arXiv subjects

Mozhgan Mohammadpour

Publications and source records attributed to Mozhgan Mohammadpour.

5 recordsLinked to original sources

arXiv2024

An explicit construction of the unitarily invariant quaternionic polynomial spaces on the sphere

The decomposition of the polynomials on the quaternionic unit sphere in $\Hd$ into irreducible modules under the action of the quaternionic unitary (symplectic) group and quaternionic scalar multiplication has been studied by several authors. Typically, these abstract decompositions into ``quaternionic spherical harmonics'' specify the irreducible representations involved and their multiplicities. The elementary constructive approach taken here gives an orthogonal direct sum of irreducibles, which can be described by some low-dimensional subspaces, to which commuting linear operators $L$ and $R$ are applied. These operators map harmonic polynomials to harmonic polynomials, and zonal polynomials to zonal polynomials. We give explicit formulas for the relevant ``zonal polynomials'' and describe the symmetries, dimensions, and ``complexity'' of the subspaces involved. Possible applications include the construction and analysis of desirable sets of points in quaternionic space, such as equiangular lines, lattices and spherical designs (cubature rules).

math.RT↗

arXiv2024

Complex spherical designs from group orbits

We consider the general question of when all orbits under the unitary action of a finite group give a complex spherical design. Those orbits which have large stabilisers are then good candidates for being optimal complex spherical designs. This is done by developing the general theory of complex designs and associated (harmonic) Molien series for group actions. As an application, we give explicit constructions of some putatively optimal real and complex spherical t-designs.

math.CO↗

arXiv2019

Constructing high order spherical designs as a union of two of lower order

We show how the variational characterisation of spherical designs can be used to take a union of spherical designs to obtain a spherical design of higher order (degree, precision, exactness) with a small number of points. The examples that we consider involve taking the orbits of two vectors under the action of a complex reflection group to obtain a weighted spherical $(t,t)$-design. These designs have a high degree of symmetry (compared to the number of points), and many are the first known construction of such a design, e.g., a $32$ point $(9,9)$-design for $\mathbb{C}^2$, a $48$ point $(4,4)$-design for $\mathbb{C}^3$, and a $400$ point $(5,5)$-design for $\mathbb{C}^4$.From a real reflection group, we construct a $360$ point $(9,9)$-design for $\mathbb{R}^4$ (spherical half-design of order $18$), i.e., a $720$ point spherical $19$-design for $\mathbb{R}^4$.

math.MG↗

arXiv2017

Finite Synchrosqueezing Transform Based On The STFT

The finite STFT Synchrosqueezing transform is a time-frequency analysis method that can decompose finite complex signals into time-varying oscillatory components. This representation is sparse and invertible, allowing recovery of the original signal. The STFT Synchrosqueezing transform on finite dimensional signals has the advantage of an efficient matrix representation. This article defines the finite STFT Synchrosqueezing transform and describes some properties of this transform. We compare the finite STFT and the finite STFT Synchrosqueezing transform by applying these transform to a set of signals.

math.NA↗

arXiv2017

Gabor Tight Fusion Frames: Construction and Applications in Signal Retrieval Modulo Phase

Hilbert space fusion frames are a natural extension of Hilbert space frames, extending the notion from a set of vectors in a Hilbert space to a set of subspaces of a Hilbert space with analogous notions of overcompleteness and boundedness. As tight frames are a very important topic within standard frame theory, tight fusion frames are similarly important; however, only trivial examples of tight fusion frames are hitherto known. In this paper, we apply ideas from Gabor analysis to demonstrate a non-trivial construction of tight fusion frames. We then use this construction to further show their applicability in some cases for the retrieval of signals modulo phase.

math.FA↗