arXiv subjects

Arash Ghaani Farashahi

Publications and source records attributed to Arash Ghaani Farashahi.

At least 19 recordsLinked to original sources

arXiv2025

Fast Convolutions on $\mathbb{Z}^2\backslash SE(2)$ via Radial Translational Dependence and Classical FFT

Let $\mathbb{Z}^2\backslash SE(2)$ denote the right coset space of the subgroup consisting of translational isometries of the orthogonal lattice $\mathbb{Z}^2$ in the non-Abelian group of planar motions $SE(2)$. This paper develops a fast and accurate numerical scheme for approximation of functions on $\mathbb{Z}^2\backslash SE(2)$. We address finite Fourier series of functions on the right coset space $\mathbb{Z}^2\backslash SE(2)$ using finite Fourier coefficients. The convergence/error analysis of finite Fourier coefficients are investigated. Conditions are established for the finite Fourier coefficients to converge to the Fourier coefficients. The matrix forms of the finite transforms are discussed. The implementation of the discrete method to compute numerical approximation of $SE(2)$-convolutions with functions which are radial in translations are considered. The paper is concluded by discussing capability of the numerical scheme to develop fast algorithms for approximating multiple convolutions with functions which are radial in translations.

math.NA↗

arXiv2025

Coherent manifolds

This paper defines coherent manifolds and discusses their properties and their application in quantum mechanics. Every coherent manifold with a large group of symmetries gives rise to a Hilbert space, the completed quantum space of $Z$, which contains a distinguished family of coherent states labeled by the points of the manifold. The second quantization map in quantum field theory is generalized to quantization operators on arbitrary coherent manifolds. It is shown how the Schrödinger equation on any such completed quantum space can be solved in terms of computations only involving the coherent product. In particular, this applies to a description of Bosonic Fock spaces as completed quantum spaces of a class of coherent manifolds called Klauder spaces.

math-ph↗

arXiv2024

Non-Abelian Fourier Analysis on $\mathbfΓ\backslash SE(d)$

This paper presents a systematic study for the general theory of non-Abelian Fourier series of integrable functions on the homogeneous space $\mathbfΓ\backslash SE(d)$, where $SE(d)$ is the special Euclidean group in dimension $d$, and $\mathbf{\mathbfΓ}$ is a discrete and co-compact subgroup of $SE(d)$. Suppose that $μ$ is the finite $SE(d)$-invariant measure on the right coset space $\mathbfΓ\backslash SE(d)$, normalized with respect to Weil's formula. The analytic aspects of the proposed method works for any given orthonormal basis of the Hilbert function space $L^2(\mathbfΓ\backslash SE(d),μ)$. The paper is concluded with some convolution and Plancherel formulas.

math.FA↗

arXiv2024

Covariant Function Algebras of Invariant Characters of Normal Subgroups

This paper presents abstract harmonic analysis foundations for structure of covariant function algebras of invariant characters of normal subgroups. Suppose that $G$ is a locally compact group and $N$ is a closed normal subgroup of $G$. Let $ξ:N\to\mathbb{T}$ be a continuous $G$-invariant character, $1\le p<\infty$, and $L_ξ^p(G,N)$ be the $L^p$-space of all covariant functions of $ξ$ on $G$. We study structure of covariant convolution in $L^p_ξ(G,N)$. It is proved that $L^1_ξ(G,N)$ is a Banach $*$-algebra and $L^p_ξ(G,N)$ is a Banach $L^1_ξ(G,N)$-module. We then investigate the theory of covariant convolutions for the case of characters of canonical normal subgroups in semi-direct product groups. The paper is concluded by realization of the theory in the case of different examples.

math.FA↗

arXiv2023

Non-Abelian Fourier Series on $\mathbb{Z}^2\backslash SE(2)$

This paper discusses computational structure of coefficients of non-Abelian Fourier series on the right coset space $\mathbb{Z}^2\backslash SE(2)$ expressed in the trigonometric basis, where $SE(2)$ is the group of handedness preserving Euclidean isometries of the plane and $\mathbb{Z}^2$ denotes the discrete subgroup of translations of the orthogonal (square) lattice in $\mathbb{R}^2$. Assume that $μ$ is the finite $SE(2)$-invariant measure on the right coset space $\mathbb{Z}^2\backslash SE(2)$, normalized with respect to Weil's formula. We present a constructive computational characterization including discrete sampling of non-Abelian Fourier matrix elements on $SE(2)$ for coefficients of $μ$-square integrable functions on $\mathbb{Z}^2\backslash SE(2)$ with respect to the concrete trigonometric basis. The paper is concluded with discussion of the method for non-Abelian Fourier coefficients of convolutions on $\mathbb{Z}^2\backslash SE(2)$.

math.FA↗

arXiv2023

Asymptotically Steerable Finite Fourier-Bessel Transforms and Closure under Convolution

This paper develops a constructive numerical scheme for Fourier-Bessel approximations on disks compatible with convolutions supported on disks. We address accurate finite Fourier-Bessel transforms (FFBT) and inverse finite Fourier-Bessel transforms (iFFBT) of functions on disks using the discrete Fourier Transform (DFT) on Cartesian grids. Whereas the DFT and its fast implementation (FFT) are ubiquitous and are powerful for computing convolutions, they are not exactly steerable under rotations. In contrast, Fourier-Bessel expansions are steerable, but lose both this property and the preservation of band limits under convolution. This work captures the best features of both as the band limit is allowed to increase. The convergence/error analysis and asymptotic steerability of FFBT/ iFFBT are investigated. Conditions are established for the FFBT to converge to the Fourier-Bessel coefficient and for the iFFBT to uniformly approximate the Fourier-Bessel partial sums. The matrix form of the finite transforms is discussed. The implementation of the discrete method to compute numerical approximation of convolutions of compactly supported functions on disks is considered as well.

math.NA↗

arXiv2022

Introduction to coherent quantization

This paper is one of a series of papers on coherent spaces and their applications, defined in the recent book 'Coherent Quantum Mechanics' by the first author. The paper studies coherent quantization -- the way operators in the quantum space of a coherent space can be studied in terms of objects defined directly on the coherent space. The results may be viewed as a generalization of geometric quantization, including the non-unitary case. Care has been taken to work with the weakest meaningful topology and to assume as little as possible about the spaces and groups involved. Unlike in geometric quantization, the groups are not assumed to be compact, locally compact, or finite-dimensional. This implies that the setting can be successfully applied to quantum field theory, where the groups involved satisfy none of these properties. The paper characterizes linear operators acting on the quantum space of a coherent space in terms of their coherent matrix elements. Coherent maps and associated symmetry groups for coherent spaces are introduced, and formulas are derived for the quantization of coherent maps. The importance of coherent maps for quantum mechanics is due to the fact that there is a quantization operator that associates homomorphically with every coherent map a linear operator from the quantum space into itself. This operator generalizes to general symmetry groups of coherent spaces the second quantization procedure for free classical fields. The latter is obtained by specialization to Klauder spaces, whose quantum spaces are the bosonic Fock spaces. A coordinate-free derivation is given of the basic properties of creation and annihilation operators in Fock spaces.

math-ph↗

arXiv2021

Banach Convolution Modules of Group Algebras on Covariant Functions of Characters of Normal Subgroups

This paper investigates structure of Banach convolution modules induced by group algebras on covariant functions of characters of closed normal subgroups. Let $G$ be a locally compact group with the group algebra $L^1(G)$ and $N$ be a closed normal subgroup of $G$. Suppose that $ξ:N\to\mathbb{T}$ is a continuous character, $1\le p<\infty$ and $L_ξ^p(G,N)$ is the $L^p$-space of all covariant functions of $ξ$ on $G$. It is shown that $L^p_ξ(G,N)$ is a Banach $L^1(G)$-module. We then study convolution module actions of group algebras on covariant functions of characters for the case of canonical normal subgroups in semi-direct product groups.

math.FA↗

arXiv2021

Harmonic Analysis of Covariant Functions of Characters of Normal Subgroups

Let $G$ be a locally compact group with the group algebra $L^1(G)$ and $N$ be a closed normal subgroup of $G$. Suppose that $ξ:N\to\mathbb{T}$ is a continuous character and $L_ξ^1(G,N)$ is the $L^1$-space of all covariant functions of $ξ$ on $G$. We showed that $L^1_ξ(G,N)$ is isometrically isomorphic to a quotient space of $L^1(G)$. It is also proved that the dual space $L^1_ξ(G,N)^*$ is isometrically isomorphic to $L^\infty_ξ(G,N)$.

math.FA↗

arXiv2021

Covariant Functions of Characters of Compact Subgroups

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let $G$ be a locally compact group and $H$ be a compact subgroup of $G$. Suppose that $ξ:H\to\mathbb{T}$ is a continuous character, $1\le p<\infty$ and $L_ξ^p(G,H)$ is the set of all covariant functions of $ξ$ in $L^p(G)$. It is shown that $L^p_ξ(G,H)$ is isometrically isomorphic to a quotient space of $L^p(G)$. It is also proven that $L^q_ξ(G,H)$ is isometrically isomorphic to the dual space $L^p_ξ(G,H)^*$, where $q$ is the conjugate exponent of $p$. The paper is concluded by some results for the case that $G$ is compact.

math.FA↗

arXiv2019

Abstract Structure of Measure Algebras on Coset Spaces of Compact Subgroups in Locally Compact Groups

This paper presents a systematic operator theory approach for abstract structure of Banach measure algebras over coset spaces of compact subgroups. Let $H$ be a compact subgroup of a locally compact group $G$ and $G/H$ be the left coset space associated to the subgroup $H$ in $G$. Also, let $M(G/H)$ be the Banach measure space consists of all complex measures over $G/H$. We then introduce an operator theoretic characterization for the abstract notion of involution over the Banach measure space $M(G/H)$.

math.FA↗

arXiv2019

Discrete Spectra of Convolutions on Disks using Sturm-Liouville Theory

This paper presents a systematic study for analytic aspects of discrete spectra methods for convolution of functions supported on disks, according to the Sturm-Liouville theory. We then investigate different aspects of the presented theory in the cases of zero-value boundary condition and derivative boundary condition.

math.FA↗

arXiv2018

Fourier-Zernike Series of Convolutions on Disks

This paper presents a systematic study for analytic aspects of Fourier-Zernike series of convolutions of functions supported on disks. We then investigate different aspects of the presented theory in the cases of zero-padded functions.

math.FA↗

arXiv2018

Exploiting Partial Structural Symmetry For Patient-Specific Image Augmentation in Trauma Interventions

In unilateral pelvic fracture reductions, surgeons attempt to reconstruct the bone fragments such that bilateral symmetry in the bony anatomy is restored. We propose to exploit this "structurally symmetric" nature of the pelvic bone, and provide intra-operative image augmentation to assist the surgeon in repairing dislocated fragments. The main challenge is to automatically estimate the desired plane of symmetry within the patient's pre-operative CT. We propose to estimate this plane using a non-linear optimization strategy, by minimizing Tukey's biweight robust estimator, relying on the partial symmetry of the anatomy. Moreover, a regularization term is designed to enforce the similarity of bone density histograms on both sides of this plane, relying on the biological fact that, even if injured, the dislocated bone segments remain within the body. The experimental results demonstrate the performance of the proposed method in estimating this "plane of partial symmetry" using CT images of both healthy and injured anatomy. Examples of unilateral pelvic fractures are used to show how intra-operative X-ray images could be augmented with the forward-projections of the mirrored anatomy, acting as objective road-map for fracture reduction procedures.

cs.CV↗

arXiv2014

A new approach to the Fourier analysis on semi-direct products of groups

Let $H$ and $K$ be locally compact groups and also $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $τ$. This paper presents a novel approach to the Fourier analysis of $G_τ$, when $K$ is abelian. We define the $τ$-dual group $G_{\hatτ}$ of $G_τ$ as the semi-direct product $H\ltimes_{\hatτ}\hat{K}$, where $\hatτ:H\to Aut(\hat{K})$ defined via (\ref{A}). We prove a Ponterjagin duality Theorem and also we study $τ$-Fourier transforms on $G_τ$. As a concrete application we show that how these techniques apply for the affine group and also we compute the $τ$-dual group of Euclidean groups and the Weyl-Heisenberg groups.

math.FA↗

arXiv2012

Zak Transform for Semidirect Product of Locally Compact Groups

Let $H$ be a locally compact group and $K$ be an LCA group also let $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semidirect product of $H$ and $K$ with respect to $τ$. In this article we define the Zak transform $\mathcal{Z}_L$ on $L^2(G_τ)$ with respect to a $τ$-invariant uniform lattice $L$ of $K$ and we also show that the Zak transform satisfies the Plancherel formula. As an application we show that how these techniques apply for the semidirect product group $\mathrm{SL}(2,\mathbb{Z})\ltimes_τ\mathbb{R}^2$ and also the Weyl-Heisenberg groups.

math.FA↗

arXiv2012

Continuous Gabor transform for semi-direct product of locally compact groups

Let $H$ be a locally compact group, $K$ be an LCA group, $τ:H\to Aut(K)$ be a continuous homomorphism and $G_τ=H\ltimes_τK$ be the semi-direct product of $H$ and $K$ with respect to the continuous homomorphism $τ$. In this article we introduce the $τ\times\hatτ$-time frequency group $G_{τ\times\hatτ}$. We define the $τ\times\hatτ$-continuous Gabor transform of $f\in L^2(G_τ)$ with respect to a window function $u\in L^2(K)$ as a function defined on $G_{τ\times\hatτ}$. It is also shown that the $τ\times\hatτ$-continuous Gabor transform satisfies the Plancherel Theorem and reconstruction formula. This approach is tailored for choosing elements of $L^2(G_τ)$ as a window function. Finally, we illustrate application of these methods in the case of some well-known semi-direct product groups.

math.FA↗

arXiv2012

Convolution and involution on function spaces of homogeneous spaces

Let $G$ be a locally compact group and also let $H$ be a compact subgroup of $G$. It is shown that, if $μ$ is a relatively invariant measure on $G/H$ then there is a well-defined convolution on $L^1(G/H,μ)$ such that the Banach space $L^1(G/H,μ)$ becomes a Banach algebra. We also find a generalized definition of this convolution for other $L^p$-spaces. Finally, we show that various types of involutions can be considered on $G/H$.

math.FA↗