Approximations for Gabor and wavelet frames

Let be a frame vector under the action of a collection of unitary operators . Motivated by the recent work of Frank, Paulsen and Tiballi and some application aspects of Gabor and wavelet frames, we consider the existence and uniqueness of the best approximation by normalized tight frame vectors. We prove that for any frame induced by a projective unitary representation for a countable discrete group, the best normalized tight frame (NTF) approximation exists and is unique. Therefore it applies to Gabor frames (including Gabor frames for subspaces) and frames induced by translation groups. Similar results hold for semi-orthogonal wavelet frames.

     

Gabor frames by sampling and periodization

By sampling the window of a Gabor frame for belonging to Feichtinger’s algebra, , one obtains a Gabor frame for . In this article we present a survey of results by R. Orr and A.J.E.M. Janssen and extend their ideas to cover interrelations among Gabor frames for the four spaces , , and . Some new results about general dual windows with respect to sampling and periodization are presented as well. This theory is used to show a new result of the Kaiblinger type to construct an approximation to the canonical dual window of a Gabor frame for .      

Varying the time-frequency lattice of Gabor frames

A Gabor or Weyl-Heisenberg frame for is generated by time-frequency shifts of a square-integrable function, the Gabor atom, along a time-frequency lattice. The dual frame is again a Gabor frame, generated by the dual atom. In general, Gabor frames are not stable under a perturbation of the lattice constants; that is, even for arbitrarily small changes of the parameters the frame property can be lost.

In contrast, as a main result we show that this kind of stability does hold for Gabor frames generated by a Gabor atom from the modulation space , which is a dense subspace of . Moreover, in this case the dual atom depends continuously on the lattice constants. In fact, we prove these results for more general weighted modulation spaces. As a consequence, we obtain for Gabor atoms from the Schwartz class that the continuous dependence of the dual atom holds even in the Schwartz topology. Also, we complement these main results by corresponding statements for Gabor Riesz sequences and their biorthogonal system.

     

Stability of Gabor frames with arbitrary sampling points

Summary We study the stability of Gabor frames with arbitrary sampling points in the time-frequency plane, in several aspects. We prove that a Gabor frame generated by a window function in the Segal algebra S₀(R^d) remains a frame even if (possibly) all the sampling points undergo an arbitrary perturbation, as long as this is uniformly small. We give explicit stability bounds when the window function is nice enough, showing that the allowed perturbation depends only on the lower frame bound of the original family and some qualitative parameters of the window under consideration. For the perturbation of window functions we show that a Gabor frame generated by any window function with arbitrary sampling points remains a frame when the window function has a small perturbation in S₀(R^d) sense. We also study the stability of dual frames, which is useful in practice but has not found much attention in the literature. We give some general results on this topic and explain consequences to Gabor frames.     

On the differentiability of first integrals of two dimensional flows

By using techniques of differential geometry we answer the following open problem proposed by Chavarriga, Giacomini, Giné, and Llibre in 1999. For a given two dimensional flow, what is the maximal order of differentiability of a first integral on a canonical region in function of the order of differentiability of the flow? Moreover, we prove that for every planar polynomial differential system there exist finitely many invariant curves and singular points , such that has finitely many connected open components, and that on each of these connected sets the system has an analytic first integral. For a homogeneous polynomial differential system in , there exist finitely many invariant straight lines and invariant conical surfaces such that their complement in is the union of finitely many open connected components, and that on each of these connected open components the system has an analytic first integral.     

Stability of wavelet frames with matrix dilations

Under certain assumptions we show that a wavelet frame

in remains a frame when the dilation matrices and the translation parameters are perturbed. As a special case of our result, we obtain that if is a frame for an expansive matrix and an invertible matrix , then is a frame if and for sufficiently small 0$">.

     

Oversampling generates super-wavelets

We show that the second oversampling theorem for affine systems generates super-wavelets. These are frames generated by an affine structure on the space .

     

Centered complexity one Hamiltonian torus actions

We consider symplectic manifolds with Hamiltonian torus actions which are ``almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are ``centered" and the moment map is proper. In particular, this classifies the preimages under the moment map of all sufficiently small open sets, which is an important step towards global classification. As an application, we construct a full packing of each of the Grassmannians and by two equal symplectic balls.

     

Characterizations of function spaces on the sphere using frames

In this paper we introduce a polynomial frame on the unit sphere of , for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere , such as , and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case . We also study a related nonlinear -term approximation problem on . In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions', Amer. J. Math. 114 (1992), no. 4, 737-785).

     

Endpoint estimates for the circular maximal function

We consider the problem of endpoint estimates for the circular maximal function defined by

where is the normalized surface area measure on . Let be the closed triangle with vertices . We prove that for , there is a constant such that Furthermore, .

     

The Perron-Frobenius theorem for homogeneous, monotone functions

If is a nonnegative matrix whose associated directed graph is strongly connected, the Perron-Frobenius theorem asserts that has an eigenvector in the positive cone, . We associate a directed graph to any homogeneous, monotone function, , and show that if the graph is strongly connected, then has a (nonlinear) eigenvector in . Several results in the literature emerge as corollaries. Our methods show that the Perron-Frobenius theorem is ``really' about the boundedness of invariant subsets in the Hilbert projective metric. They lead to further existence results and open problems.

     

Lineability and spaceability of sets of functions on

We show that there is an infinite-dimensional vector space of differentiable functions on every non-zero element of which is nowhere monotone. We also show that there is a vector space of dimension of functions every non-zero element of which is everywhere surjective.

     

Convergence of two-dimensional weighted integrals

A two-dimensional weighted integral in is proposed as a tool for analyzing higher-dimensional unweighted integrals, and a necessary and sufficient condition for the finiteness of the weighted integral is obtained.

     

Frame representations and Parseval duals with applications to Gabor frames

Let be a frame for a Hilbert space . We investigate the conditions under which there exists a dual frame for which is also a Parseval (or tight) frame. We show that the existence of a Parseval dual is equivalent to the problem whether can be dilated to an orthonormal basis (under an oblique projection). A necessary and sufficient condition for the existence of Parseval duals is obtained in terms of the frame excess. For a frame induced by a projective unitary representation of a group , it is possible that can have a Parseval dual, but does not have a Parseval dual of the same type. The primary aim of this paper is to present a complete characterization for all the projective unitary representations such that every frame (with a necessary lower frame bound condition) has a Parseval dual of the same type. As an application of this characterization together with a result about lattice tiling, we prove that every Gabor frame (again with the same necessary lower frame bound condition) has a Parseval dual of the same type if and only if the volume of the fundamental domain of is less than or equal to .

     

Infinitely many radial solutions of a variational problem related to dispersion-managed optical fibers

We consider a non-local variational problem whose critical points are related to bound states in certain optical fibers. The functional is given by , and relying on the regularizing properties of the solution to the free Schrödinger equation, it will be shown that has infinitely many critical points.

     

Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation

The aim of this article is twofold. First we consider the wave equation in the hyperbolic space and obtain the counterparts of the Strichartz type estimates in this context. Next we examine the relationship between semilinear hyperbolic equations in the Minkowski space and in the hyperbolic space. This leads to a simple proof of the recent result of Georgiev, Lindblad and Sogge on global existence for solutions to semilinear hyperbolic problems with small data. Shifting the space-time Strichartz estimates from the hyperbolic space to the Minkowski space yields weighted Strichartz estimates in which extend the ones of Georgiev, Lindblad, and Sogge.

     

Oblique projections and frames

We characterize those frames on a Hilbert space which can be represented as the image of an orthonormal basis by an oblique projection defined on an extension of . We show that all frames with infinite excess and frame bounds are of this type. This gives a generalization of a result of Han and Larson which only holds for normalized tight frames.

     

Tight wavelet frames generated by three symmetric -spline functions with high vanishing moments

In this note, we show that one can derive from any -spline function of order ( ) an MRA tight wavelet frame in that is generated by the dyadic dilates and integer shifts of three compactly supported real-valued symmetric wavelet functions with vanishing moments of the highest possible order .

     

The distribution of prime ideals of imaginary quadratic fields

Let be a primitive positive definite quadratic form with integer coefficients. Then, for all there exist such that is prime and

This is deduced from another result giving an estimate for the number of prime ideals in an ideal class of an imaginary quadratic number field that fall in a given sector and whose norm lies in a short interval.

     

Necessary conditions for Schatten class localization operators

We study time-frequency localization operators of the form , where is the symbol of the operator and are the analysis and synthesis windows, respectively. It is shown in an earlier paper by the authors that a sufficient condition for , the Schatten class of order , is that belongs to the modulation space and the window functions to the modulation space . Here we prove a partial converse: if for every pair of window functions with a uniform norm estimate, then the corresponding symbol must belong to the modulation space . In this sense, modulation spaces are optimal for the study of localization operators. The main ingredients in our proofs are frame theory and Gabor frames. For and , we recapture earlier results, which were obtained by different methods.