Every frame is a sum of three (but not two) orthonormal bases—and other frame representations

We show that every frame for a Hilbert space H can be written as a (multiple of a) sum of three orthonormal bases for H. We next show that this result is best possible by including a result of Kalton: A frame can be represented as a linear combination of two orthonormal bases if and only if it is a Riesz basis. We further show that every frame can be written as a (multiple of a) sum of two tight frames with frame bounds one or a sum of an orthonormal basis and a Riesz basis for H. Finally, every frame can be written as a (multiple of a) average of two orthonormal bases for a larger Hilbert space. Acknowledgements and Notes. This research was supported by NSF DMS 9701234. Part of this research was conducted while the author was a visitor at the “Workshop on Linear Analysis and Probability”, Texas A&M University.     

Localized Frames Are Finite Unions of Riesz Sequences

It is shown that every localized frame is a finite union of Riesz sequences. This is a partial solution to a question of Feichtinger.     

Angle criteria for frame sequences and frames containing a Riesz basis

We obtain a condition implying that the union of two frame sequences is also a frame sequence. Christensen found a condition for this in terms of orthogonal projections. We phrase our condition by use of the angle between closed subspaces. Also a lower bound formula is obtained. We then apply the results to find conditions for a frame containing a Riesz basis to be a Riesz frame.     

On Weighted Spaces of Harmonic and Holomorphic Functions

Weighed spaces of harmonic and holomorphic functions on theunit disc are studied. We show that for all radial weights whichare not decreasing too fast the space of harmonic functionsis isomorphic to c₀. For the weights that we consider we completelycharacterize those spaces of holomorphic functions which areisomorphic to c₀. Moreover, we determine when the Riesz projection,mapping the weighted space of harmonic functions onto the correspondingspace of holomorphic functions, is bounded.     

A note on regular martingales in Riesz spaces

We consider regular martingales (i.e., martingales which can be written as differences of two positive martingales) in the measure-free setting of Riesz spaces. The aim of this note is to show that the space of such martingales is a Riesz space. We derive an analogue of a result of Troitsky's on regular norm bounded martingales in Banach lattices.     

Riesz basis for strongly continuous groups

Given a Hilbert space and the generator of a strongly continuous group on this Hilbert space. If the eigenvalues of the generator have a uniform gap, and if the span of the corresponding eigenvectors is dense, then these eigenvectors form a Riesz basis (or unconditional basis) of the Hilbert space. Furthermore, we show that none of the conditions can be weakened. However, if the eigenvalues (counted with multiplicity) can be grouped into subsets of at most K elements, and the distance between the groups is (uniformly) bounded away from zero, then the spectral projections associated to the groups form a Riesz family. This implies that if in every range of the spectral projection we construct an orthonormal basis, then the union of these bases is a Riesz basis in the Hilbert space.     

Exponential Riesz Bases, Discrepancy of Irrational Rotations and BMO

We study the basis property of systems of exponentials with frequencies belonging to ‘simple quasicrystals’. We show that a diophantine condition is necessary and sufficient for such a system to be a Riesz basis in L ² on a finite union of intervals. For the proof we extend to BMO a theorem of Kesten about the discrepancy of irrational rotations of the circle.     

A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbortransition probability that yields a recurrent random walk andshow that, although such trees have no positive potentials,many of the standard results of potential theory can be transferredto this setting. We accomplish this by defining a non-negativefunction H, harmonic outside the root e and vanishing only ate, and a substitute notion of potential which we call H-potential.We define the flux of a superharmonic function outside a finiteset of vertices, give some simple formulas for calculating theflux and derive a global Riesz decomposition theorem for superharmonicfunctions with a harmonic minorant outside a finite set. Wediscuss the connection of the H-potentials with other notionsof potentials for recurrent Markov chains in the literature.     

Uncertainty principles for orthonormal sequences

The aim of this paper is to provide complementary quantitative extensions of two results of H.S. Shapiro on the time-frequency concentration of orthonormal sequences in L²(R). More precisely, Shapiro proved that if the elements of an orthonormal sequence and their Fourier transforms are all pointwise bounded by a fixed function in L²(R) then the sequence is finite. In a related result, Shapiro also proved that if the elements of an orthonormal sequence and their Fourier transforms have uniformly bounded means and dispersions then the sequence is finite. This paper gives quantitative bounds on the size of the finite orthonormal sequences in Shapiro's uncertainty principles. The bounds are obtained by using prolate spheroïdal wave functions and combinatorial estimates on the number of elements in a spherical code. Extensions for Riesz bases and different measures of time-frequency concentration are also given.     

Prime tight frames

We introduce a class of finite tight frames called prime tight frames and prove some of their elementary properties. In particular, we show that any finite tight frame can be written as a union of prime tight frames. We then characterize all prime harmonic tight frames and use thischaracterization to suggest effective analysis and synthesis computation strategies for such frames. Finally, we describe all prime frames constructed from the spectral tetris method, and, as a byproduct, we obtain a characterization of when the spectral tetris construction works for redundancies below two.     

A p Interpolation in the Unit Ball

We study interpolating sequences in the unit ball for A^–pwith p > 0, the Banach space of holomorphic functions f with(1 – |z|²)^p |f(z)| bounded. The finite unions of A^–p-interpolatingsequences are characterized by a Carleson type condition.     

Broue's Conjecture for the Hall-Janko Group and its Double Cover

Broué's abelian defect conjecture suggests a deep linkbetween the module categories of a block of a group algebraand its Brauer correspondent, viz. that they should be derivedequivalent. We are able to verify Broué's conjecturefor the Hall–Janko group, even its double cover 2.J₂,as well as for U₃(4) and Sp₄(4). In fact we verify Rickard'srefinement to Broué's conjecture and show that the derivedequivalence can be chosen to be a splendid equivalence for theseexamples. 2000 Mathematical Subject Classification: 20C20, 20C34.     

A transform method for linear evolution PDEs on a finite interval

We study initial boundary value problems for linear scalar evolutionpartial differential equations, with spatial derivatives ofarbitrary order, posed on the domain {t > 0, 0 < x <L}. We show that the solution can be expressed as an integralin the complex k-plane. This integral is defined in terms ofan x-transform of the initial condition and a t-transform ofthe boundary conditions. The derivation of this integral representationrelies on the analysis of the global relation, which is an algebraicrelation defined in the complex k-plane coupling all boundaryvalues of the solution. For particular cases, such as the case of periodic boundaryconditions, or the case of boundary value problems for even-orderPDEs, it is possible to obtain directly from the global relationan alternative representation for the solution, in the formof an infinite series. We stress, however, that there existinitial boundary value problems for which the only representationis an integral which cannot be written as an infinite series.An example of such a problem is provided by the linearized versionof the KdV equation. Similarly, in general the solution of odd-orderlinear initial boundary value problems on a finite intervalcannot be expressed in terms of an infinite series.     

A decomposition theorem for frames and the Feichtinger Conjecture

In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in -Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is -independent for -sequences.

     

On the dynamics and recursive properties of multidimensional symbolic systems

We study the (sub)dynamics of multidimensional shifts of finite type and sofic shifts, and the action of cellular automata on their limit sets. Such a subaction is always an effective dynamical system: i.e. it is isomorphic to a subshift over the Cantor set the complement of which can be written as the union of a recursive sequence of basic sets. Our main result is that, to varying degrees, this recursive-theoretic condition is also sufficient. We show that the class of expansive subactions of multidimensional sofic shifts is the same as the class of expansive effective systems, and that a general effective system can be realized, modulo a small extension, as the subaction of a shift of finite type or as the action of a cellular automaton on its limit set (after removing a dynamically trivial set). As applications, we characterize, in terms of their computational properties, the numbers which can occur as the entropy of cellular automata, and construct SFTs and CAs with various interesting properties.     

Finite-element wavelets on manifolds

We construct locally supported, continuous wavelets on manifolds that are given as the closure of a disjoint union of generalsmooth parametric images of an n-simplex. The wavelets are provento generate Riesz bases for Sobolev spaces H^s () when s (–1,3/2), if not limited by the global smoothness of . These resultsgeneralize the findings from Dahmen & Stevenson (1999) SIAMJ. Numer. Anal., 37, 319–352, where it was assumed thateach parametrization has a constant Jacobian determinant. Thewavelets can be arranged to satisfy the cancellation propertyof, in principle, any order, except for wavelets with supportsthat extend to different patches, which generally satisfy thecancellation property of only order 1.     

A Note on a Maximal Function Over Arbitrary Sets of Directions

Let M_S be the universal maximal operator over unit vectors ofarbitrary directions. This operator is not bounded in L²(R²).We consider a sequence of operators over sets of finite equidistributeddirections converging to M_S. We provide a new proof of N. Katz'sbound for such operators. As a corollary, we deduce that M_Sis bounded from some subsets of L² to L². These subsets arecomposed of positive functions whose Fourier transforms havea logarithmic decay or which are supported on a disc. 1991 MathematicsSubject Classification 42B25.     

The Structure of Extended Real-valued Metric Spaces

An extended metric on a set X is a distance function that satisfies the usual properties of a metric except that it can assume values of infinity, in addition to nonnegative real values. Given a metrizable space we exhibit a universal space for all extended metric spaces compatible with the topology. Defining a set in an extended metric space to be bounded if it is contained in a finite union of open balls, we characterize those bornologies on X that can be realized as bornologies of metrically bounded sets. We also consider a second possible definition of bounded set in this setting.