Orthonormality of wavelet system on the Heisenberg group and twisted wavelet system on $\mathbb{C}$

The aim of this paper is to obtain necessary and sufficient conditions for the orthonormality of wavelet system arising out of left translations and nonisotropic dilations on the Heisenberg group $\mathbb{H}$. A similar problem is also discussed for the twisted wavelet system on $\mathbb{C}$.