A necessary and sufficient condition for dual Weyl-Heisenberg frames to be compactly supported

In this note we consider continuous-time Weyl-Heisenberg (Gabor) frame expansions with rational oversampling. We present a necessary and sufficient condition on a compactly supported function g(t) generating a Weyl-Heisenberg frame for L² () for its minimal dual (Wexler-Razdual) ⁰ (t) to be compactly supported. We furthermore provide a necessary and sufficient condition for a band-limited function g(t) generating a Weyl-Heisenberg frame for L² () to have a band-limited minimal dual ⁰ (t). As a consequence of these conditions, we show that in the cases of integer oversampling and critical sampling a compactly supported (band-limited) g(t) has a compactly supported (band-limited) minimal dual ⁰(t) if and only if the Weyl-Heisenberg frame operator is a multiplication operator in the time (frequency) domain. Our proofs rely on the Zak transform, on the Zibulski-Zeevi representation of the Weyl-Heisenberg frame operator, and on the theory of polynomial matrices.on leave from Department of Communications, Vienna University of TechnologyThis work was supported in part by FWF grants P10531-ÖPH, P12228-TEC, and J1629-TEC.     

ON LAGRANGE INTERPOLATION TO ｜x｜^α（1 〈α 〈 2） WITH EQUALLY SPACED NODES

S.M. Lozinskii proved the exact convergence rate at the zero of Lagrange interpolation polynomials to |x| based on equidistant nodes in [−1,1], In 2000, M. Rever generalized S.M. Lozinskii’s result to |x|^α(0≤α≤1). In this paper we will present the exact rate of convergence at the point zero for the interpolants of |x|^α(1<α<2).     

Discrete Gabor transforms: The Gabor-Gram matrix approach

The fundamental problem ofdiscrete Gabor transforms is to compute a set ofGabor coefficients in efficient ways. Recent study on the subject is an indirect approach: in order to compute the Gabor coefficients, one needs to find an auxiliary bi-orthogonal window function γ. We are seeking a direct approach in this paper. We introduce concepts ofGabor-Gram matrices and investigate their structural properties. We propose iterative methods to compute theGabor coefficients. Simple solutions for critical sampling, certain oversampling, and undersampling cases are developed. Acknowledgements and Notes. The author was with University of Connecticut, Storrs, CT 06269-3009.     

Iterative Algorithms to Approximate Canonical Gabor Windows: Computational Aspects

In this article we investigate the computational aspects of some recently proposed iterative methods for approximating the canonical tight and canonical dual window of a Gabor frame (g, a, b). The iterations start with the window g while the iteration steps comprise the window g, the k-th iterand γ_k, the frame operators S and S_k corresponding to (g, a, b) and (γ_k, a, b), respectively, and a number of scalars. The structure of the iteration step of the method is determined by the envisaged convergence order m of the method. We consider two strategies for scaling the terms in the iteration step: Norm scaling, where in each step the windows are normalized, and initial scaling where we only scale in the very beginning. Norm scaling leads to fast, but conditionally convergent methods, while initial scaling leads to unconditionally convergent methods, but with possibly suboptimal convergence constants. The iterations, initially formulated for time-continuous Gabor systems, are considered and tested in a discrete setting in which one passes to the appropriately sampled-and-periodized windows and frame operators. Furthermore, they are compared with respect to accuracy and efficiency with other methods to approximate canonical windows associated with Gabor frames.     

Refinable Functions with Exponential Decay: An?Approach via Cascade Algorithms

Refinable functions with exponential decay arise from applications such as the Butterworth filters in signal processing. Refinable functions with exponential decay also play an important role in the study of Riesz bases of wavelets generated from multiresolution analysis. A fundamental problem is whether the standard solution of a refinement equation with an exponentially decaying mask has exponential decay. We investigate this fundamental problem by considering cascade algorithms in weighted L _p spaces (1≤p≤∞). We give some sufficient conditions for the cascade algorithm associated with an exponentially decaying mask to converge in weighted L _p spaces. Consequently, we prove that the refinable functions associated with the Butterworth filters are continuous functions with exponential decay. By analyzing spectral properties of the transition operator associated with an exponentially decaying mask, we find a characterization for the corresponding refinable function to lie in weighted L ₂ spaces. The general theory is applied to an interesting example of bivariate refinable functions with exponential decay, which can be viewed as an extension of the Butterworth filters.     

The Besov and Triebel-Lizorkin spaces with high order on the Lipschitz curves

The Besov spacesB _p ^α,μ (Γ) and Triebel-Lizorkin spacesF _p ^α,μ (Γ) with high order α∈R on a Lipschitz curve Γ are defined. when 1≤p≤∞, 1≤q≤∞. To compare to the classical case a difference characterization of such spaces in the case |α|<1 is given also. The author is supported in part by the Foundation of Zhongshan University Advanced Research Centre and NSF of China.     

Harnack Inequalities for some Lévy Processes

In this paper we prove Harnack inequality for nonnegative functions which are harmonic with respect to random walks in ℝ ^d . We give several examples when the scale invariant Harnack inequality does not hold. For any α ∈ (0,2) we also prove the Harnack inequality for nonnegative harmonic functions with respect to a symmetric Lévy process in ℝ ^d with a Lévy density given by $c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}$c|x|^{-d-\alpha}1_{\{|x|\leq 1\}}+j(|x|)1_{\{|x|>1\}}, where 0 ≤ j(r) ≤ cr ^{− d − α} , ∀ r > 1, for some constant c. Finally, we establish the Harnack inequality for nonnegative harmonic functions with respect to a subordinate Brownian motion with subordinator with Laplace exponent ϕ(λ) = λ ^α/2ℓ(λ), λ > 0, where ℓ is a slowly varying function at infinity and α ∈ (0,2).     

Frequency-Domain Bounds for Nonnegative,Unsharply Band-Limited Functions

In this paper we present a technique for proving bounds of the Boas-Kac-Lukosz type for unsharply restricted functions with nonnegative Fourier transforms. Hence we consider functions F(x) ≥ 0, the Fourier transform f(u) of which satisfies |f(u)| ≤ ε for all u in a subset of (-∞,-1] ⋃ [1,∞), and are interested in bounds on |f(u)| for |u| ≤ 1. This technique gives rise to several "epsilonized" versions of the Boas-Kac-Lukosz bound (which deals with the case f(u) = 0, |u| ≥ 1). For instance, we find that |f(u)| ≤ L(u) + O(ε^2/3), where L(u) is the Boas-Kac-Lukosz bound, and show by means of an example that this version is the sharpest possible with respect to its behaviour as a function of ε as ε ↓ 0. The technique also turns out to be sufficiently powerful to yield the best bound as ε ↓ 0 in various other cases with less severe restrictions on f.     

Dilations and Completions for Gabor Systems

Let be a full rank time-frequency lattice in ℝ ^d ×ℝ ^d . In this note we first prove that any dual Gabor frame pair for a Λ-shift invariant subspace M can be dilated to a dual Gabor frame pair for the whole space L ²(ℝ ^d ) when the volume v(Λ) of the lattice Λ satisfies the condition v(Λ)≤1, and to a dual Gabor Riesz basis pair for a Λ-shift invariant subspace containing M when v(Λ)>1. This generalizes the dilation result in Gabardo and Han (J. Fourier Anal. Appl. 7:419–433, [2001]) to both higher dimensions and dual subspace Gabor frame pairs. Secondly, for any fixed positive integer N, we investigate the problem whether any Bessel–Gabor family G(g,Λ) can be completed to a tight Gabor (multi-)frame G(g,Λ)∪(∪ _j=1 ^N G(g _j ,Λ)) for L ²(ℝ ^d ). We show that this is true whenever v(Λ)≤N. In particular, when v(Λ)≤1, any Bessel–Gabor system is a subset of a tight Gabor frame G(g,Λ)∪G(h,Λ) for L ²(ℝ ^d ). Related results for affine systems are also discussed. Communicated by Chris Heil.     

Spectral properties of a pseudorelativistic system of two particles with finite masses

The discrete spectrum of the Hamiltonian of a pseudorelativistic system of two particles with finite masses is investigated for a fixed total system momentum p and interaction potentials having the form Z|r₁₂|^−γ for large |r₁₂|, where Z<0 and 0<γ≤2. The leading spectral asymptotic term is derived for γ<2. For γ=2, conditions are found for the finiteness and infiniteness of the discrete spectrum that depend (for some Z) on the fixed momentum p, which radically distinguishes the situation in question from the nonrelativistic case. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 121, No. 2, pp. 297–306, November, 1999.     

Global existence and uniform decay for the coupled Klein-Gordon-Schr?dinger equations

This paper is concerned to the existence, uniqueness and uniform decay for the solutions of the coupled Klein-Gordon-Schr?dinger damped equations where ω is a bounded domain of R ⁿ , n≤ 3, F : R ²→R is a C ¹-function; γ, β; θ are constants such that γ, β > 0 and 1 ≤ 2θ≤ 2. Received January 1999 – Accepted October 1999     

Characterization ofL p (R n ) using Gabor frames

We characterize L^p norms of functions onR ⁿ for 1<p<∞ in terms of their Gabor coefficients. Moreover, we use the Carleson-Hunt theorem to show that the Gabor expansions of L^p functions converge to the functions almost everywhere and in L^p for 1<p<∞. In L¹ we prove an analogous result: the Gabor expansions converge to the functions almost everywhere and in L¹ in a certain Cesàro sense. Consequently, we are able to establish that a large class of Gabor families generate Banach frames for L^p (R ⁿ) when 1≤p<∞.     

Characterization and Computation of Canonical Tight Windows for Gabor Frames

Let (\gnm)_{n,m ? \Zst}(\gnm)_{n,m\in\Zst} be a Gabor frame for \LtR\LtR for given window gg. We show that the window \ho = \SQI g\ho=\SQI g that generates the canonically associated tight Gabor frame minimizes ||g-h||\|g-h\| among all windows hh generating a normalized tight Gabor frame. We present and prove versions of this result in the time domain, the frequency domain, the time-frequency domain, and the Zak transform domain, where in each domain the canonical \ho\ho is expressed using functional calculus for Gabor frame operators. Furthermore, we derive a Wiener--Levy type theorem for rationally oversampled Gabor frames. Finally, a Newton-type method for a fast numerical calculation of \ho\ho is presented. We analyze the convergence behavior of this method and demonstrate the efficiency of the proposed algorithm by some numerical examples.     

Zeros in Tables of Characters for the Groups <Emphasis Type="Italic">S</Emphasis><Subscript>n</Subscript> and <Emphasis Type="Italic">A</Emphasis><Subscript>n</Subscript>. II

Let P(n) be the set of all partitions of a natural number n. In the representation theory of symmetric groups, for every partition α ∈ P(n), the partition h(α) ∈ P(n) is defined so as to produce a certain set of zeros in the character table for S_n. Previously, the analog f(α) of h(α) was obtained pointing out an extra set of zeros in the table mentioned. Namely, h(α) is greatest (under the lexicographic ordering ≤) of the partitions β of n such that χ^α(g_β) ≠ 0, and f(α) is greatest of the partitions γ of n that are opposite in sign to h(α) and are such that χ^α(g_γ) ≠ 0, where χ^α is an irreducible character of S_n, indexed by α, and g_β is an element in the conjugacy class of S_n, indexed by β. For α ∈ P(n), under some natural restrictions, here, we construct new partitions h′(α) and f′(α) of n possessing the following properties. (A) Let α ∈ P(n) and n ⩾ 3. Then h′(α) is identical is sign to h(α), χ^α(g_h′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of h(α), and h′(α) < γ < h(α). (B) Let α ∈ P(n), α ≠ α′, and n ⩾ 4. Then f′(α) is identical in sign to f(α), χ^α(g_f′(α)) ≠ 0, but χ^α(g_γ) = 0 for all γ ∈ P(n) such that the sign of γ coincides with one of f(α), and f′(α) < γ < f(α). The results obtained are then applied to study pairs of semiproportional irreducible characters in A_n. Supported by RFBR grant No. 04-01-00463. __________ Translated from Algebra i Logika, Vol. 44, No. 6, pp. 643–663, November–December, 2005.     

Non-commutative Clarkson Inequalities for n-Tuples of Operators

Let A ₀, ... , A _n−1 be operators on a separable complex Hilbert space , and let α₀,..., α _n−1 be positive real numbers such that 1. We prove that for every unitarily invariant norm,

Functional Gabor frame multipliers

A Gabor frame multiplier is a bounded operator that maps normalized tight Gabor frame generators to normalized tight Gabor frame generators. While characterization of such operators is still unknown, we give a complete characterization for the functional Gabor frame multipliers. We prove that a L^∞ -function h is a functional Gabor frame multiplier (for the time-frequency lattice aℤ × bℤ) if and only if it is unimodular and is a-periodic. Along the same line, we also characterize all the Gabor frame generators g (resp. frame wavelets ψ) for which there is a function ∈ L^∞(ℝ) such that {wg_mn} (resp. ωψ_k,ℝ) is a normalized tight frame.     

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.     

A remark on the coefficients of bounded polynomials

Let be such that |p(e^iq)|≤1 for ϕ∈R and |p(1)|=a∈[0,1]. An inequality of Dewan and Govil for the sum |a_v|+|a_n|, 0≤u<v≤n is sharpened.     

Non-group Ranks in Finite Full Transformation Semigroups

T _n be the full transformation semigroup on a finite set. Both rank and idempotent rank of the semigroup K(n,r) = {α∈T _n : | im α | ≤r, 2 ≤ r ≤ n - 1. In this paper we prove that the non-group rank, defined as the cardinality of a minimal generating set of non-group elements, of K(n,r) is S(n,r) , the Stirling number of the second kind.     

On the embedding problem with noncommutative kernel of order p4. VI

In the case of number fields the embedding problem of a p-extension with non-Abelian kernel of order p⁴ is studied. The two kernels of order 3⁴ with generators α, γ and relations α⁹ = 1, [α,α]³=1,[α,αγγ]==1,[αγγ]=α³,γ³=1 or γ³=α³ and the kernel of order 2⁴ with generators α, β, γ and relations α⁴=1 β²,[αγ]=1, [α,γ]=1,[βγ]=α² are considered. For kernels of odd order the embedding problem is always solvable. For the kernel of order 16 the solvability conditions are reduced to those for the associated problems at the Archimedean points, and to the compatibility condition. Bibliography: 9 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 227, 1995, pp. 74–82.