B-splines on the Heisenberg Group

Abstract The aim of this paper is to introduce a class of B -splines $${\phi }_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> on the Heisenberg group $${\mathbb {H}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> and prove certain fundamental properties. The non-commutative group structure of $${\mathbb {H}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>H</mml:mi> </mml:math> leads to very different results for $${\phi }_n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>ϕ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:math> from those of the classical B -splines on the real line $${\mathbb {R}}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>R</mml:mi> </mml:math> .