arXiv subjects

Keith F. Taylor

Publications and source records attributed to Keith F. Taylor.

5 recordsLinked to original sources

arXiv2022

Harmonic Analysis on the Affine Group of the Plane

For any natural number $n$, the group $G_n$ of all invertible affine transformations of $n$-dimensional Euclidean space has, up to equivalence, just one square-integrable representation and the left regular representation of $G_n$ is a multiple of this square-integrable representation. We provide a concrete realization $σ_2$ of this square-integrable representation of $G_2$ acting on the Hilbert space $L^2\big(\widehat{\mathbb{R}^2}\times\widehat{\mathbb{R}}\big)$. We explicitly decompose the Hilbert space $L^2(G_2)$ as a direct sum of left invariant closed subspaces on each of which the left regular representation acts as a representation equivalent to $σ_2$.

math.RT↗

arXiv2022

A Two Plus One Dimensional Continuous Wavelet Transform

The group $G_2$ of invertible affine transformations of $\mathbb{R}^2$ has, up to equivalence, one square--integrable representation. Two new realizations of this representation are presented and novel continuous wavelet transforms, acting on functions of two plus one variables, are derived.

math.FA↗

arXiv2020

Decomposing the wavelet representation for shifts by wallpaper groups

The wavelet group and wavelet representation associated with shifts coming from a two dimensional crystal symmetry group $Γ$ and dilations by powers of 3, are defined and studied. The main result is an explicit decomposition of the $3Γ-$wavelet representation into irreducible representations of the wavelet group.

math.FA↗

arXiv2015

Projections in $L^1(G)$; the unimodular case

We consider the issue of describing all self-adjoint idempotents (projections) in $L^1(G)$ when $G$ is a unimodular locally compact group. The approach is to take advantage of known facts concerning subspaces of the Fourier-Stieltjes and Fourier algebras of $G$ and the topology of the dual space of $G$. We obtain an explicit description of any projection in $L^1(G)$ which happens to also lie in the coefficient space of a finite direct sum of irreducible representations. This leads to a complete description of all projections in $L^1(G)$ for $G$ belonging to a class of groups that includes $SL(2,R)$ and all almost connected nilpotent locally compact groups.

math.RT↗

arXiv2001

A Direct Integral Decomposition of the Wavelet Representation

In this paper we use the concept of wavelet sets as introduced by X. Dai and D. Larson, to decompose the wavelet representation of the discrete group associated to an arbitrary $n \times n$ integer dilation matrix as a direct integral of irreducible monomial representations. In so doing we generalize a result of F. Martin and A. Valette in which they show that the wavelet representation is weakly equivalent to the regular representation for the Baumslag-Solitar groups.

math.FA↗