Showing papers in "Transactions of the American Mathematical Society in 1991"

Prescribing curvature on compact surfaces with conical singularities

Abstract: We study the Berger-Nirenberg problem on surfaces with conical singularities, i.e, we discuss conditions under which a function on a Riemann surface is the Gaussian curvature of some conformal metric with a prescribed set of singularities of conical types.

Multiplicity of solutions for elliptic problems with critical exponent or with a nonsymmetric term

Abstract: We study the existence of solutions for the following nonlinear degenerate elliptic problems in a bounded domain Q c RN -div(1Vulp-2Vu) = lulp -2U + AUIq2 2u, A > O, where p* is the critical Sobolev exponent, and ulI = 0. By using critical point methods we obtain the existence of solutions in the following cases: If p 0 such that for all A > AO there exists a nontrivial solution. If max(p, p* pl(p 1)) 0. If 1 < q < p there exists A, such that, for 0 < A < AIl, there exist infinitely many solutions. Finally, we obtain a multiplicity result in a noncritical problem when the associated functional is not symmetric.

A cubic counterpart of Jacobi’s identity and the AGM

Abstract: We produce exact cubic analogues of Jacobi's celebrated theta function identity and of the arithmetic-geometric mean iteration of Gauss and Legendre. The iteration in question is $a_n+1 := a_n + 2b_n / 3$ and b_n+1 := [formula cannot be replicated]. The limit of this iteration is identified in terms of the hypergeometric function ₂F₁ (1/3, 2/3; 1 ; ·), which supports a particularly simple cubic transformation.

Fractal drum, inverse spectral problems for elliptic operators and a partial resolution of the Weyl-Berry conjecture

Abstract: Let Í2 be a bounded open set of E\" (n > 1) with \"fractal\" boundary T . We extend Hermann Weyl's classical theorem by establishing a precise remainder estimate for the asymptotics of the eigenvalues of positive elliptic operators of order 2m (m > 1) on Í2 . We consider both Dirichlet and Neumann boundary conditions. Our estimate—which is expressed in terms of the Minkowski rather than the Hausdorff dimension of Y—specifies and partially solves the Weyl-Berry conjecture for the eigenvalues of the Laplacian. Berry's conjecture—which extends to \"fractals\" Weyl's conjecture—is closely related to Kac's question \"Can one hear the shape of a drum?\"; further, it has significant physical applications, for example to the scattering of waves by \"fractal\" surfaces or the study of porous media. We also deduce from our results new remainder estimates for the asymptotics of the associated \"partition function\" (or trace of the heat semigroup). In addition, we provide examples showing that our remainder estimates are sharp in every possible \"fractal\" (i.e., Minkowski) dimension. The techniques used in this paper belong to the theory of partial differential equations, the calculus of variations, approximation theory and—to a lesser extent—geometric measure theory. An interesting aspect of this work is that it establishes new connections between spectral and \"fractal\" geometry.

On the existence of conformal measures

Abstract: A general notion of conformal measure is introduced and some basic properties are studied. Sufficient conditions for the existence of these measures are obtained, using a general construction principle. The geometric properties of conformal measures relate equilibrium states and Hausdorff measures. This is shown for invariant subsets of S under expanding maps.

A representation theorem for cyclic analytic two-isometries

Abstract: A bounded linear operator T on a complex separable Hilbert space Z is called a 2-isometry if T*2T2-2T* T+I = O . We say that T is analytic if nn,0 Tnt = (0) . In this paper we show that every cyclic analytic 2-isometry can be represented as multiplication by z on a Dirichlet-type space D(8). Here ,ze denotes a finite positive Borel measure on the unit circle. For two measures ,u and v the 2-isometries obtained as multiplication by z on D(8) and D(v) are unitarily equivalent if and only if ,u = v . We also investigate similarity and quasisimilarity of these 2-isometries, and we apply our results to the invariant subspaces of the Dirichlet shift.

bounds for spectral multipliers on nilpotent groups

Abstract: A criterion is given for the LP boundedness of a class of spec- tral multiplier operators associated to left-invariant, homogeneous subelliptic second-order differential operators on nilpotent Lie groups, generalizing a theo- rem of Hormander for radial Fourier multipliers on Euclidean space. The order of differentiability required is half the homogeneous dimension of the group, improving previous results in the same direction.

Representations of knot groups in SU(2)

Abstract: This paper is a study of the structure of the space R(K) of representations of classical knot groups into SU(2) . Let R(K) equal the set of conjugacy classes of irreducible representations. In gI, we interpret the relations in a presentation of the knot group in terms of the geometry of SU(2); using this technique we calculate R(K) for K equal to the torus knots, twist knots, and the Whitehead link. We also determine a formula for the number of binary dihedral representations of an arbitrary knot group. We prove, using techniques introduced by Culler and Shalen, that if the dimension of R(K) is greater than 1, then the complement in SJ of a tubular neighborhood of K contains closed, nonboundary parallel, incompressible surfaces. We also show how, for certain nonprime and doubled knots, R(K) has dimension greater than one. In gII, we calculate the Zariski tangent space, Tp(R(K)), for an arbitrary knot K, at a reducible representation p, using a technique due to Weil. We prove that for all but a finite number of the reducible representations, dimTp(R(K)) = 3. These nonexceptional representations possess neighborhoods in R(K) containing only reducible representations. At the exceptional representations, which correspond to real roots of the Alexander polynomial, dim Tp(R(K)) = 3 + 2k for a positive integer k . In those examples analyzed in this paper, these exceptional representations can be expressed as limits of arcs of irreducible representations. We also give an interpretation of these "extra" tangent vectors as representations in the group of Euclidean isometries of the plane. 0. INTRODUCTION This paper is a study of the space R(K) of representations of the fundamental group of the complement of a knot K into the Lie group SU(2) . The major results are as follows: (1) a calculation of the topological type of R(K) for K a torus knot, a twist knot, or the Whitehead link, (2) a determination of the Zariski tangent space to R(K) at a reducible representation for an arbitrary knot K, and (3) a proposition and some examples relating the dimension of R(K) to incompressible surfaces in the complement of K. While the idea of representing 3-manifold groups in SU(2) is a fairly recent one, the representation of these groups in other groups has a substantial history. We will not attempt to chronicle this history in detail, but will comment here on Received by the editors January 18, 1989 and, in revised form, July 25, 1989. 1980 Mathematics Subject Classification ( 1985 Revision). Primary 57M05, 57M25.

Quantitative stability of variational systems. I. The epigraphical distance

Abstract: This paper proposes a global measure for the distance between the elements of a variational system (parametrized families of optimization problems).

On the braid index of alternating links

Abstract: We show that, at least for an alternating fibered link or 2-bridge link L, there is an exact formula which expresses the braid index b(L) of L as a function of the 2-variable generalization PL(I, m) of the Jones polynomial

Crossed simplicial groups and their associated homology

Abstract: We introduce a notion of crossed simplicial group, which generalizes Connes' notion of the cyclic category. We show that this concept has several equivalent descriptions and give a complete classification of these structures. We also show how many of Connes' results can be generalized and simplified in this framework. A simplicial set (resp. group) is a family of sets (resp. groups) {Gn}n>0 together with maps (resp. group homomorphisms) which satisfy some well-known universal formulas. The geometric realization of a simplicial set is a space and the geometric realization of a simplicial group is a topological group. We define a crossed simplicial group as a simplicial set 6\\ = {Gn}n>0 such that the Gn 's are groups and the faces and degeneracies are crossed group homorphisms, that is, satisfy a formula like f(gg') = f(g)(g-f)(g') (see §1 for the precise definition). A simplicial group is thus the particular case of a trivial action. The geometric realization of a crossed simplicial group is still a topological group. The reason for introducing such objects comes from cyclic homology whose definition, as given by Connes in [C], relies on the existence of a certain category A (denoted AC in this paper ) satisfying some special properties. In fact these properties are equivalent to the following assertion: the standard simplicial circle can be endowed with the structure of a crossed simplicial group Ct {Cn}n>0 with Cn = Z/n + l (cyclic groups). In [L] we remarked that the family of dihedral groups {Dn+l}n>0 (resp. quaternion groups {ô„+1}„>o ) forms a crossed simplicial group (but not a simplicial group). The notion of crossed simplicial group provides a useful conceptual framework for studying these basic examples. In this paper we investigate the existence of other families of groups bearing a crossed simplicial group structure. In particular we show that it is the case for the family of symmetric groups St {Sn+l}n>0. Then in 3.6 we give a complete classification theorem: Received by the editors May 28, 1988. 1980 Mathematics Subject Classification (1985 Revision). Primary 18D05, 18F25, 18G30. © 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page

$q$-tensor space and $q$-Weyl modules

Abstract: We obtain the irreducible representations of the q-Schur algebra, motivated by the fact that these representations give all the irreducible representations of GLn(q) in the nondescribing characteristic The irreducible polynomial representations of the general linear groups in the describing characteristic are a special case of this construction The theory of polynomial representations of general linear groups is equivalent to the representation theory of Schur algebras (see Green's book [5] and the bibliography therein) In [4], we defined q-analogues of Schur algebras When q = 1, these are the usual Schur algebras, and when q is a prime power, representations of q-Schur algebras give a substantial part of the representation theory of finite general linear groups in the nondescribing characteristic case, including all irreducible representations of these groups and important information about decomposition numbers It is natural to ask what features of the classical Schur algebras have qanalogues In this paper, we define q-analogues of tensor space, of Weyl modules, and of weight spaces, thereby generalizing the main reslllts which appear in Green's book [5] For example, we classify the irreducible modules for qSchur algebras, we determine bases for q-Weyl modules compatible with weight spaces, and we give results on composition multiplicities of the irreducible modules in q-Weyl modules The proofs are largely self-contained, so by specializing q to 1, we recover the corresponding results in [5] This paper, therefore, is relevant to the representation theory of symmetric groups and to the representation theory of general linear groups in the describing and in the nondescribing characteristics 1 THE q-SCHUR ALGEBRA Let r be a natural number, let R be an integral domain, and let q be a unit in R We denote the symmetric group on r letters by er The Hecke algebra Z is the R-free R-algebra with basis {Twlw E er} where the multiplication is determined by the following rule If a = (i, i + 1) is a basic transposition in Received by the editors August 15, 1989 1980 Mathematics Subject Classification ( 198 5 Revision) Primary 1 6A64, 1 6A6 5 ; Secondary 20C30 This research was supported in part by NSF Grant No DMS-8802290 The authors gratefully acknowledge support received from NATO Grant No 0222/87 (r) 1991 American Mathematical Society 0002-9947/91 $100 + $25 per page

The construction of analytic diffeomorphisms for exact robot navigation on star worlds

Abstract: The authors consider the construction of navigation functions on configuration spaces whose geometric expressiveness is rich enough for navigation amidst real-world obstacles. They describe a general methodology which extends the construction of navigation functions on sphere worlds to any smoothly deformable space. According to this methodology, the problem of constructing a navigation function is reduced to the construction of a transformation mapping a given space into its model sphere world. The transformation must satisfy certain regularity conditions guaranteeing invariance of the navigation function properties. The authors demonstrate this idea by constructing navigation functions on star worlds: n-dimensional star shaped subsets of E/sup n/ punctured by any finite number of smaller disjoint n-dimensional stars. This construction yields automatically a bounded torque feedback control law which is guaranteed to guide the robot to destination point from almost every initial position without hitting any obstacle. >

Stable patterns in a viscous diffusion equation

Abstract: We consider a pseudoparabolic regularization of a forward-backward nonlinear diffusion equation ut = A(f(u) + vut), motivated by the problem of phase separation in a viscous binary mixture. The function f is nonmonotone, so there are discontinuous steady state solutions corresponding to arbitrary arrangements of phases. We find that any bounded measurable steady state solution u(x) satisfying f(u) = constant, f'(u(x)) > 0 a.e. is dynamically stable to perturbations in the sense of convergence in measure. In particular, smooth solutions may achieve discontinuous asymptotic states. Furthermore, stable states need not correspond to absolute minimizers of free energy, thus violating Gibbs' principle of stability for phase mixtures.

Tilings of the torus and the Klein bottle and vertex-transitive graphs on a fixed surface

Abstract: We describe all regular tiings of the torus and the Klein bottle. We apply this to describe, for each orientable (respectively nonorientable) surface S, all (but finitely many) vertex-transitive graphs which can be drawn on S but not on any surface of smaller genus (respectively crosscap number). In particular, we prove the conjecture of Babai that, for each g > 3, there are only finitely many vertex-transitive graphs of genus g. In fact, they all have order 2, there are only finitely many groups that act on the surface of genus g . We also derive a nonorientable version of Hurwitz' theorem.

The associated Askey-Wilson polynomials

Abstract: We derive some contiguous relations for very well-poised 8 7 series and use them to construct two linearly independent solutions of the three-term recurrence relation of the associated Askey-Wilson polynomials. We then use these solutions to find explicit representations of two families of associated Askey-Wilson polynomials. We identify the corresponding continued fractions as quotients of two very well-poised 807 series and find the weight functions.

Shadows of convex bodies

Abstract: It is proved that if C is a convex body in Rn then C has an affine image C (of nonzero volume) so that if P is any 1-codimensional orthogonal projection, |PCa > 1Žcl(n-l)/n It is also shown that there is a pathological body, K, all of whose orthogonal projections have volume about VX/ times as large as IKI(n 1)/n 0. INTRODUCTION The problems discussed in this paper concern the areas of shadows (orthogonal projections) of convex bodies and, to a lesser extent, the surface areas of such bodies. If C is a convex body in iRn and 0 a unit vector, P. C will denote the orthogonal projection of C onto the 1-codimensional space perpendicular to 0. Volumes and areas of convex bodies and their surfaces will be denoted with I 1. The relationship between shadows and surface areas of convex bodies is expressed in Cauchy's well-known formula. For each n E N, let vn be the volume of the n-dimensional Euclidean unit ball and let a = an-, be the rotationally invariant probability on the unit sphere Sn. Cauchy's formula states that if C is a convex body in Rn then its surface area is =ac I S'P6CI dcr(O). vnI Jn I The classical isoperimetric inequality in Rn states that any body has surface area at least as large as a Euclidean ball of the same volume. The first section of this paper is devoted to the proof of a "local" isoperimetric inequality showing that all bodies have large shadows rather than merely large surface area (or average shadow). The principal motivation for this result is its relationship to a conjecture of Vaaler and the important problems surrounding it. This theorem and its connection with Vaaler's conjecture are described at the beginning of § 1. An important role is played in the theory of convex bodies by the so-called "projection body" of a convex body. It is easily seen, by considering polytopes, Received by the editors September 12, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 52A20, 52A40. The author was supported in part by NSF DMS-8807243. ( 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page

Linearization of bounded holomorphic mappings on Banach spaces

Abstract: The main result in this paper is the following linearization theorem. For each open set U in a complex Banach space E, there is a complex Banach space Goo (U) and a bounded holomorphic mapping gu: U -G? (U) with the following universal property: For each complex Banach space F and each bounded holomorphic mapping f: U -* F , there is a unique continuous linear operator Tf: Goo (U) -F such that Tf ? gu = f . The correspondence f Tf is an isometric isomorphism between the space Ho (U; F) of all bounded holomorphic mappings from U into F , and the space L(G?? (U); F) of all continuous linear operators from G? (U) into F. These properties characterize G??'(U) uniquely up to an isometric isomorphism. The rest of the paper is devoted to the study of some aspects of the interplay between the spaces H?O(U;F) and L(G?(U);F). This paper consists of five sections. In ? 1 we establish our notation and terminology. In ?2 we prove the aforementioned linearization theorem. In ?3 we translate certain properties of a mapping f E H' (U; F) into properties of the corresponding operator Tf E L(G'(U); F). We show, for instance, that f has a relatively compact range if and only if Tf is a compact operator. In ?4 we give a seminorm characterization of the unique locally convex topology T y on H?(U; F) such that the correspondence f --* Tf is a topological isomorphism between the spaces (H (U; F), TY) and (L(G ( U); F), T,), where TC denotes the compact-open topology. Finally, in ? 5 we use the preceding results to establish necessary and sufficient conditions for the spaces G? (U) and H? (U) to have the approximation property. These are holomorphic analogues of classical results of A. Grothendieck [8], and complement results of R. Aron and M. Schottenloher [2]. We show, in particular, that if U is a balanced, bounded, open set in a complex Banach space E, then G? (U) has the approximation property if and only if E has the approximation property. We also show that if U is an arbitrary open set in a complex Banach space E, then H? (U) has the approximation property if and only if, for each complex Banach space F, each mapping in H?(U; F) with a relatively compact range can be uniformly approximated on U by mappings in H?(U; F) with finite-dimensional range. Since it is still unknown whether Received by the editors April 10, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46G20, 46E15; Secondary 46E10. (D 1991 American Mathematical Society 0002-9947/91 $1.00+ $.25 per page

Maxwell’s equations in a periodic structure

Abstract: Consider a diffraction of a beam of particles in R3 when the dielectric coefficient is a constant cl above a surface S and a constant e2 below a surface S, and the magnetic permeability is constant throughout RIi. S is assumed to be periodic in the xl direction and of the form x1 = f1(s), X3 = f3(s), x2 arbitrary. We prove that there exists a unique solution to the time-harmonic Maxwell equations in R 3 having the form of refracted waves for x3 > 1 and of transmitted waves for -X3 >> 1 if and only if there exists a unique solution to a certain system of two coupled Fredholm equations. Thus, in particular, for all the E 's, except for a discrete number, there exists a unique solution to the Maxwell equations. INTRODUCTION In this paper we consider the Maxwell equations for time harmonic solutions in the entire space R3 with piecewise constant dielectric coefficient having jump across a periodic surface. The magnetic permeability ,u is assumed to be constant whereas the dielectric coefficient e is given by e = 8, above a surface S: x3 = f(xl) and e = e2 below the surface S; e, and 82 are different constants. If S is a half-space {x3 = 0} then the solution Eo Ho can be computed explicitly. We assume in this paper that S is periodic, i.e., f(xl + L) =f(xl) VX1 E R (L > 0). We wish to find a solution E, H such that E Eo and H Ho are superpositions of "transmitted' waves (0.1) in {x3 A} where A >maxlfl. In ?? 1-7 we assume that f E C2 and we reduce the solution of the Maxwell equations to a Fredholm system of four integral equations; in ?8 we reduce it further to a Fredholm system of two integral equations. Thus for all but a discrete sequence of values of the physical parameters there exists a unique solution to the integral equations, yielding a solution of the Maxwell equations; Received by the editors January 25, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 78A10, 78A45; Secondary 35P25, 47A40.

Weighted Sobolev-Poincaré inequalities and pointwise estimates for a class of degenerate elliptic equations

Abstract: In this paper we prove a Sobolev-Poincare inequality for a class of function spaces associated with some degenerate elliptic equations. These estimates provide us with the basic tool to prove an invariant Harnack inequality for weak positive solutions. In addition, Holder regularity of the weak solutions follows in a standard way. Let y = Z,7 ai(aij9 a ) be a second-order degenerate elliptic operator in divergence form with measurable coefficients. In this paper we shall obtain pointwise estimates for the weak solutions of Su = 0 (H61der continuity of the weak solutions and Harnack inequality for nonnegative solutions). Let us recall that the original results for elliptic operators were obtained by De Giorgi, Nash, and Moser. An extensive bibliography about the degenerate case can be found in [FLI, FL2, FS]. To introduce the results of the present paper, let us recall some recent results. In [FL1, FL2] a suitable metric d is associated with the differential operator Y in such a way that we obtain a new geometry which is natural for the degenerate operator as the Euclidean geometry is natural for the Laplace operator (or, more precisely, as a suitable Riemannian geometry is natural for a secondorder elliptic operator). In the smooth case, this idea is contained in many papers: we refer to [FP, NSW]. The basic results in [FL 1, FL2] are obtained via a precise description of this geometry under suitable technical hypotheses on the coefficients whose aim is to give a nonsmooth formulation of the Hormander hypoellipticity condition for sum-of-squares operators. We note that the same idea is used in [NSW, S, J, V] to obtain pointwise estimates for sum-of-squares operators. On the other hand, a different class of degenerate elliptic operators is considered in [FKS]: instead of a geometrical degeneracy, a measure degeneracy is allowed. A typical example of this class is given by Yu = div(wo(x)Vu), where cl is a weight function belonging to the A2-class of Muckenhoupt. Unified results for a class containing both the operators in [FL 1] and in [FKS] have Received by the editors August 1, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 46E35, 35J70. Partially supported by G.N.A.F.A. of C.N.R. and M.U.R.S.T., Italy. ( 1991 American Mathematical Society 0002-9947/91 $1.00 + $.25 per page

Commutation methods applied to the mKdV-equation

Abstract: An explicit construction of solutions of the modified Korteweg-de Vries equation given a solution of the (ordinary) Korteweg-de Vries equation is provided. Our theory is based on commutation methods (i.e., N = 1 supersymmetry) underlying Miura's transformation that links solutions of the two evolution equations.

A spectral commutant lifting theorem

Abstract: ©1991 American Mathematical Society. First published in Transactions of the American Mathematical Society, Vol. 325, No. 2, June 1991 by the American Mathematical Society

Convex optimization and the epi-distance topology

Abstract: Let 17(X) denote the proper, lower semicontinuous, convex functions on a Banach space X, equipped with the completely metrizable topology T of uniform convergence of distance functions on bounded sets. A function f in 17(X) is called well-posed provided it has a unique minimizer, and each minimizing sequence converges to this minimizer. We show that well-posedness of f e r(X) is the minimal condition that guarantees strong convergence of approximate minima of T-approximating functions to the minimum of f . Moreover, we show that most functions in (17(X), Taw ) are well-posed, and that this fails if 17(X) is topologized by the weaker topology of Mosco convergence, whenever X is infinite dimensional. Applications to metric projections are also given, including a fundamental characterization of approximative compactness.

On the existence and uniqueness of positive solutions for competing species models with diffusion

Abstract: In this paper, we consider strictly positive solutions of competing species systems with diffusion under Dirichlet boundary conditions. We obtain a good understanding of when strictly positive solutions exist, obtain new nonuniqueness results and a number of other results, showing how complicated these equations can be. In particular, we consider how the shape of the underlying domain affects the behaviour of the equations. The purpose of this paper is to obtain much better results on the existence and uniqueness of strictly positive stationary (that is time-independent) solutions of

Chebyshev polynomials in several variables and the radial part of the Laplace-Beltrami operator

Abstract: Chebyshev polynomials of the first and the second kind in n variables :1 Z25 n Zn are introduced. The Yariables : n Z2 ... ' Zn are the characters of the representations of SL(n + 1, C) corresponding to the fundamental weights. The Chebyshev polynomials are eigenpolynomials of a second order linear partial differential operator which is in fact the radial part of the Laplace-Beltrami operator on certain symmetric spaces. We give an explicit expression of this operator in the coordinates z1 ff Z2 S . . . ) Zn and then show how many results in the literature on differential equations satisfied by Chebyshev polynomials in several variables follow immediately from well-known results on the radial part of the Laplace-Beltrami operator. Related topics like orthogonality, symmetry relations, generating functions and recurrence relations are also discussed. Finally we note that the Chebyshev polynomials are a special case of a more general class of orthogonal polynomials in several variables.

Stability of travelling wave solutions of diffusive predator-prey systems

Abstract: The stability of travelling wave solutions of singularly perturbed, diffusive predator-prey systems is proved by showing that the linearized operator about such a solution has no unstable spectrum and that the translation eigenvalue at A = 0 is simple. The proof illustrates the application of some recently developed geometric and topological methods for counting eigenvalues.

The symbolic representation of billiards without boundary condition

Abstract: We consider a dynamical system with elastic reflections in the whole plane and show that such a dynamical system can be represented as a symbolic flow over a mixing subshift of finite type. This fact enables us to prove an analogue of the prime number theorem for the closed orbits of such a dynamical system.

Ramsey theorems for knots, links and spatial graphs

Abstract: An embedding f: G —► R of a graph G into R is said to be linear if each edge f(e) (e G E(G)) is a straight line segment. It will be shown that for any knot or link type k , there is a finite number R(k) such that every linear embedding of the complete graph Kn with at least R(k) vertices (n > R(k)) in R contains a knot or link equivalent to k .

Inner amenable locally compact groups

Abstract: In this paper we study the relationship between amenability, inner amenability and property P of a von Neumann algebra. We give necessary conditions on a locally compact group G to have an inner invariant mean m such that m(V) = 0 for some compact neighborhood V of G invariant under the inner automorphisms. We also give a sufficient condition on G (satisfied by the free group on two generators or an I.C.C. discrete group with Kazhdan's property T, e.g., SL(n, Z), n > 3) such that each linear form on L 2(G) which is invariant under the inner automorphisms is continuous. A characterization of inner amenability in terms of a fixed point property for left Banach G-modules is also obtained.

On the convergence of moment problems

Abstract: We study the problem of estimating a nonnegative density, given a finite number of moments. Such problems arise in numerous practical applications. As the number of moments increases, the estimates will always converge weak * as measures, but need not converge weakly in L₁. This is related to the existence of functions on a compact metric space which are not essentially Riemann integrable (in some suitable sense). We characterize the type of weak convergence we can expect in terms of Riemann integrability, and in some cases give error bounds. When the estimates are chosen to minimize an objective function with weakly compact level sets (such as the Bolzmann-Shannon entropy) they will converge weakly in L₁. When an Lp norm (1 < p < ∞) is used as the objective, the estimates actually converge in norm. These results provide theoretical support to the growing popularity of such methods in practice.