Rigidity of causal maps
We show that order-invariant injective maps on the noncompactly causal symmetric space $SO_0 (1,n)/SO_0 (1,n-1)$ belong to $O(1,n)^+$.
math.DG↗
arXiv subjects
Nils Byrial Andersen
Publications and source records attributed to Nils Byrial Andersen.
A connection between the Uncertainty Principles on the real line and on the circle
The purpose of this short note is to exhibit a new connection between the Heisenberg Uncertainty Principle on the line and the Breitenberger Uncertainty Principle on the circle, by considering the commutator of the multiplication and difference operators on Bernstein functions
math.FA↗
Asymptotics for the Radon transform on hyperbolic spaces
Let G/H be a hyperbolic space over R C or H, and let K be a maximal compact subgroup of G. Let D denote a certain explicit invariant differential operator, such that the non-cuspidal discrete series belong to the kernel of D. For any L^2-Schwartz function f on G/H, we prove that the Abel transform A(Df) of Df is a Schwartz function. This is an extension of a result established in [2] for K-finite and K\cap H-invariant functions.
math.RT↗
Cuspidal discrete series for projective hyperbolic spaces
We have in [1] proposed a definition of cusp forms on semisimple symmetric spaces $G/H$, involving the notion of a Radon transform and a related Abel transform. For the real non-Riemannian hyperbolic spaces, we showed that there exists an infinite number of cuspidal discrete series, and at most finitely many non-cuspidal discrete series, including in particular the spherical discrete series. For the projective spaces, the spherical discrete series are the only non-cuspidal discrete series. Below, we extend these results to the other hyperbolic spaces, and we also study the question of when the Abel transform of a Schwartz function is again a Schwartz function.
math.RT↗
Cuspidal discrete series for semisimple symmetric spaces
We propose a notion of cusp forms on semisimple symmetric spaces. We then study the real hyperbolic spaces in detail, and show that there exists both cuspidal and non-cuspidal discrete series. In particular, we show that all the spherical discrete series are non-cuspidal.
math.RT↗
On the Fourier transform of Schwartz functions on Riemannian Symmetric Spaces
Consider the (Helgason-) Fourier transform on a Riemannian symmetric space G/K. We give a simple proof of the L^p-Schwartz space isomorphism theorem (0 <p \le 2) for K-finite functions. The proof is a generalization of J.-Ph. Anker's proof for K-invariant functions.
math.RT↗
Local spectral radius formulas for a class of unbounded operators on Banach spaces
We exhibit a general class of unbounded operators in Banach spaces which can be shown to have the single-valued extension property, and for which the local spectrum at suitable points can be determined. We show that a local spectral radius formula holds, analogous to that for a globally defined bounded operator on a Banach space with the single-valued extension property. An operator of the class under consideration can occur in practice as (an extension of) a differential operator which, roughly speaking, can be diagonalised on its domain of smooth test functions via a discrete transform, such that the diagonalising transform establishes an isomorphism of topological vector spaces between the domain of the differential operator, in its own topology, and a sequence space. We give concrete examples of (extensions of) such operators (constant coefficient differential operators on the d-torus, Jacobi operators, the Hermite operator, Laguerre operators) and indicate further perspectives.
math.SP↗
Local spectral radius formulas on compact Lie groups
We determine the local spectrum of a central element of the complexified universal enveloping algebra of a compact connected Lie group at a smooth function as an element of L^p(G). Based on this result we establish a corresponding local spectral radius formula.
math.RT↗
Real Paley-Wiener theorems and local spectral radius formulas
We systematically develop real Paley-Wiener theory for the Fourier transform on R^d for Schwartz functions, L^p-functions and distributions, in an elementary treatment based on the inversion theorem. As an application, we show how versions of classical Paley-Wiener theorems can be derived from the real ones via an approach which does not involve domain shifting and which may be put to good use for other transforms of Fourier type as well. An explanation is also given why the easily applied classical Paley-Wiener theorems are unlikely to be able to yield information about the support of a function or distribution which is more precise than giving its convex hull, whereas real Paley-Wiener theorems can be used to reconstruct the support precisely, albeit at the cost of combinatorial complexity. We indicate a possible application of real Paley-Wiener theory to partial differential equations in this vein and furthermore we give evidence that a number of real Paley-Wiener results can be expected to have an interpretation as local spectral radius formulas. A comprehensive overview of the literature on real Paley-Wiener theory is included.
math.FA↗
Elementary proofs of Paley-Wiener theorems for the Dunkl transform on the real line
We give an elementary proof of the Paley-Wiener theorem for smooth functions for the Dunkl transforms on the real line, establish a similar theorem for L^2-functions and prove identities in the spirit of Bang for L^p-functions. The proofs seem to be new also in the special case of the Fourier transform.
math.CA↗