Gabor pairs,and a discrete approach to wave-front sets

We introduce admissible lattices and Gabor pairs to define discrete versions of wave-front sets with respect to Fourier–Lebesgue and modulation spaces. We prove that these wave-front sets agree with each other and with corresponding wave-front sets of “continuous type”. This implies that the coefficients of a Gabor frame expansion of f are parameter dependent, and describe the wave-front set of f.     

Pointwise Summability of Gabor Expansions

A general summability method, the so-called θ-summability method is considered for Gabor series. It is proved that if the Fourier transform of θ is in a Herz space then this summation method for the Gabor expansion of f converges to f almost everywhere when f∈L ₁ or, more generally, when f∈W(L ₁,ℓ _∞) (Wiener amalgam space). Some weak type inequalities for the maximal operator corresponding to the θ-means of the Gabor expansion are obtained. Hardy-Littlewood type maximal functions are introduced and some inequalities are proved for these.     

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χ _E are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory. This work was supported by National Natural Science Foundation of China (Grant No. 10671008), Beijing Natural Science Foundation (Grant No. 1092001), PHR (IHLB) and the project sponsored by SRF for ROCS, SEM of China     

Gabor frame sets for subspaces

This paper investigates Gabor frame sets in a periodic subset \mathbb S\mathbb S of \mathbb R\mathbb R. We characterize tight Gabor sets in \mathbb S\mathbb S, and obtain some necessary/sufficient conditions for a measurable subset of \mathbb S\mathbb S to be a Gabor frame set in \mathbb S\mathbb S. We also characterize those sets \mathbb S\mathbb S admitting tight Gabor sets, and obtain an explicit construction of a class of tight Gabor sets in such \mathbb S\mathbb S for the case that the product of time-frequency shift parameters is a rational number. Our results are new even if \mathbb S=\mathbb R\mathbb S=\mathbb R.     

Beurling Dimension of Gabor Pseudoframes for Affine Subspaces

Pseudoframes for subspaces have been recently introduced by Li and Ogawa as a tool to analyze lower dimensional data with arbitrary flexibility of both the analyzing and the dual sequence. In this paper we study Gabor pseudoframes for affine subspaces by focusing on geometrical properties of their associated sets of parameters. We first introduce a new notion of Beurling dimension for discrete subsets of ℝ ^d by employing a certain generalized Beurling density. We present several properties of Beurling dimension including a comparison with other notions of dimension showing, for instance, that our notion includes the mass dimension as a special case. Then we prove that Gabor pseudoframes for affine subspaces satisfy a certain Homogeneous Approximation Property, which implies invariance under time–frequency shifts of an approximation by elements from the pseudoframe. The main result of this paper is a classification of Gabor pseudoframes for affine subspaces by means of the Beurling dimension of their sets of parameters. This provides us, in particular, with a Nyquist dimension which separates sets of parameters of pseudoframes from those of non-pseudoframes and which links a fixed value to sets of parameters of pseudo-Riesz sequences. These results are even new for the special case of Gabor frames for an affine subspace.      

Improved Convergence Result for the Discrete Gradient and Secant Methods for Nonsmooth Optimization

We study a generalization of the nonderivative discrete gradient method of Bagirov et al. for minimizing a locally Lipschitz function f on ℝ ⁿ . We strengthen the existing convergence result for this method by showing that it either drives the f-values to −∞ or each of its cluster points is Clarke stationary for f, without requiring the compactness of the level sets of f. Our generalization is an approximate bundle method, which also subsumes the secant method of Bagirov et al.     

Micro-Local Analysis with Fourier Lebesgue Spaces. Part?I

Let ω,ω ₀ be appropriate weight functions and q∈[1,∞]. We introduce the wave-front set, WF_FL^q(w)(f)\mathrm{WF}_{\mathcal{F}L^{q}_{(\omega)}}(f) of f ? S￠f\in \mathcal{S}' with respect to weighted Fourier Lebesgue space FL^q_(w)\mathcal{F}L^{q}_{(\omega )}. We prove that usual mapping properties for pseudo-differential operators Op (a) with symbols a in S^(w₀)_r,0S^{(\omega _{0})}_{\rho ,0} hold for such wave-front sets. Especially we prove that

Julia sets and periodic kneading sequences

We construct abstract Julia sets homeomorphic to Julia sets for complex polynomials of the form f _c (z) = z ² + c, having an associated periodic kneading sequence of the form [`(a*)]{\overline{\alpha\ast}} which is not a period n-tupling. We show that there is a single simply-defined space of “itineraries” which contains homeomorphic copies of all such Julia sets in a natural combinatorial way, with dynamical properties which are derivable directly from the combinatorics. This also leads to a natural definition of abstract Julia sets even for those kneading sequences which are not realized by any polynomial f _c , with similar dynamical properties.     

Optimal control under persistent disturbances

Let f be an orientation-preserving Morse-Smale diffeomorphism of an n-dimensional (n ≥ 3) closed orientable manifold M ⁿ . We show the possibility of representing the dynamics of f in a “source-sink” form. The roles of the “source” and “sink” are played by invariant closed sets one of which, A _f , is an attractor, and the other, R _f , is a repeller. Such a representation reveals new topological invariants that describe the embedding (possibly, wild) of stable and unstable manifolds of saddle periodic points in the ambient manifold. These invariants have allowed us to obtain a classification of substantial classes of Morse-Smale diffeomorphisms on 3-manifolds. In this paper, for any n ≥ 3, we describe the topological structure of the sets A _f and R _f and of the space of orbits that belong to the set M ⁿ \ (A _f ∪ R _f ).     

Edge of the Wedge Theory in Hypo-analytic Manifolds

Abstract

This article studies microlocal regularity properties of the distributions fon a strongly noncharacteristic submanifold Eof a hypo-analytic manifold Mthat arise as the boundary values of solutions on wedges in Mwith edge E. The hypo-analytic wave-front set of fin the sense of Baouendi-Chang-Treves is constrained as a consequence of the fact that fextends as a solution to a wedge.     

Joint continuity of <Emphasis Type="Italic">K</Emphasis><Subscript><Emphasis Type="Italic">h</Emphasis></Subscript><Emphasis Type="Italic">C</Emphasis>-functions with values in moore spaces

We introduce the notion of categorical cliquish mapping and show that, for each K _h C-mapping f: X × Y → Z, where X is a topological space, Y is a space with the first axiom of countability, and Z is a Moore space, with categorical-cliquish horizontal y-sections f _y , the sets C _y (f) are residual G _δ-type sets in X for every y ∈ Y. Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 11, pp. 1539–1547, November, 2008.     

On product-cordial index sets and friendly index sets of 2-regular graphs and generalized wheels

A vertex labeling f : V → Z2 of a simple graph G = (V, E) induces two edge labelings f+ , f*: E → Z2 defined by f+ (uv) = f(u)+f(v) and f*(uv) = f(u)f(v). For each i∈Z2 , let vf(i) = |{v ∈ V : f(v) = i}|, e+f(i) = |{e ∈ E : f+(e) = i}| and e*f(i)=|{e∈E:f*(e)=i}|. We call f friendly if |vf(0)-vf(1)|≤ 1. The friendly index set and the product-cordial index set of G are defined as the sets{|e+f(0)-e+f(1)|:f is friendly} and {|e*f(0)-e*f(1)| : f is friendly}. In this paper we study and determine the connection between the friendly index sets and product-cordial index sets of 2-regular graphs and generalized wheel graphs.     

Partition of graphs with condition on the connectivity and minimum degree

C. Thomassen and M. Szegedy proved the existence of a functionf(s, t) such that the points of anyf(s, t)-connected graph have a decomposition into two non-empty sets such that the subgraphs induced by them ares-connected andt-connected, respectively. We prove, thatf(s, t) ≦ 4s+4t − 13 and examine a similar problem for the minimum degree.     

A topological invariant for pairs of maps

In this paper we develop the notion of contact orders for pairs of continuous self-maps (f, g) from ℝⁿ, showing that the set Con(f, g) of all possible contact orders between f and g is a topological invariant (we remark that Con(f, id) = Per(f)). As an interesting application of this concept, we give sufficient conditions for the graphs of two continuous self-maps from ℝ intersect each other. We also determine the ordering of the sets Con(f, 0) and Con(f, h), for h ∈ Hom(ℝ) such that f ∘ h = h ∘ f. For this latter set we obtain a generalization of Sharkovsky’s theorem.     

On some coding and mixing properties for a class of chaotic systems

We study certain ergodic properties of equilibrium measures of hyperbolic non-invertible maps f on basic sets with overlaps Λ. We prove that if the equilibrium measure of a Holder potential , is 1-sided Bernoulli, then f is expanding from the point of view of a pointwise section dimension of . If the measure of maximal entropy μ ₀ is 1-sided Bernoulli, then f is shown to be distance expanding on Λ; and if is 1-sided Bernoulli for f expanding, then must be the measure of maximal entropy. These properties are very different from the case of hyperbolic diffeomorphisms. Another result is about the non 1-sided Bernoullicity for certain equilibrium measures for hyperbolic toral endomorphisms. We also prove the non-existence of generating Rokhlin partitions for measure-preserving endomorphisms in several cases, among which the case of hyperbolic non-expanding toral endomorphisms with Haar measure. Nevertheless the system is shown to have always exponential decay of correlations on Holder observables and to be mixing of any order.     

Zero sets of holomorphic functions in the bidisk

We characterize in geometric terms the zero sets of holomorphic functionsf in the bidisk such that log |f|∈L ^p (D ²) for 1<p<∞. Partially supported by the DGCYT grant PB95-0956-C02-02 and grant 1996-SGR-26.     

Stable Gabor Phase Retrieval and Spectral Clustering

We consider the problem of reconstructing a signal f from its spectrogram, i.e., the magnitudes |V_φf| of its Gabor transform Such problems occur in a wide range of applications, from optical imaging of nanoscale structures to audio processing and classification. While it is well-known that the solution of the above Gabor phase retrieval problem is unique up to natural identifications, the stability of the reconstruction has remained wide open. The present paper discovers a deep and surprising connection between phase retrieval, spectral clustering, and spectral geometry. We show that the stability of the Gabor phase reconstruction is bounded by the reciprocal of the Cheeger constant of the flat metric on ℝ², conformally multiplied with |V_φf|. The Cheeger constant, in turn, plays a prominent role in the field of spectral clustering, and it precisely quantifies the “disconnectedness” of the measurements V_φf. It has long been known that a disconnected support of the measurements results in an instability—our result for the first time provides a converse in the sense that there are no other sources of instabilities. Due to the fundamental importance of Gabor phase retrieval in coherent diffraction imaging, we also provide a new understanding of the stability properties of these imaging techniques: Contrary to most classical problems in imaging science whose regularization requires the promotion of smoothness or sparsity, the correct regularization of the phase retrieval problem promotes the “connectedness” of the measurements in terms of bounding the Cheeger constant from below. Our work thus, for the first time, opens the door to the development of efficient regularization strategies. © 2018 Wiley Periodicals, Inc.     

Cluster sets for sobolev functions and quasiminimizers

In this paper, we study cluster sets and essential cluster sets for Sobolev functions and quasiharmonic functions (i.e., continuous quasiminimizers). We develop their basic theory with a particular emphasis on when they coincide and when they are connected. As a main result, we obtain that if a Sobolev function u on an open set Ω has boundary values f in Sobolev sense and f |_∂Ω is continuous at x ₀ ∈ ∂Ω, then the essential cluster set (u, x ₀,Ω) is connected. We characterize precisely in which metric spaces this result holds. Further, we provide some new boundary regularity results for quasiharmonic functions. Most of the results are new also in the Euclidean case.