Paley_ Wiener Theorems for The Fourier Transform depending on the Heisenberg group: مبرهنات بالي _ وينر لتحويل فورييه اعتماداً على زمرة هايزنبرغ

In this paper, we talk about Heisenberg group, the most known example from the lie groups. After that, we discuss the representation theory of this group and the relationship between the representation theory of the Heisenberg group and the position and momentum operators and momentum operators relationship between the representation theory of the Heisenberg group and the position and momentum that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schrodinger picture. That is, all the representations we considered are realized in the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator-valued function, and other facts and properties. In our research, we depended on new formulas for some mathematical concepts such as Fourier Transform and Weyl transform. The main aim of our research is to introduce the Paley_ Wiener theorem for the Fourier transform on the Heisenberg group. We will show that the classical Paley_ Wiener theorem for the Euclidean Fourier transform characterizes compactly supported functions in terms of the behaviour of their Fourier transforms and Weyl transform. And we are interested in establishing results for the group Fourier transform and the Weyl transform.

The role of Heisenberg group in Harmonic analysis: دور زمرة هايزنبرغ في التحليل التوافقي

In this paper we talk about Heisenberg group, the most know example from the lie groups. After that we discuss the representation theory of this group, and the relationship between the representation theory of the Heisenberg group and the position and momentum operatorsو and momentum operators.ors. ielationship between the representation theory of the Heisenberg group and the position and momen, that shows how we will make the connection between the Heisenberg group and physics. we have considered only the Schr dinger picture. That is, all the representations we considered are realized on the Hilbert space . we define the group Fourier transform on the Heisenberg group as an operator valued function, and other facts and properties. The main aim of our research is having the formula of Schr dinger Representation that connect physics with the Heisenberg group. Depending on this Representation we will study new formulas for some mathematical concepts such us Fourier Transform and .

The Stochastic Resonance of Noise Frequency Modulation Signal Using Motion-Group Fourier Transform

In this paper stochastic resonance was studied in radar driven by noise frequency modulation signal. According to the intrinsic relations between the stochastic differential and the radar jamming signal processing, the stochastic calculus was used in the radar jamming signal processing in this paper. The noise frequency modulation signal was particularly analyzed. The Fokker-Planck equation of noise frequency modulation was presented and the Motion-Group Fourier Transform was used by converting the partial differential equation into the variable coefficient homogenous linear differential equations. Then the solutions were given.

Inversion of the Radon Transform on the Free Nilpotent Lie Group of Step Two

Let F2n;2 be the free nilpotent Lie group of step two on 2n generators, and let P denote the affine automorphism group of F2n;2. In this article the theory of continuous wavelet transformon F2n;2 associated with P is developed, and then a type of radial wavelet is constructed. Secondly, the Radon transform on F2n;2 is studied, and two equivalent characterizations of the range for Radon transform are given. Several kinds of inversion Radon transform formulae are established. One is obtained from the Euclidean Fourier transform; the others are from the group Fourier transform. By using wavelet transforms we deduce an inversion formula of the Radon transform, which does not require the smoothness of functions if the wavelet satisfies the differentiability property. In particular, if n = 1, F2;2 is the 3-dimensional Heisenberg group H1, the inversion formula of the Radon transform is valid, which is associated with the sub-Laplacian on F2;2. This result cannot be extended to the case n ≥ 2.

High performance of amidoxime/amine functionalized polypropylene for uranyl (VI) from aqueous solution

Amidoxime-based adsorbents are widely investigated as the main adsorbent in the recovery of uranium from seawater. However, the adsorption rate and loading capacity of such adsorbents should be further improved due to the economic viability consideration. In this paper, the adsorption properties of amidoxime-based adsorbent were enhanced through the cografting of amino groups. The acrylonitrile (AN) and glycidyl methacrylate (GMA) were firstly cografted onto polypropylene fibers by preirradiation grafting technique. Then the cografted fibers were treated with ethylenediamine to convert GMA to amino group, following treated with hydroxylamine to convert AN groups to amidoxime (AO) group. Fourier transform infrared spectroscopy, scanning electron microscopy and contact angle measurement were used to characterize the structure of the adsorbent. The results showed that the amidoxime and amino groups had been successfully grafted onto the polypropylene fibers. For AO fiber, high adsorption rate was observed within the first 30 min and the plateau value of 42.3% uranium loading (0.0904mg/g) was reached at around 30 min. The AO/amine fibers exhibited a higher adsorption rate. The adsorption equilibrium for AO/amine fiber was attained within 20 min, resulting in the adsorption of 93.3% uranium loading (0.191mg/g). Additionally, AO/amine fibers can avoid pH reducing for better adsorption efficiency.

Diffusive wavelets on groups and homogeneous spaces

We explain the basic ideas behind the concept of diffusive wavelets on spheres in the language of representation theory of Lie groups and within the framework of the group Fourier transform given by the Peter-Weyl decomposition of L2() for a compact Lie group . After developing a general concept for compact groups and their homogeneous spaces, we give concrete examples for tori, which reflect the situation on ℝn, and for 2 and 3 spheres.

VARIANTS OF MIYACHI’S THEOREM FOR NILPOTENT LIE GROUPS

We formulate and prove two versions of Miyachi’s theorem for connected, simply connected nilpotent Lie groups. This allows us to prove the sharpness of the constant 1/4 in the theorems of Hardy and of Cowling and Price for any nilpotent Lie group. These theorems are proved using a variant of Miyachi’s theorem for the group Fourier transform.

A CLASS OF STRONGLY SINGULAR RADON TRANSFORMS ON THE HEISENBERG GROUP

We primarily consider here the $L^2$ mapping properties of a class of strongly singular Radon transforms on the Heisenberg group $\mathbb{H}^n$; these are convolution operators on $\mathbb{H}^n$ with kernels of the form $M(z,t)=K(z)\delta_0(t)$, where $K$ is a strongly singular kernel on $\mathbb{C}^n$. Our results are obtained by using the group Fourier transform and uniform asymptotic forms for Laguerre functions due to Erdélyi.We also discuss the behaviour of related twisted strongly singular operators on $L^2(\mathbb{C}^n)$ and obtain results in this context independently of group Fourier transform methods. Key to this argument is a generalization of the results for classical strongly singular integrals on $L^2(\mathbb{R}^d)$.

A convolution type characterization forLp- multipliers for the Heisenberg group

It is well known that ifmis anLp- multiplier for the Fourier transform onℝn(1<p<∞), then there exists a pseudomeasureσsuch thatTmf =σ*f. A similar result is proved for the group Fourier transform on the Heisenberg groupHn. Though this result is already known in generality for amenable groups, a simple proof is provided in this paper.

AnLp-Lq-Version of Morgan's Theorem for then-Dimensional Euclidean Motion Group

We establish anLp-Lq-version of Morgan's theorem for the group Fourier transform on then-dimensional Euclidean motion groupM(n).