不规则多生成子Gabor框架及其对偶

对给定的φ_0,…,φ_(r-1)∈L~2(R)和a_0,b_0,…,a_(r-1)1,b_(r-1)>0,本文考虑不规则多生成子Gabor系统{E_(mb_l)T_(na_l)φ_l,m,n∈Z,l=0,…,r-1}.本文给出了该系统成为L~2(R)框架的充要条件;得到了不规则多生成子Gabor框架与其对偶之间关系的刻画.特别地,给出了一类多生成子Gabor框架及其对偶的显式构造.     

On Algebras of Skew Polynomials Generated by Quadratic Homogeneous Relations

We consider the algebras, with two generators a and b, generated by the quadratic relations ba = α² + βab + γb², where the coefficients α, β, and γ belong to an arbitrary field F of characteristic 0. We find conditions for such an algebra to be expressed as a skew polynomial algebra with generator b over the polynomial ring F [a]. These conditions are equivalent to the existence of the Poincaree-Birkhoff-Witt basis, i. e., basis of the form {a^m, bⁿ}. Bibliography: 16 titles.__________Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 301, 2003, pp. 144–171.     

Dilation of Dual Frame Pairs in Hilbert C*-Modules

We investigate the geometric properties for Hilbert C*-modular frames. We show that any dual frame pair in a Hilbert C*-module is an orthogonal compression of a Riesz basis and its canonical dual for some larger Hilbert C*-module. This generalizes the Hilbert space dual frame pair dilation theory due to Casazza, Han and Larson to dual Hilbert C*-modular frame pairs. Additionally, for any Hilbert C*-modular dual frame pair induced by a group of unitary operators, we show that there is a dilated dual pair which inherits the same group structure.     

On Pairs of Dual Consequence Operations

In the paper, the authors discuss two kinds of consequence operations characterized axiomatically. The first one are consequence operations of the type Cn ⁺ that, in the intuitive sense, are infallible operations, always leading from accepted (true) sentences of a deductive system to accepted (true) sentences of the deductive system (see Tarski in Monatshefte für Mathematik und Physik 37:361–404, 1930, Comptes Rendus des Séances De la Société des Sciences et des Lettres de Varsovie 23:22–29, 1930; Pogorzelski and Słupecki in Stud Logic 9:163–176, 1960, Stud Logic 10:77–95, 1960). The second kind are dual consequence operations of the type Cn ⁻ that can be regarded as anti-infallible operations leading from non-accepted (rejected, false) sentences of a deductive system to non-accepted (rejected, false) sentences of the system (see Słupecki in Funkcja Łukasiewicza, 33–40, 1959; Wybraniec-Skardowska in Teoria zdań odrzuconych, 5–131, Zeszyty Naukowe Wyższej Szkoły Inżynierskiej w Opolu, Seria Matematyka 4(81):35–61, 1983, Ann Pure Appl Logic 127:243–266, 2004, in On the notion and function of rejected propositions, 179–202, 2005). The operations of the types Cn ⁺ and Cn ⁻ can be ordinary finitistic consequence operations or unit consequence operations. A deductive system can be characterized in two ways by the following triple:

On the Geometry of Dual Pairs

The set of dual pairs of any norm v equivalent to a Hilbert norm is shown to be naturally homeomorphic to the sphere of the Hilbert space. The proof begins with a known result showing the representability of every vector as a sum of two orthogonal vectors, one coming from a cone and the other from its dual (a generalization of representation by orthogonal subspaces). The key theorem, showing that every non-zero vector has a positive multiple which is the sum of two v-dual vectors, follows from this and in turn provides the required homeomorphism. One consequence of this topological equivalence is the arc-connectedness of the numerical range determined by v.     

On Representation of a Solution to the Cauchy Problem by a Fourier Series in Sobolev-Orthogonal Polynomials Generated by Laguerre Polynomials

We consider the problem of representing a solution to the Cauchy problem for an ordinary differential equation as a Fourier series in polynomials l _r,k ^α (x) (k = 0, 1,...) that are Sobolev-orthonormal with respect to the inner product

$$\left\langle {f,g} \right\rangle = \sum\limits_{v = 0}^{r - 1} {{f^{(v)}}(0){g^{(v)}}} (0) + \int\limits_0^\infty {{f^{(r)}}(t)} {g^{(r)}}(t){t^\alpha }{e^{ - t}}dt$$

, and generated by the classical orthogonal Laguerre polynomials L _k ^α (x) (k = 0, 1,...). The polynomials l _r,k ^α (x) are represented as expressions containing the Laguerre polynomials L _n ^α?r (x). An explicit form of the polynomials l _r,k+r ^α (x) is established as an expansion in the powers x ^r+l , l = 0,..., k. These results can be used to study the asymptotic properties of the polynomials l _r,k ^α (x) as k→∞and the approximation properties of the partial sums of Fourier series in these polynomials.

     

On the Dual Mean-Value Conjecture for Complex Polynomials

Mathematical Notes -     

Frame expansions for Gabor multipliers

Discrete Gabor multipliers are composed of rank one operators. We shall prove, in the case of rank one projection operators, that the generating operators for such multipliers are either Riesz bases (exact frames) or not frames for their closed linear spans. The same dichotomy conclusion is valid for general rank one operators under mild and natural conditions. This is relevant since discrete Gabor multipliers have an emerging role in communications, radar, and waveform design, where redundant frame decompositions are increasingly applicable.     

B-Spline Approximations of the Gaussian,their Gabor Frame Properties,and Approximately Dual Frames

We prove that Gabor systems generated by certain scaled B-splines can be considered as perturbations of the Gabor systems generated by the Gaussian, with a deviation within an arbitrary small tolerance whenever the order N of the B-spline is sufficiently large. As a consequence we show that for any choice of translation/modulation parameters \(a,b>0\) with \(ab<1\), the scaled version of \(B_N\) generates Gabor frames for N sufficiently large. Considering the Gabor frame decomposition generated by the Gaussian and a dual window, the results lead to estimates of the deviation from perfect reconstruction that arise when the Gaussian is replaced by a scaled B-spline, or when the dual window of the Gaussian is replaced by certain explicitly given and compactly supported linear combinations of the B-splines. In particular, this leads to a family of approximate dual windows of a very simple form, leading to “almost perfect reconstruction” within any desired error tolerance whenever the product ab is sufficiently small. In contrast, the known (exact) dual windows have a very complicated form. A similar analysis is sketched with the scaled B-splines replaced by certain truncations of the Gaussian. As a consequence of the approach we prove (mostly known) convergence results for the considered scaled B-splines to the Gaussian in the \(L^p\)-spaces, as well in the time-domain as in the frequency domain.     

Wold-Type Decompositions for Pairs of Commutative Semigroups Generated by Isometries

Functional Analysis and Its Applications - In this paper we analyze connections between Wold-type decompositions of bi/isometries and pairs of semigroups of isometries, where at least one of...     

A Characterization of MRA Based Wavelet Frames Generated by the Walsh Polynomials

Extension Principles play a significant role in the construction of MRA based wavelet frames and have attracted much attention for their potential applications in various scientific fields. A novel and simple procedure for the construction of tight wavelet frames generated by the Walsh polynomials using Extension Principles was recently considered by Shah in [Tight wavelet frames generated by the Walsh poly-nomials, Int. J. Wavelets, Multiresolut. Inf. Process., 11(6) (2013), 1350042]. In this paper, we establish a complete characterization of tight wavelet frames generated by the Walsh polynomials in terms of the polyphase matrices formed by the polyphase components of the Walsh polynomials.     

Zeros of Gegenbauer-Sobolev Orthogonal Polynomials: Beyond Coherent Pairs

Iserles et al. (J. Approx. Theory 65:151–175, 1991) introduced the concepts of coherent pairs and symmetrically coherent pairs of measures with the aim of obtaining Sobolev inner products with their respective orthogonal polynomials satisfying a particular type of recurrence relation. Groenevelt (J. Approx. Theory 114:115–140, 2002) considered the special Gegenbauer-Sobolev inner products, covering all possible types of coherent pairs, and proves certain interlacing properties of the zeros of the associated orthogonal polynomials. In this paper we extend the results of Groenevelt, when the pair of measures in the Gegenbauer-Sobolev inner product no longer form a coherent pair. This research is supported by grants from CNPq and FAPESP.     

Dual Pairs of Hopf *-Algebras

If A is a Hopf *-algebra, the dual space A' is again a *-algebra.There is a natural subalgebra A° of A' that is again a Hopf*-algebra. In many interesting examples, A° will be largeenough (to separate points of A). More generally, one can considera pair (A, B) of Hopf *-algebras and a bilinear form on A xB with conditions such that, if the pairing is non-degenerate,one algebra can be considered as a subalgebra of the dual ofthe other. In these notes, we study such pairs of Hopf *-algebras. We startfrom the notion of a Hopf *-algebra A and its reduced dual A°.We give examples of pairs of Hopf *-algebras, and discuss theproblem of non-degeneracy. The first example is an algebra pairedwith itself. The second example is the pairing of a Hopf *-algebra(due to Jimbo) and the twisted SU(n) of Woronowicz. We alsodiscuss the notion of the quantum double of Drinfeld in thisframework of dual pairs.     

Basic Results of Gabor Frame on Local Fields

Basic facts for Gabor frame {Eu(m)bTu(n)ag}m,n∈p on local field are investigated. Accurately, that the canonical dual of frame {Eu(m)bTu(n)ag}m,n∈p also has the Gabor structure is showed; that the product ab decides whether it is possible for {Eu(m)bTu(n)ag}m,n∈p to be a frame for L2(K) is discussed; some necessary conditions and two sufficient conditions of Gabor frame for L2(K) are established. An example is finally given.     

伪仿射对偶框架

对于给定冗余的仿射框架X（Ф）={Ф1,…,ФL）,存在对偶Y（ψ）={ψ1,…,ψL）,这个对偶可能不是仿射框架,因为它不满足Bessel条件．这样的对偶我们称为伪仿射对偶,并给出伪仿射对偶框架的一种构造公式．     

Integral Representations of the Wilson Polynomials and the Continuous Dual Hahn Polynomials

We give representations of the Wilson polynomials and the continuous dual Hahn polynomials in terms of multidimensional generalizations of Barnes type integrals. Motivation is to study the Barnes type integrals from the viewpoint of the de Rham theory and holonomic systems.     

Dualities and Dual Pairs in Heyting Algebras

We extract the abstract core of finite homomorphism dualities using the techniques of Heyting algebras and (combinatorial) categories.