Matrix-valued Gabor frames over LCA groups for operators

In practice, the time variable cannot be negative. The space $$L^2({\mathbb {R}}_+)$$ of square integrable functions defined on the right half real line $${\mathbb {R}}_+$$ models the causal signal space. This paper focuses on a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$. The density theorem for Gabor systems in $$L^2(\mathbb {R})$$ states a necessary and sufficient condition for the existence of complete Gabor systems or Gabor frames in $$L^2({\mathbb {R}})$$ in terms of the index set alone-independently of window functions. The space $$L^2({\mathbb {R}}_+)$$ admits no nontrivial Gabor system since $${\mathbb {R}}_+$$ is not a group according to the usual addition. In this paper, we introduce a class of dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ and the notion of $$\Theta $$-transform matrix. Using the $$\Theta $$-transform matrix method we obtain the density theorem of the dilation-and-modulation systems in $$L^2({\mathbb {R}}_+)$$ under the condition that $$\log _ba$$ is a positive rational number, where a and b are the dilation and modulation parameters respectively. Precisely, we prove that a necessary and sufficient condition for the existence of such a complete dilation-and-modulation system or dilation-and-modulation system frame in $$L^2({\mathbb {R}}_+)$$ is that $$\log _ba \le 1$$. Simultaneously, we obtain a $$\Theta $$-transform matrix-based expression of all complete dilation-and-modulation systems and all dilation-and-modulation system frames in $$L^2(\mathbb {R}_+)$$.

We consider pseudo-differential operators and Gabor frames on locally compact commutative Hausdorff groups. Let [Formula: see text] be an open subgroup of a locally compact commutative Hausdorff group [Formula: see text] and [Formula: see text] be a topological isomorphism, then we define the cross [Formula: see text]-ambiguity function [Formula: see text] and cross [Formula: see text]-Wigner function [Formula: see text] and a pseudo-differentiable operator [Formula: see text] with the symbol [Formula: see text]. Employing properties of [Formula: see text], we establish characteristics of pseudo-differentiable operators. Also, assume that [Formula: see text] and [Formula: see text], then the coefficient operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text]; and similarly, the adjoint operator [Formula: see text] uniquely, continuously extends to the linear operator [Formula: see text], where [Formula: see text] is the restriction of [Formula: see text] to [Formula: see text].

Gabor systems play an important role in various practical problems related to signal and image processing.In this paper, we study the computability of Gabor frames in L 2 .Computable version of some existing results related to Gabor systems are obtained.Also, computability of operators associated with Gabor frames is discussed and the computability of Zak transform is defined.Finally, we discuss the existence of the computable dual frames associated with the Gabor Systems.

We consider the construction of tight Gabor frames (h, a=1, b=1) from Gabor systems (g, a=1, b=1) with g a window having few zeros in the Zak transform domain via the operation h=Z -1(Zg/|Zg|), where Z is the standard Zak transform. We consider this operation with g the Gaussian, the hyperbolic secant, and for g belonging to a class of positive, even, unimodal, rapidly decaying windows of which the two-sided exponential is a typical example. All these windows g have the property that Zg has a single zero, viz. at (1/2,\1/2), in the unit square [0,1)2. The Gaussian and hyperbolic secant yield a frame for any a,b > 0 with ab < 1, and we show that so does the two-sided exponential. For these three windows it holds that S a -1/2 g h as a 1, where S a is the frame operator corresponding to the Gabor frame (g,a,a). It turns out that the h’s corresponding to g’s of the above type look and behave quite similarly when scaling parameters are set appropriately. We give a particular detailed analysis of the h corresponding to the two-sided exponential. We give several representations of this h, and we show that , and is continuous and differentiable everywhere except at the half-integers, etc., and we pay particular attention to the cases that the time constant of the two-sided exponential g tends to . We also consider the cases that the time constants of the Gaussian and of the hyperbolic secant tend to 0 or to . It so turns out that h thus obtained changes from the box function into its Fourier transform when the time constant changes from 0 to

In this paper, we propose a new type of a neural network which is inspired by Gabor systems from harmonic analysis. In this regard, we construct a class of sparsely connected neural networks utilizing the concept of time–frequency shifts, and we show that their approximation error rates can be tied to the number of modulations in the corresponding Gabor frame and to the smoothness of the input function. Furthermore, we show that such networks are easily implementable and we illustrate their performance with some numerical examples.

Short-time Fourier transform and superoscillations

The material presented so far naturally splits in two parts: a functional analytic treatment of frames in general Hilbert spaces, and a more direct approach to structured frames like Gabor frames and wavelet frames. In this final chapter we make connections to abstract harmonic analysis and show how we can gain insight about frames via the theory for group representations. More precisely, we show how the orthogonality relations for square-integrable group representations lead to series expansions of the elements in the underlying Hilbert space; on a concrete level, this gives an alternative approach to Gabor systems and wavelet systems. Feichtinger and Gröchenig proved that the group-theoretic setup even allows us to obtain series expansions in a large scale of Banach spaces, a result which leads Gröchenig to define frames in Banach spaces. By removing some of the conditions we obtain p-frames, first studied separately by Aldroubi, Sun, and Tang.

Generalized frame called g-frame was first proposed using a sequence of adjointable operators to deal with all the existing frames as a united object. In fact, the g-frame is an extension of ordinary frames. Generalized frames with adjointable operators called K-g-frame is a generalization of a g-frame. It can be used to reconstruct elements from the range of a adjointable operator K. K-g-frames have a certain advantage compared with g-frames in practical applications. This paper is devoted to study some properties of K-g-frame in Hilbert C∗ -module, we characterize the concept of K-g-frame by quotient maps. Also discus some result of the dual K-g-Bessel sequences of K-g-frame in Hilbert C∗ -module. Our results are more general than those previously obtained. It is shown that the results we obtained can immediately lead to the existing corresponding results in Hilbert Spaces.

Presently a direct analytical method is available for the digit-by-digit extraction of the square root of a given positive real number. To calculate the nthroot of a given positive real number one may use trial and error method, iterative method, etc. When one desires to determine the nth root, it is found thatsuch methods are inherent with certain weaknesses like the requirement of an initial guess, a large number of arithmetic operations and several iterativesteps for convergence, etc. There has been no direct method for the determination of the nth root of a given positive real number. This paper focusesattention on developing a numerical algorithm to determine the digit-by-digit extraction of the nth root of a given positive real number up to any desiredaccuracy. Examples are provided to illustrate the algorithm.

In the eighteenth century, Landen, Lagrange and Gauss studied a function of two positive real numbers that has become known as the arithmetic-geometric mean (AGM). In the nineteenth century, Borchardt generalized the AGM to a function of any 2n(n = 1,2,3,…) positive real numbers. In this paper, we generalize the AGM to a function of any even number of positive real numbers. If M(a, b) is the original AGM then M(a, b) is concave in the pair (a, b) of positive numbers and log M(eα, eβ) is convex in the pair (α,β) of real numbers; all our generalizations of the AGM behave similarly. We generalize this analysis extensively.

If λ \lambda is a positive real number strictly less than log ⁡ 3 \operatorname {log}3 , there is a positive number V λ V_\lambda such that every orientable hyperbolic 3 3 -manifold of volume greater than V λ V_\lambda admits λ \lambda as a Margulis number. If λ &gt; ( log ⁡ 3 ) / 2 \lambda &gt;(\operatorname {log}3)/2 , such a V λ V_\lambda can be specified explicitly and is bounded above by \[ λ ( 6 + 880 log ⁡ 3 − 2 λ log ⁡ 1 log ⁡ 3 − 2 λ ) , \lambda \bigg (6+\frac {880}{\operatorname {log}3-2\lambda } \operatorname {log}{1\over \operatorname {log}3-2\lambda }\bigg ), \] where log \operatorname {log} denotes the natural logarithm. These results imply that for λ &gt; log ⁡ 3 \lambda &gt;\operatorname {log}3 , an orientable hyperbolic 3 3 -manifold that does not have λ \lambda as a Margulis number has a rank- 2 2 subgroup of bounded index in its fundamental group and in particular has a fundamental group of bounded rank. Again, the bounds in these corollaries can be made explicit if λ &gt; ( log ⁡ 3 ) / 2 \lambda &gt;(\operatorname {log}3)/2 .

Let {fl(n) n Z} be a sequence of positive numbers with/3(0) 1 and sup{fl(n + 1)/fl(n) n Z} < .We then define the Hilbert space L2(fl) by L2(fl) L2(Z, f12).We denote f L2(fl) by f E= f(n)z with the summand f(n)z indicatin that f(n) is the n-th term of the sequence.Thus we have Ilfll :=-[f(n)12(fl(n))2.We also use z 1.The set {Z n Z} can be thought of as an othogonal basis for L2(/3).For f, g L2(/3) we define the formal product h fg by h .l (n)z where'(n)= E (k)g(n-k) k= if all the latter sums converge.Now let L(fl) {o L2(fl) of L2(/3) for all f LE(fl)}.Then for o L(fl) we can define the linear map M r LE(fl) --LE(fl) by Mr(f) of.If we let wn fl(n / 1)/fl(n), we have the following theorem.THEOREM (Shields, [3]).For o L(fl), M r is a bounded linear operator on LE(fl) and Mz is unitarily equivalent to the bilateral weighted shift T with weight sequence {Wn n Z}.Furthermore, under this unitary equiv- alence, {M r o L(fl)} corresponds to the commutant of T.One could also start with a bilateral weighted shift with a weight sequence of positive terms and then define {fl(n) n Z} by:In this paper when we say weighted shift, we are referring to a bilateral weighted shift with all terms of the weight sequence being positive.For the most part, we will be studying weighted shifts with periodic weight sequences.Theorem 2 provides some information about the adjoints of operators in a certain subalgebra of L(/3), and Theorem 3 is a spectral inclusion theorem for shifts with periodic weight sequences.

In this paper we investigate the behaviour of the solutions of the following close-to-cyclic system of two difference equations with maximum: x n + 1 = max { A , y n p x n − 1 q } , y n + 1 = max { B , x n p y n − 1 q } where n = 0 , 1 , … , the coefficients A , B are positive real numbers, thevexponents p , q are positive real numbers such that p &gt; q + 1 , and the initial values x − 1 , x 0 , y − 1 , y 0 , are positive real numbers.

Titchmarsh determined the spectrum of the Schrδdinger energy operator associated with the hydrogen atom, i.e. the operator — Δ — 1/r. He showed, in particular, that its essential spectrum consists of the positive real axis. On the other hand, Agudo-Wolf and Birman formulated overlapping criteria for a potential, which ensured that addition of such a potential does not change the essential spectrum of — Δ. These criteria do not admit the potential 1/r and a criterion admitting it is formulated in the forthcoming work of Balslev where he also considers operators in Lp spaces. In this paper we slightly extend this Balslev criterion, in case the operator is a Schrδdinger operator. Our proofs are different, inasmuch as we capitalize on the representation of the kernel of the unperturbed resolvent. Then we make essential use of a result of Friedrichs which gives a bound for the norm of an integral operator. A criterion which ensures that the essential spectra of two self adjoint operators are equal is the following: for an appropriate complex number ζ the difference of their resolvents is compact. We shall refer to this property by saying that these operators are resolvent congruent. This property requires less and ensures less, then the one introduced independently by de Branges [18] and Birman-Krein [11] in connection with the perturbation of the continuous spectrum. It is implied by one which we called relative compactness by Gokhberg-Krei n [24] and for convenience this is shown in §1. For semi-bounded operators still another property was introduced by Birman [9] who showed that his property also implies this one. In § 1 we recall the classical notion of the essential spectrum, and formulate some general operator theoretic facts, which are used in the subsequent sections. Section 2 contains two theorems. In the first one we show, in particular, that the local square integrability of 1/r and the fact that

Ultrasound image reconstruction can be posed as an inverse problem, with image pixels as parameters to a model of wave propagation that are estimated from raw ultrasound channel data. Such framework allows one to make assumptions about channel signals and image properties in form of regularization that can be used to improve reconstruction accuracy in the presence of electronic and acoustic noise. A simple model assumes spherical wave propagation from point sources in a homogeneous medium, similar to delay-and-sum (DAS) beamforming; solving such problems numerically can require many iterations and can result in blurred images when traditional L2 norm regularization is used. We use Fast Iterative Shrinkage Thersholding Algorithm (FISTA) to reduce the number of iterations to less than 10, and to apply L1-norm regularization to the data, which improves edge sharpness and image contrast. We demonstrate the concept using FIELD II simulated ultrasound signals from point, speckle, and anechoic targets with controlled levels of electronic and speckle noise. The FISTA-reconstructed point targets show reduced sidelobe levels by 15 dB compared to the traditional DAS image, and the reduction in full-width at half maximum by a factor of 2. We also implement iterative reconstruction in omega-k frequency domain, which allows factoring the forward and adjoint operators at each spatial frequency and paves the way for an intuitive and more memory efficient, multi-threaded implementation of the method.