On the Right-Definite and Left-Definite Spectral Theory of the Legendre Polynomials

In this paper, we further develop the left-definite and right-definite spectral theory associated with the self-adjoint differential operator A in L²(-1,1), generated from the classical second-order Legendre differential equation, having the sequence of Legendre polynomials as eigenfunctions. Specifically, we determine the first three left-definite spaces associated with the pair (L²(-1,1),A). As a consequence of these results, we determine the explicit domain of both the associated left-definite operator A₁, first observed by Everitt, and the self-adjoint operator A^1/2. In addition, we give a new characterization of the domain D(A) of A and, as a corollary, we present a new proof of the Everitt-Mari result which gives optimal global smoothness of functions in D(A).     

Lower Bounds for n-Term Approximations of Plane Convex Sets and Related Topics

In this paper, we establish lower bounds for n-term approximations in the metric of L ²⁽I ² ) of characteristic functions of plane convex subsets of the square I ² with respect to arbitrary orthogonal systems. It is shown that, as n, these bounds cannot decrease more rapidly than .     

Berezin Symbols and Schatten--von Neumann Classes

In terms of Berezin symbols, we give several criteria for operators to belong to the Schatten--von Neumann classes . In particular, for functions of model operators, we give a complete answer to a question posed by Nordgren and Rosenthal.     

Adjoint L-Values and Primes of Congruence for Hilbert Modular Forms

Let f be a primitive Hilbert modular cusp form of arbitrary level and parallel weight k, defined over a totally real number field F. We define a finite set of primes that depends on the weight and level of f, the field F, and the torsion in the boundary cohomology groups of the Borel–Serre compactification of the underlying Hilbert-Blumenthal variety. We show that, outside , any prime that divides the algebraic part of the value at s=1 of the adjoint L-function of f is a congruence prime for f. In special cases we identify the boundary primes in terms of expressions of the form , where is a totally positive unit of F.     

A description of Hilbert -modules in which all closed submodules are orthogonally closed

Let , be -algebras and a full Hilbert --bimodule such that every closed right submodule is orthogonally closed, i.e., . Then there are families of Hilbert spaces , such that and are isomorphic to -direct sums , resp. , and is isomorphic to the outer direct sum .

     

On lower bounds for theL 2-Wasserstein metric in a Hilbert space

We provide two families of lower bounds for theL ²-Wasserstein metric in separable Hilbert spaces which depend on the basis chosen for the space. Then we focus on one of these families and we provide a necessary and sufficient condition for the supremum in it to be attained. In the finite dimensional case, we identify the basis which provides the most accurate lower bound in the family.Research partially supported by the Spanish DGICYT under grants PB91-0306-02-00, 01 and 02.     

S~1 × S~1-Borsuk-Ulam Theorem on Product Spaces

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On the discrete-time regulator problem in infinite-dimensional spaces

The linear, discrete-time regulator problem is considered in infinite-dimensional spaces without posing in advance any positivity conditions on quadratic criterion. The convergence of the finite-time optimum solution is studied, when time increases to infinity with a stable, stabilizable, and detectable system.The author thanks Professors P. Karttunen and H. Koivo for helpful discussions regarding this note.     

Solution of nonlinear optimal control problems in Hilbert spaces by means of linear programming techniques

Optimal control problems in Hilbert spaces are considered in a measure-theoretical framework. Instead of minimizing a functional defined on a class of admissible trajectory-control pairs, we minimize one defined on a set of measures; this set is defined by the boundary conditions and the differential equation of the problem. The new problem is an infinite-dimensionallinear programming problem; it is shown that it is possible to approximate its solution by that of a finite-dimensional linear program of sufficiently high dimensions, while this solution itself can be approximated by a trajectory-control pair. This pair may not be strictly admissible; if the dimensionality of the finite-dimensional linear program and the accuracy of the computations are high enough, the conditions of admissibility can be said to be satisfied up to any given accuracy. The value given by this pair to the functional measuring the performance criterion can be about equal to theglobal infimum associated with the classical problem, or it may be less than this number. It appears that this method may become a useful technique for the computation of optimal controls, provided the approximations involved are acceptable.     

Some remarks on spherically invariant distributions

We consider the notion of a spherically invariant measure on a real and separable Hilbert space and study some equivalence properties of measures of this type. The latter are of interest in statistical communication theory.