Jacobi-like forms,differential equations,and Hecke operators

We construct a map from the space of Jacobi-like forms [image omitted]() for a discrete subgroup [image omitted] to the space [image omitted] of sequences of meromorphic functions satisfying certain conditions determined by some linear ordinary differential operators and prove that the Hecke operator actions on [image omitted]() and on [image omitted] are compatible with respect to this map.     

Automorphisms of Mn,partially ordered by the star order

Let Mn be the space of all n × n matrices with coefficients in [image omitted] or [image omitted], where n ≥ 3. The star order on Mn is defined by [image omitted] iff [image omitted], where A* is the Hermitian adjoint (i.e., the conjugate transpose) of A. We characterize surjective mappings Φ on Mn such that [image omitted] iff [image omitted]. The tools we use are the Fundamental theorem of projective geometry, Wigner's theorem, and the Penrose decomposition, which we need to describe the main result as well.     

Harmonic analysis in infinite dimension

We study the Hermite transform onL ²() where is a Gaussian measure on a Lusin locally convex spaceE. We are then lead to a Hilbert space () of analytic functions onE which is also a natural range for the Laplace transform. LetB be a convenient Hilbert-Schmidt operator on the Cameron-Martin spaceH of . There exists a natural sequence Cap _n of capacities onE associated toB. This implies the Kondratev-Yokoi theorem about positive linear forms on the Hida test-functions space.     

Fourier transformation of Sato's hyperfunctions

A new generalized function space in which all Gelfand-Shilov classes (α>1) of analytic functionals are embedded is introduced. This space of ultrafunctionals does not possess a natural nontrivial topology and cannot be obtained via duality from any test function space. A canonical isomorphism between the spaces of hyperfunctions and ultrafunctionals on R^k is constructed that extends the Fourier transformation of Roumieu-type ultradistributions and is naturally interpreted as the Fourier transformation of hyperfunctions. The notion of carrier cone that replaces the notion of support of a generalized function for ultrafunctionals is proposed. A Paley-Wiener-Schwartz-type theorem describing the Laplace transformation of ultrafunctionals carried by proper convex closed cones is obtained and the connection between the Laplace and Fourier transformations is established.     

Geodesic currents and Teichmüller space

Consider a hyperbolic surface X of infinite area. The Liouville map assigns to any quasiconformal deformation of X a measure on the space of geodesics of the universal covering X? of X. We show that the Liouville map is a homeomorphism from the Teichmüller space onto its image, and that the image is closed and unbounded. The set of asymptotic rays to consists of all bounded measured laminations on X. Hence, the set of projective bounded measured laminations is a natural boundary for . The action of the quasiconformal mapping class group on continuously extends to this boundary for .     

Meromorphic functions on compact Riemann surfaces and value sharing

Let X be a smooth and connected Riemann surface of genus g 0 and f: X P1, h: X P1 non-constant meromorphic functions on X. Fix an integer n 4 and assume the existence of n distinct points a1, , an P1 such that [image omitted] (set-theoretically) for every i. Here we prove that either f = h or [image omitted].     

Analytic functions of bounded mean oscillation and the Bloch space

In this paper we show that the closure of the space BMOA of analytic functions of bounded mean oscillation in the Bloch spaceB is the image P(U) of space of all continuous functions on the maximal ideal space ofH under the Bergman projection P. It is proved that the radial growth of functions in P(U) is slower than the iterated logarithm studied by Makarov. So some geometric conditions are given for functions in P(U), which we can easily use to construct many Bloch functions not in P(U).     

Invariant subspaces of the quasinilpotent DT-operator

In [4] we introduced the class of DT-operators, which are modeled by certain upper triangular random matrices, and showed that if the spectrum of a DT-operator is not reduced to a single point, then it has a nontrivial, closed, hyperinvariant subspace. In this paper, we prove that also every DT-operator whose spectrum is concentrated on a single point has a nontrivial, closed, hyperinvariant subspace. In fact, each such operator has a one-parameter family of them. It follows that every DT-operator generates the von Neumann algebra of the free group on two generators.     

Composition of hypersingular integral operators

By using the notion of the S-asymptotic of distributions, we give several assertions on the ordinary asymptotic behaviour of the distributional Weierstrass transform. The obtained results are ap-plied on the Cauchy problem for the heat equation     

The QRPS algorithm: A generalization of the QR algorithm for the singular values decomposition of rectangular matrices

A generalization of the QR algorithm proposed by Francis [2] for square matrices is introduced for the singular values decomposition of arbitrary rectangular matrices. Geometrically the algorithm means the subsequent orthogonalization of the image of orthonormal bases produced in the course of the iteration. Our purpose is to show how to get a series of lower triangular matrices by alternate orthogonal-upper triangular decompositions in different dimensions and to prove the convergence of this series.     

On a first order nonlinear differential superordination

In this article we shall extend the nonlinear superordination [image omitted] to the superordination of the form [image omitted]     

An Erdős-Révész type law of the iterated logarithm for stationary Gaussian processes

Summary Let {X(t),t 0} be a stationary Gaussian process withEX(t)=0,EX ²(t)=1 and covariance function satisfying (i)r(t) = 1 2212;C |t | + o (|t|)ast0 for someC>0, 0<2; (ii)r(t)=0(t ^–2) as t for some >0 and (iii) sup_ts|r(t)|<1 for eachs>0. Put (t)= sup {s:0 s t,X(s) (2logs)^1/2}. The law of the iterated logarithm implies a.s. This paper gives the lower bound of (t) and obtains an Erds-Rèvèsz type LIL, i.e., a.s. if 0<<2 and . Applications to infinite series of independent Ornstein-Uhlenbeck processes and to fractional Wiener processes are also given.Research supported by the Fok Yingtung Education Foundation of China and by Charles Phelps Taft Postdoctoral Fellowship of the University of Cincinnati     

Linearly ordered compacta and Banach spaces with a projectional resolution of the identity

We construct a compact linearly ordered space Kω₁ of weight ℵ₁, such that the space C(Kω₁) is not isomorphic to a Banach space with a projectional resolution of the identity, while on the other hand, Kω₁ is a continuous image of a Valdivia compact and every separable subspace of C(Kω₁) is contained in a 1-complemented separable subspace. This answers two questions due to O. Kalenda and V. Montesinos.     

Uniqueness in the characteristic Cauchy problem of the Klein–Gordon equation and tame restrictions of generalized functions

We show that every tempered distribution, which is a solution of the (homogenous) Klein–Gordon equation, admits a “tame” restriction to the characteristic (hyper)surface {x ⁰ + x ⁿ = 0} in (1 + n)-dimensional Minkowski space and is uniquely determined by this restriction. The restriction belongs to the space which we have introduced in (Ullrich in J. Math. Phys. 45, 2004). Moreover, we show that every element of appears as the “tame” restriction of a solution of the (homogeneous) Klein–Gordon equation.     

Commutative Jacobi fields in Fock space

A jacobi field is understood to be a family (Ã()) of commuting selfadjoint operatorsÃ() acting in a Fock space, having a Jacobi structure, and depending linearly on the test functions . In this article, we give a spectral representation of such a family and outline its applications to the theory of distributions on an infinite dimensional space.This article is dedicated to the memory of my dear teacher Mark G. KreinThe work is partially supported by Fundamental Research Foundation of Ukraine, grant 1.4/62.     

Fast and stable QR eigenvalue algorithms for generalized companion matrices and secular equations

Summary We introduce a class of n×n structured matrices which includes three well-known classes of generalized companion matrices: tridiagonal plus rank-one matrices (comrade matrices), diagonal plus rank-one matrices and arrowhead matrices. Relying on the structure properties of , we show that if A then A=RQ , where A=QR is the QR decomposition of A. This allows one to implement the QR iteration for computing the eigenvalues and the eigenvectors of any A with O(n) arithmetic operations per iteration and with O(n) memory storage. This iteration, applied to generalized companion matrices, provides new O(n²) flops algorithms for computing polynomial zeros and for solving the associated (rational) secular equations. Numerical experiments confirm the effectiveness and the robustness of our approach.The results of this paper were presented at the Workshop on Nonlinear Approximations in Numerical Analysis, June 22 – 25, 2003, Moscow, Russia, at the Workshop on Operator Theory and Applications (IWOTA), June 24 – 27, 2003, Cagliari, Italy, at the Workshop on Numerical Linear Algebra at Universidad Carlos III in Leganes, June 16 – 17, 2003, Leganes, Spain, at the SIAM Conference on Applied Linear Algebra, July 15 – 19, 2003, Williamsburg, VA and in the Technical Report [8]. This work was partially supported by MIUR, grant number 2002014121, and by GNCS-INDAM. This work was supported by NSF Grant CCR 9732206 and PSC CUNY Awards 66406-0033 and 65393-0034.     

Nonlinear eigenvalue approximation

Summary For each in some domainD in the complex plane, letF() be a linear, compact operator on a Banach spaceX and letF be holomorphic in . Assuming that there is a so thatI–F() is not one-to-one, we examine two local methods for approximating the nonlinear eigenvalue . In the Newton method the smallest eigenvalue of the operator pencil [I–F(),F()] is used as increment. We show that under suitable hypotheses the sequence of Newton iterates is locally, quadratically convergent. Second, suppose 0 is an eigenvalue of the operator pencil [I–F(),I] with algebraic multiplicitym. For fixed leth() denote the arithmetic mean of them eigenvalues of the pencil [I–F(),I] which are closest to 0. Thenh is holomorphic in a neighborhood of andh()=0. Under suitable hypotheses the classical Muller's method applied toh converges locally with order approximately 1.84.     

Overcompleteness of Sequences of Reproducing Kernels in Model Spaces

We give necessary conditions and sufficient conditions for sequences of reproducing kernels (k_Θ(·, λ_n))_{n ≥ 1} to be overcomplete in a given model space K_Θ^p where Θ is an inner function in H^∞, p ∈ (1, ∞), and where (λ_n)_{n ≥ 1} is an infinite sequence of pairwise distinct points of Under certain conditions on Θ we obtain an exact characterization of overcompleteness. As a consequence we are able to describe the overcomplete exponential systems in L² (0, a).     

The generalized superoptimal preconditioner

For any given n-by-n matrix A, a specific circulant preconditioner t_F(A) introduced by Tyrtyshnikov [E. Tyrtyshnikov, Optimal and super-optimal circulant preconditioners, SIAM J. Matrix Anal. Appl. 13 (1992) 459-473] is defined to be the solution of