A. Plastino, M. C. Rocca
We appeal to a complex q-Fourier transform as a generalization of the (real) one analyzed in [Milan J. Math. {\bf 76} (2008) 307]. By recourse to tempered ultra-distributions we are able to show that the q-Gaussian distribution can be nicely obtained, overcoming all troubles that afflict its real counterpart.
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We develop a comprehensive theory of reflectionless canonical systems with an arbitrary Dirichlet-regular Widom spectrum with the Direct Cauchy Theorem property. This generalizes, to an infinite gap setting, the constructions of finite gap quasiperiodic (algebro-geometric) solutions of stationary integrable hierarchies. Instead of theta functions on a compact Riemann surface, the construction is based on reproducing kernels of character-automorphic Hardy spaces in Widom domains with respect to Martin measure. We also construct unitary character-automorphic Fourier transforms which generalize t…
A systematic analytic approach to the evaluation of the eigenvalues and eigenvectors of the 5D discrete number operator is formulated. This approach is essentially based on the use of the symmetricity of 5D discrete Fourier transform operator with respect to the discrete reflection operator.
Lorentz Transformations from Reflections:Some Applications
我们指出，通过展示两个例子，并提到第三个，它有时是有用的考虑洛伦兹变换产生的超平面或线反射。一个例子是关于连接两个给定的4-向量的提升的构造，另一个例子是关于V. Moretti的正交洛伦兹矩阵的极分解的闵可夫斯基几何理解。
(Anti)symmetric multivariate exponential functions and corresponding Fourier transforms
定义并研究了对称化和反对称化多元指数函数。它们被定义为元素为单变量指数函数的矩阵的行列式和反行列式。这些函数是拉普拉斯算子在满足一定边界条件的相应基本区上的本征函数。对于对称和反对称的多元指数函数，有相应的傅里叶变换。有三种类型的这种傅里叶变换：展开成相应的傅里叶级数，积分傅里叶变换，和多元有限傅里叶变换。求出了积分傅里叶变换的本征函数。
Uncertainty Principles for the Fourier Transforms in Quantum Calculus
本文研究了$q$ -傅里叶正弦变换的一些性质，导出了在\cite{FB}中研究的$q$ -傅里叶余弦变换和$q$ -傅里叶正弦变换的海森堡测不准原理的$q$ -类似物。
Fourier transform and related integral transforms in superspace
本文研究了经典Fourier变换、分数阶Fourier变换和Radon变换在超空间上的推广。以前，超空间中的傅里叶变换已经被研究过了，但是具有不同的内核。在这项工作中，费米子部分的傅立叶内核有一个自然的辛结构，使用克利福德分析方法。研究了这三种变换的几个基本性质。使用适当的推广的埃尔米特多项式的超空间（见[H。De Bie，F. Sommen，Hermite and Gegenbauer polynomials in superspace using Clifford analysis，J.Phys.A 40（2007）10441-10456]），构造了用于傅里叶变换的本征函数基。
We study Fourier theory on quantum Euclidean space. A modified version of the general definition of the Fourier transform on a quantum space is used and its inverse is constructed. The Fourier transforms can be defined by their Bochner's relations and a new type of q-Hankel transforms using the first and second q-Bessel functions. The behavior of the Fourier transforms with respect to partial derivatives and multiplication with variables is studied. The Fourier transform acts between the two representation spaces for the harmonic oscillator on quantum Euclidean space. By using this property it…
The present chapter [submitted for publication in "Fourier Transforms, Theory and Applications", G. Nikolic (Ed.), InTech (Open Access Publisher), Vienna, 2011] is concerned with the introduction and study of a quadratic discrete Fourier transform. This Fourier transform can be considered as a two-parameter extension, with a quadratic term, of the usual discrete Fourier transform. In the case where the two parameters are taken to be equal to zero, the quadratic discrete Fourier transform is nothing but the usual discrete Fourier transform. The quantum quadratic discrete Fourier transform plays…
In this paper, we prove that given a cut-and-project scheme $(G, H, \mathcal{L})$ and a compact window $W \subseteq H$, the natural projection gives a bijection between the Fourier transformable measures on $G \times H$ supported inside the strip $\cL \cap (G \times W)$ and the Fourier transformable measures supported inside $\oplam(W)$, and relate their Fourier transforms. We use this formula to relate the Fourier transforms of the measures, and explain how one can use this relation to re-derive some known results about Fourier analysis of measures with Meyer set support.
In this review, an overview is given of several recent generalizations of the Fourier transform, related to either the Lie algebra sl_2 or the Lie superalgebra osp(1|2). In the former case, one obtains scalar generalizations of the Fourier transform, including the fractional Fourier transform, the Dunkl transform, the radially deformed Fourier transform and the super Fourier transform. In the latter case, one has to use the framework of Clifford analysis and arrives at the Clifford-Fourier transform and the radially deformed hypercomplex Fourier transform. A detailed exposition of all these tr…
It is suggest that a new fractal model for the Yang-Fourier transforms of discrete approximation based on local fractional calculus and the Discrete Yang-Fourier transforms are investigated in detail.
We define a planar para algebra, which arises naturally from combining planar algebras with the idea of $\mathbb{Z}_{N}$ para symmetry in physics. A subfactor planar para algebra is a Hilbert space representation of planar tangles with parafermionic defects, that are invariant under para isotopy. For each $\mathbb{Z}_{N}$, we construct a family of subfactor planar para algebras which play the role of Temperley-Lieb-Jones planar algebras. The first example in this family is the parafermion planar para algebra (PAPPA). Based on this example, we introduce parafermion Pauli matrices, quaternion re…
Fourier and fractional-Fourier transformations are widely used in theoretical physics. In this paper we make quantum perspectives and generalization for the fractional Fourier transformation (FrFT). By virtue of quantum mechanical representation transformation and the method of integration within normal ordered product (IWOP) of operators, we find the key point for composing FrFT, and reveal the structure of FrFT. Following this procedure, a full family of generalized fractional transformations are discovered with the usual FrFT as one special case. The eigen-functions of arbitrary GFrT are de…
We propose an alternative definition for a Tsallis entropy composition-inspired Fourier transform, which we call "$τ_q$-Fourier transform". We comment about the underlying "covariance" on the set of algebraic fields that motivates its introduction. We see that the definition of the $τ_q$-Fourier transform is automatically invertible in the proper context. Based on recent results in Fourier analysis, it turns that the $τ_q$-Fourier transform is essentially unique under the assumption of the exchange of the point-wise product of functions with their convolution.
We explore a framework for complex classical fields, appropriate for describing quantum field theories. Our fields are linear transformations on a Hilbert space, so they are more general than random variables for a probability measure. Our method generalizes Osterwalder and Schrader's construction of Euclidean fields. We allow complex-valued classical fields in the case of quantum field theories that describe neutral particles. From an analytic point-of-view, the key to using our method is reflection positivity. We investigate conditions on the Fourier representation of the fields to ensure th…
We find a formula that relates the Fourier transform of a radial function on $\mathbf{R}^n$ with the Fourier transform of the same function defined on $\mathbf{R}^{n+2}$. This formula enables one to explicitly calculate the Fourier transform of any radial function $f(r)$ in any dimension, provided one knows the Fourier transform of the one-dimensional function $t\to f(|t|)$ and the two-dimensional function $(x_1,x_2)\to f(|(x_1,x_2)|)$. We prove analogous results for radial tempered distributions.
We discuss different representations of the White noise spaces $(E)_β$, $0 \leq β< 1$ by introducing generalized Wick tensors. As an application we state a generalization of the Mehler formula for the Ornstein-Uhlenbeck semigroup.
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