Gyroharmonic analysis on relativistic gyrogroups

Einstein, Möbius, and Proper Velocity gyrogroups are relativistic gyrogroups that appear as three different realizations of the proper Lorentz group in the real Minkowski space-time \bkR^{n,1}. Using the gyrolanguage we study their gyroharmonic analysis. Although there is an algebraic gyroisomorphism between the three models we show that there are some differences between them. Our study focus on the translation and convolution operators, eigenfunctions of the Laplace-Beltrami operator, Poisson transform, Fourier-Helgason transform, its inverse, and Plancherel's Theorem. We show that in the limit of large $t,$ $t \rightarrow +\infty,$ the resulting gyroharmonic analysis tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis

Harmonic analysis on the Möbius gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball $B_t^n$, a model of the n-dimensional real hyperbolic space. The Poincaré ball $B_t^n$ is the open ball of the Euclidean n-space $R^n$ with radius $t>0$, centered at the origin of $R^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb{R}^n$. For any $t>0$ and an arbitrary parameter $\sigma \in R$ we study the $(\sigma,t)$-translation, the $( \sigma,t)$-convolution, the eigenfunctions of the $(\sigma,t)$-Laplace-Beltrami operator, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and the associated Plancherel's Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on $B_t^n$ tends to the standard Euclidean harmonic analysis on $R^n$, thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on $B_t^n$

Number fields with prescribed norms

We study the distribution of extensions of a number field $k$ with fixed abelian Galois group $G$, from which a given finite set of elements of $k$ are norms. In particular, we show the existence of such extensions. Along the way, we show that the Hasse norm principle holds for $100\%$ of $G$-extensions of $k$, when ordered by conductor. The appendix contains an alternative purely geometric proof of our existence result.Comment: 35 pages, comments welcome

Harmonic analysis on the Einstein gyrogroup

In this paper we study harmonic analysis on the Einstein gyrogroup of the open ball of R$^n$, $n \in N,$ centered at the origin and with arbitrary radius $t \in R^+,$ associated to the generalised Laplace-Beltrami operator L_{\sigma,t} = \disp \left( 1 - \frac{\|x\|^2}{t^2} \right) \!\left( \Delta - \sum_{i,j=1}^n \frac{x_i x_j}{t^2} \frac{\partial^2}{\partial x_i \partial x_j} - \frac{\kappa}{t^2} \sum_{i=1}^n x_i \frac{\partial}{\partial x_i} + \frac{\kappa(2-\kappa)}{4t^2} \right) where $\kappa=n+\sigma$ and $\sigma \in {\mathbb R}$ is an arbitrary parameter. The generalised harmonic analysis for $L_{\sigma,t}$ gives rise to the $(\sigma,t)$-translation, the $(\sigma,t)$-convolution, the $(\sigma,t)$-spherical Fourier transform, the $(\sigma,t)$-Poisson transform, the $(\sigma,t)$-Helgason Fourier transform, its inverse transform and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the resulting hyperbolic harmonic analysis tends to the standard Euclidean harmonic analysis on $R^n,$ thus unifying hyperbolic and Euclidean harmonic analysis

Harmonic analysis on the proper velocity gyrogroup

Revista sem período de embargo.In this paper we study harmonic analysis on the Proper Velocity (PV) gyrogroup using the gyrolanguage of analytic hyperbolic geometry. PV addition is the relativistic addition of proper velocities in special relativity and it is related with the hyperboloid model of hyperbolic geometry. The generalized harmonic analysis depends on a complex parameter $z$ and on the radius $t$ of the hyperboloid and comprises the study of the generalized translation operator, the associated convolution operator, the generalized Laplace-Beltrami operator and its eigenfunctions, the generalized Poisson transform and its inverse, the generalized Helgason-Fourier transform, its inverse and Plancherel's Theorem. In the limit of large $t,$ $t \rightarrow +\infty,$ the generalized harmonic analysis on the hyperboloid tends to the standard Euclidean harmonic analysis on ${\mathbb R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis

The Hasse norm principle for abelian extensions

We study the distribution of abelian extensions of bounded discriminant of a number field k which fail the Hasse norm principle. For example, we classify those finite abelian groups G for which a positive proportion of G-extensions of k fail the Hasse norm principle. We obtain a similar classification for the failure of weak approximation for the associated norm one tori. These results involve counting abelian extensions of bounded discriminant with infinitely many local conditions imposed, which we achieve using tools from harmonic analysis, building on work of Wright