B-splines on the Heisenberg group

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 $L_{(-n\mathrm{.}-\frac{n}{2},-{\alpha }_n)}{\varphi }_n(x,y,t)=L_{(-n\mathrm{.}-\frac{n}{2},-{\alpha }_n)}{\varphi }_n(-x,-y,-t)$, for $n\ge 2$, 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.