Tight Gabor Frames Associated with Non-separable Lattices and the Hyperbolic Secant

We consider Gabor systems generated by a window given by the hyperbolic secant function. We show that such a system forms a Parseval frame for L ²(?) when the translations and modulations of the window are associated with certain non-separable lattices in ?² which we explicitly describe. We also study the more general problem of characterizing the positive Borel measures μ on ?²ⁿ with the property that the short-time Fourier transform defines an isometric embedding from L ²(? ⁿ ) to L _μ ² (?²ⁿ ) when the window belongs to the Schwartz class and, in particular, we characterize the extreme points of this set. In the case where the window is the hyperbolic secant function, we consider the situation where the measure is discrete with constant weights and supported on a non-separable lattice yielding a Parseval frame. We provide arithmetic conditions on the parameters defining the lattice characterizing when the associated measure is an extreme point.     

多元正交小波基

本文构造了一组Haar型的多元正交小波基,它们不能用张量积的方法产生.     

Gabor Frames over Irregular Lattices

We give necessary and sufficient conditions for gW(L ,¹) to generate a Gabor frame over certain irregular lattices.     

Time-Frequency Mean and Variance Sequences of Orthonormal Bases

We show that there exists an orthonormal basis for such that and are bounded sequences. We also show that there does not exist any orthonormal basis for , and being bounded sequences. This is motivated by a question posed by H.S. Shapiro on the mean and variance sequences associated to orthonormal bases.     

Gabor Time-Frequency Lattices and the Wexler-Raz Identity

Gabor time-frequency lattices are sets of functions of the form generated from a given function by discrete translations in time and frequency. They are potential tools for the decomposition and handling of signals that, like speech or music, seem over short intervals to have well-defined frequencies that, however, change with time. It was recently observed that the behavior of a lattice can be connected to that of a dual lattice Here we establish this interesting relationship and study its properties. We then clarify the results by applying the theory of von Neumann algebras. One outcome is a simple proof that for to span the lattice must have at least unit density. Finally, we exploit the connection between the two lattices to construct expansions having improved convergence and localization properties.     

Some Methods for Constructing Nonseparable,Orthonormal, Compactly Supported Wavelet Bases

We first show that by combining monodimensional filter banks one can obtain nonseparable filter banks. We then give necessary conditions for these filter banks to generate orthonormal and regular wavelets. Finally, we establish that some of these filter banks lead to arbitrarily smooth, nonseparable, orthonormal, compactly supported wavelet bases.     

Quantum Harmonic Analysis on Lattices and Gabor Multipliers

Journal of Fourier Analysis and Applications - We calculate the norm of the Fourier operator from $$L^p(X)$$ to $$L^q({hat{X}})$$ when X is an infinite locally compact abelian group that is,...     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned.     

Haar Type Orthonormal Wavelet Bases in R2

K.-H. Grochenig and A. Haas asked whether for every expanding integer matrix A ∈ M_n(ℤ) there is a Haar type orthonormal wavelet basis having dilation factor A and translation lattice ℤⁿ. They proved that this is the case when the dimension n = 1. This article shows that this is also the case when the dimension n = 2.     

Korobov Spaces with Respect to Digital Orthonormal Bases

 For an orthonormal basis (ONB) of we define classes of functions according to the order of decay of the Fourier coefficients with respect to the considered ONB . The rate is expressed in the real parameter α. We investigate the following problem: What is the order of decay, if any, when we consider with respect to another ONB ? If the function is expressable as an absolutely convergent Fourier series with respect to , we give bounds for the new order of decay, which we call . Special attention is given to digital orthonormal bases (dONBs) of which the Walsh and Haar systems are examples treated in the present paper. Bounding intervals and in several cases explicit values for are given for the case of dONBs. An application to quasi-Monte Carlo numerical integration is mentioned. (Received 21 February 2000; in revised form 19 October 2000)     

The Uniformity of Non-Uniform Gabor Bases

There have been extensive studies on non-uniform Gabor bases and frames in recent years. But interestingly there have not been a single example of a compactly supported orthonormal Gabor basis in which either the frequency set or the translation set is non-uniform. Nor has there been an example in which the modulus of the generating function is not a characteristic function of a set. In this paper, we prove that in the one dimension and if we assume that the generating function g(x) of an orthonormal Gabor basis is supported on an interval, then both the frequency and the translation sets of the Gabor basis must be lattices. In fact, the Gabor basis must be the trivial one in the sense that |g(x)|=c(x) for some fundamental interval of the translation set. We also give examples showing that compactly supported non-uniform orthonormal Gabor bases exist in higher dimensions.     

正则剩余格上的模糊理想基

首先,在正则剩余格中引入模糊理想基的概念,介绍了模糊理想基的一些重要性质,并且利用这些性质,给出了模糊理想基的三种等价形式;其次,结合具体实例讨论了模糊理想基与模糊理想的关系;最后,给出由模糊集生成模糊理想基及由模糊理想基生成模糊理想的方法.     

The Matrix-Valued Riesz Lemma and Local Orthonormal Bases in Shift-Invariant Spaces

We use the matrix-valued Fejér–Riesz lemma for Laurent polynomials to characterize when a univariate shift-invariant space has a local orthonormal shift-invariant basis, and we apply the above characterization to study local dual frame generators, local orthonormal bases of wavelet spaces, and MRA-based affine frames. Also we provide a proof of the matrix-valued Fejér–Riesz lemma for Laurent polynomials.     

Planarity, Symmetry and Counting Tilings

We axiomatize the geometrical properties of T-tetromino to what we call generalised k-T-tetromino. Using this set of axioms, we show that the number of tilings of a 2kn × 2km rectangular region is given by T(L _n,m ; 3, 3) if and only if the tile is a k-T-tetromino. This generalizes a result of Korn and Pak (Theor Comp Sci 319:3–27, 2004).     

Polyboxes,Cube Tilings and Rigidity

A non-empty subset A of X=X ₁×???×X _d is a (proper) box if A=A ₁×???×A _d and A _i ?X _i for each i. Suppose that for each pair of boxes A, B and each i, one can only know which of the three states takes place: A _i =B _i , A _i =X _i ?B _i , A _i ?{B _i ,X _i ?B _i }. Let F and G be two systems of disjoint boxes. Can one decide whether ∪F=∪G? In general, the answer is ‘no’, but as is shown in the paper, it is ‘yes’ if both systems consist of pairwise dichotomous boxes. (Boxes A, B are dichotomous if there is i such that A _i =X _i ?B _i .) Several criteria that enable to compare such systems are collected. The paper includes also rigidity results, which say what assumptions have to be imposed on F to ensure that ∪F=∪G implies F=G. As an application, the rigidity conjecture for 2-extremal cube tilings of Lagarias and Shor is verified.     

Refinement of Vahlen's Theorem for Minkowski Bases of Three-Dimensional Lattices

In the paper, the analog of Vahlen's theorem for Minkowski bases of three-dimensional lattices is sharpened.     

A Decay Result for Certain Windows Generating Orthogonal Gabor Bases

We consider tight Gabor frames (h,a=1,b=1) at critical density with h of the form Z ⁻¹(Zg/|Zg|). Here Z is the standard Zak transform and g is an even, real, well-behaved window such that Zg has exactly one zero, at , in [0,1)². We show that h and its Fourier transform have maximal decay as allowed by the Balian-Low theorem. Our result illustrates a theorem of Benedetto, Czaja, Gadziński, and Powell, case p=q=2, on sharpness of the Balian-Low theorem.      

Parallelogram Tilings, Worms, and Finite Orientations

This paper studies properties of tilings of the plane by parallelograms. In particular, it is established that in parallelogram tilings using a finite number of shapes all tiles occur in only finitely many orientations.     

Triangular Dissections, Aperiodic Tilings, and Jones Algebras

The Brattelli diagram associated with a given bicolored Dynkin-Coxeter graph of type A_n determines planar fractal sets obtained by infinite dissections of a given triangle. All triangles appearing in the dissection process have angles that are multiples of π/(n + 1). There are usually several possible infinite dissections compatible with a given n but a given one makes use of n/2 triangle types if n is even. Jones algebra with index [4 cos²(π/(n + 1))]⁻¹ (values of the discrete range) act naturally on vector spaces associated with those fractal sets. Triangles of a given type are always congruent at each step of the dissection process. In the particular case n = 4, there are isometric and the whole structure lead, after proper inflation, to aperiodic Penrose tilings. The "tilings" associated with other values of the index are discussed and shown to be encoded by equivalence classes of infinite sequences (with appropriate constraints), using n/2 digits (if- n is even) and generalizing the Fibonacci numbers.