In this paper, we establish two approximation theorems for the multidimensional fractional Fourier transform via appropriate convolutions. As applications, we study the boundary and initial problems of the Laplace and heat equations with chirp functions. Furthermore, we obtain the general Heisenberg inequality with respect to the multidimensional fractional Fourier transform.

In the past 20 years, the connected dominating set (CDS) as a virtual backbone network has been widely used in the wireless networks. Many researchers have been devoted to designing approximate algorithms for CDS problem since constructing the minimum CDS (MCDS) is NP-hard problem. Different from the most existing algorithms with two phases, we employ greedy strategy to design a centralized algorithm GR_CDS in only one phase to get MCDS, with the time complexity of $O(({\mathrm{\Delta }}^{\mathrm{2}}+\log n)n)$ . Afterwards, another algorithm P_CDS is designed for pruning redundant nodes in the obtained MCDS with the time complexity of $O(n^{\mathrm{2}})$ .