Narrow Results

Based on our previous paper [H.-Y. Fan and H.-L. Lu, Int. J. Mod. Phys.19, 799 (2005)] and the two mutually-conjugate entangled state representations, we derive the convolution theorem of complex fractional Fourier transformation in the context of quantum mechanics. This approach seems convenient and neat.

The fractional Fourier transform (FRFT) is a generalized form of the Fourier transform (FT), it is another important class of time–frequency analysis tool in signal processing. In this paper, we study the two-dimensional (2D) FRFT in the polar coordinates setting. First, Parseval theorem of the 2D FRFT in the polar coordinates is obtained. Then, according to the relationship between 2D FRFT and fractional Hankel transform (FRHT), the convolution theorem for the 2D FRFT in polar coordinates is obtained. It shows that the FRFT of the convolution of two functions is the product of their respective FRFTs. Moreover, the fast algorithm for the convolution theorem of the 2D FRFT is discussed. Finally, the sampling theorem for signal is explored.

In this paper, the discrete octonion linear canonical transform (DOCLCT) is presented. According to the definition of DOCLCT, its shift transform and modulation transform are explored. In addition, based on a new convolution operator, we obtain the convolution theorem of DOCLCT. The DOCLCT can be derived by the three-dimensional (3D) linear canonical transform. So, the fast algorithm of 3D DOCLCT is discussed. Finally, Heisenberg’s uncertainty principle associated with DOCLCT is established.