Positivity aspects of the Fantappiè transform

A Hilbert space approach to the classical Fantappiè transform, based on the concept of Gel'fand triples of locally convex spaces, leads to a novel proof of Martineau-Aizenberg duality theorem. A study of Fantappiè transforms of positive measures on the unit ball inC ⁿ relates ideas of realization theory of multivariate linear systems, locally convex duality and pluripotential theory. This is applied to obtain von Neumann type estimates on the joint numerical range of tuples of Hilbert space operators. Partially supported by National Science Foundation Grants DMS 0070639 and DMS 0322255. Partially supported by National Science Foundation Grant DMS 0100367.     

Polynomial representations of the orthogonal groups

This paper is concerned with realizations of the irreducible representations of the orthogonal group and construction of specific bases for the representation spaces. As is well known, Weyl's branching theorem for the orthogonal group provides a labeling for such bases, called Gelfand-Žetlin labels. However, it is a difficult problem to realize these representations in a way that gives explicit orthogonal bases indexed by these Gelfand-–etlin labels. Thus, in this paper the irreducible representations of the orthogonal group are realized in spaces of polynomial functions over the general linear groups and equipped with an invariant differentiation inner product, and the Gelfand-Žetlin bases in these spaces are constructed explicitly. The algorithm for computing these polynomial bases is illustrated by a number of examples. Partially supported by a grant from the Department of Energy. Partially supported by NSF grant No. MCS81-02345.     

Compactly supported box-spline wavelets

A general procedure for constructing multivariate non-tensor-product wavelets that generate an orthogonal decomposition ofL ²(R)^s,s s≥1, is described and applied to yield explicit formulas for compactly supported spline-wavelets based on the multiresolution analysis ofL ²(R)^s 1≤s≤3, generated by any box spline whose direction set constitutes a unimodular matrix. In particular, when univariate cardinal B-splines are considered, the minimally supported cardinal spline-wavelets of Chui and Wang are recovered. A refined computational scheme for the orthogonalization of spaces with compactly supported wavelets is given. A recursive approximation scheme for “truncated” decomposition sequences is developed and a sharp error bound is included. A condition on the symmetry or anti-symmetry of the wavelets is applied to yield symmetric box-spline wavelets. Partially supported by ARO Grant DAAL 03-90-G-0091 Partially supported by NSF Grant DMS 89-0-01345 Partially supported by NATO Grant CRG 900158.     

Processes of Meixner type

Canonical form, self-decomposability, and the Esscher transforms of Meixner processes are discussed. Mixed Meixner processes are defined and characterized as Markov processes or semimartingales. Ornstein-Uhlenbeck and selfsimilar processes of Meixner type are also described. Partially supported by the Lithuanian State Science and Studies Foundation. Institute of Mathematics and Informatics, Akademijos 4; Vilnius University, Naugarduko 24, 2600 Vilnius, Lithuania. Published in Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 40–51, January–March, 1999.     

A Riesz basis for Bargmann-Fock space related to sampling and interpolation

It is shown that the Bargmann-Fock spaces of entire functions, A^p (C),p≧1 have a bounded unconditional basis of Wilson type [DJJ] which is closely related to the reproducing kernel. From this is derived a new sampling and interpolation result for these spaces. Partially supported by grant AFOSR 90-0311.     

Distances between measures from 1-dimensional projections as implied by continuity of the inverse radon transform

Summary Distances between measures on IR ^d are determined from distances between their 1-dimensional projections. The method employed involves considering the 1-dimensional projections to be the Radon transform of the measures. Crucial to the main theorem is a continuity result for the inverse Radon transform. Focus is restricted to the Prohorov, dual bounded Lipschitz and d _k metrics which metrize weak convergence of probability measures. These metrics are related to each other and to the Sobolev norms. The d _k results extend from measures to generalized functions.Partially supported by NSF Grant No. MCS-81-01895Partially supported by NSF Grant No. MCS-82-01627 and support from the Mellon Foundation     

Critical Gevrey index for hypoellipticity of parabolic operators and Newton polygones

Summary In this paper we give geometrical expressions of the (non) hypoellipticity in Gevrey spaces of parabolic operators via Newton polygones. We also determine the critical Gevrey class for which the hypoellipticity holds.Partially supported by GNAFA, CNR, Italy.Partially supported by JSPS, Japan and a grant MM-410/94 with MES, Bulgaria.Partially supported by Chuo University special research fund.     

Approximation of infinite matrices by matricial Haar polynomials

The main goal of this paper is to extend the approximation theorem of contiuous functions by Haar polynomials (see Theorem A) to infinite matrices (see Theorem C). The extension to the matricial framework will be based on the one hand on the remark that periodic functions which belong toL ^∞ (T) may be one-to-one identified with Toeplitz matrices fromB(l ₂) (see Theorem 0) and on the other hand on some notions given in the paper. We mention for instance:ms—a unital commutative subalgebra ofl ^∞,C(l ₂) the matricial analogue of the space of all continuous periodic functionsC(T), the matricial Haar polynomials, etc. In Section 1 we present some results concerning the spacems—a concept important for this generalization, the proof of the main theorem being given in the second section. Partially supported by EUROMMAT ICA1-CT-2000-70022. Partially supported by V-Stabi-RUM/1022123. Partially supported by EUROMMAT ICA1-CT-2000-70022 and V-Stabi-RUM/1022123.     

COMPLEX MONGE-AMPEREEQUATIONS ON GENERAL DOMAINS

Abstract. The smooth solutions of the Dirichlet problems for the complex Monge-Ampere equa-tions on general smooth domains are found ,provided that there exists a C3 strictly plurisub har-monic subsolution with prescribed boundary value. It is the smooth version of an existence theo-rem given by Bedford and Taylor.     

A vector valued transference of Fourier-Bessel multipliers onto Hankel multipliers with applications

In this note we prove a vector valued transference theorem relating Fourier-Bessel multipliers and Hankel multipliers. An application of such a transference theorem allows to show that results of Córdoba [3] and Romera [4] can be deduced from a recent result of Balodis and Córdoba [1, Theorem 3]. Partially supported by DGICYT Grant PB 97-1489 (Spain). Partially supported by KBN grant # 2 PO3A 034 20.     

Operators on harmonic Bergman spaces

We study the commutator of the multiplication and harmonic Bergman projection, Hankel and Toeplitz operators on the harmonic Bergman spaces. The same type operators have been well studied on the analytic Bergman spaces. The main difficulty of this study is that the bounded harmonic function space is not an algebra! In this paper, we characterize theL ^p boundedness and compactness of these operators with harmonic symbols. Results about operators in Schatten classes, the cut-off phenomenon and general symbols are also included.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

On a class of finite dimensional contractive perturbations of restricted shifts of finite multiplicity

We study a class of finite-dimensional contractive perturbations of shift operators of finite multiplicity restricted to left invariant subspaces of vectorialH ² spaces. We determine their spectra in terms of the characteristic function of the unperturbed operator and the perturbation. Partially supported by the Batsheva de Rothschild Fund for the Advancement of Science and Technology.     

The Heckman–Opdam Markov processes

We introduce and study the natural counterpart of the Dunkl Markov processes in a negatively curved setting. We give a semimartingale decomposition of the radial part, and some properties of the jumps. We prove also a law of large numbers, a central limit theorem, and the convergence of the normalized process to the Dunkl process. Eventually we describe the asymptotic behavior of the infinite loop as it was done by Anker, Bougerol and Jeulin in the symmetric spaces setting in (Iberoamericana 18: 41–97, 2002). Partially supported by the European Commission (IHP Network HARP 2002–2006).     

Geometric analysis of conditional independence on Wiener space

Summary In this paper we study the conditional independence of sigma-fields on the Wiener space using the tools of the Stochastic Calculus of Variations. Particular emphasis is given to the relation between the splitting of the (random) tangent spaces associated to the sigma-fields and the conditional independence.Partially supported by the CICYT grant n^o PB86-0238     

Excision in Bivariant Periodic Cyclic Cohomology: A Categorical Approach

We extend Cuntz and Quillen's excision theorem for algebras and pro-algebras in arbitrary Q-linear categories with tensor product.CONICET Researcher and ICTP Associate. Partially supported by grant UBACyTX066.IMCAIMCA-PUCP. Partially supported by CONCYTEC grant CS017-2002-OAJ.     

A Second Main Theorem of Nevanlinna Theory for Meromorphic Functions on Complex Submanifolds in C<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

We show a second main theorem of Nevalinna theory for meromorphic functions on complex submanifolds in C ⁿ . This has a similar form to the classical one and has a remainder term including Ricci curvature. We also give a concrete computation of the remainder term in the case of nonsingular algebraic submanifolds. Partially supported by the Grant-in-Aid for Scientific Research (C), Japan Society for the Promotion of Science.     

Regularity of the degenerate Monge-Ampère equation on compact Kähler manifolds

We study the C^1,1 and Lipschitz regularity of the solutions of the degenerate complex Monge-Ampère equation on compact K?hler manifolds. In particular, in view of the local regularity for the complex Monge-Ampère equation, the obtained C^1,1 regularity is a generalization of the Yau theorem which deals with the nondegenerate case. Received: 18 April 2002 / Published online: 24 February 2003 Partially supported by KBN Grant #2 P03A 028 19     

Sufficient conditions for bounded point evaluations

A geometric condition for the existence of bounded evaluations is given. Using this criterion for sets of finite perimeter, it is shown that P²(wdm₂) has analytic bounded point evaluations, when w is of bounded variation with compact support. This theorem is then related to previous work on bounded point evaluations.Partially supported by a grant from the Research Grants Committee of the University of Alabama.     

Standard position for objects in hyperbolic space

It is shown that, by applying hyperbolic isometries, one may arrangen points in hyperbolic (n–1)-space in a certain standard position. Similar results are developed for complex hyperbolic space, as well as hyperbolic spaces defined over nearly arbitrary fields. The algebraic basis of the paper is the determination of the structure of a double coset space which occurs in representation theory.Partially supported by NSA grant # MDA-904-96-0018.Partially supported by NSA grant # MDA904-95-1-1089.     

Sobolev type embeddings in the limiting case

We use interpolation methods to prove a new version of the limiting case of the Sobolev embedding theorem, which includes the result of Hansson and Brezis-Wainger for W _n ^k/k as a special case. We deal with generalized Sobolev spaces W _A ^k , where instead of requiring the functions and their derivatives to be in L_n/k, they are required to be in a rearrangement invariant space A which belongs to a certain class of spaces “close” to L_n/k. We also show that the embeddings given by our theorem are optimal, i.e., the target spaces into which the above Sobolev spaces are shown to embed cannot be replaced by smaller rearrangement invariant spaces. This slightly sharpens and generalizes an, earlier optimality result obtained by Hansson with respect to the Riesz potential operator. In memory of Gene Fabes. Acknowledgements and Notes This research was supported by Technion V.P.R. Fund-M. and C. Papo Research Fund.