristic, and certain families of Lagrangian subvarieties in the cotangent bundle to X. The correspondence is based on the reduction to positive characteristic. This article is based on the 5th Takagi Lectures that the author delivered at the University of Tokyo on October 4 and 5, 2008.     

On convergence of sections of sequences in Banach spaces

An elementary proof of the (known) fact that each element of the Banach spaceℓ _w ^p (X) of weakly absolutelyp-summable sequences (if 1≤p<∞) in the Banach spaceX is the norm limit of its sections if and only if each element ofℓ _w ^p (X) is a norm null sequence inX, is given. Little modification to this proof leads to a similar result for a family of Orlicz sequence spaces. Some applications to spaces of compact operators on Banach sequence spaces are considered.