The M-band cardinal orthogonal scaling function

The M-band symmetric cardinal orthogonal scaling function with compact support is of interest in several applications such as sampling theory, signal processing, computer graphics, and numerical algorithms. In this paper, we provide a complete mathematical analysis for the M  -band symmetric cardinal orthogonal scaling function. Firstly, we generalize some results of the cardinal orthogonal scaling function from the special case M=2M=2 to the most general case M⩾2M⩾2. Also, we find some new results. Secondly, we obtain the characterizations of the M-band symmetric cardinal orthogonal scaling function and revisit some known examples to prove our theory.