B-splines on the Heisenberg group

TL;DR

This paper introduces B-splines on the Heisenberg group, analyzing their properties, translation systems, and duals, revealing unique non-symmetries and establishing conditions for frames and Riesz sequences.

Contribution

It develops the theory of B-splines on the Heisenberg group, including their fundamental properties, translation systems, and duality conditions, which are novel extensions of classical spline theory.

Findings

No symmetric sequence of left translates exists for n ≥ 2.

Conditions for translation systems to form frames or Riesz sequences are established.

A sufficient condition for the existence of oblique duals of translation systems is provided.

Abstract

In this paper, we introduce a class of $B$-splines on the Heisenberg group $\mathbb{H}$ and study their fundamental properties. Unlike the classical case, we prove that there does not exist any sequence $\{\alpha_n\}_{n\in\mathbb{N}}$ such that $L_{(-n.-\frac{n}{2},-\alpha_n)}\phi_n(x,y,t)=L_{(-n.-\frac{n}{2},-\alpha_n)}\phi_n(-x,-y,-t)$, for $n\geq 2$, where $L_{(x,y,t)}$ denotes the left translation on $\mathbb{H}$. We further investigate the problem of finding an equivalent condition for the system of left translates to form a frame sequence or a Riesz sequence in terms of twisted translates. We also find a sufficient condition for obtaining an oblique dual of the system $\{L_{(2k,l,m)}g:k,l,m\in\mathbb{Z}\}$ for a certain class of functions $g\in L^2(\mathbb{H})$. These concepts are illustrated by some examples. Finally, we make some remarks about $B$-splines regarding these results.