Orthogonal M-band compactly supported scaling functions with sampling property

In general case, the interpolant in the Waiter's wavelet sampling theorem is not necessarily compactly supported. Requiring that it is compactly supported is equivalent to requiring that the corresponding scaling function has the sampling property. Our focus in this paper is on considering the case where the scaling function is not only compactly supported, but also orthogonal and of the sampling property. This paper makes a parameterization of two regular unitary M-band sampling scaling filters of the length 3M, constructs a 3-band sampling scaling function and show that it is not only compactly supported, but also orthogonal and continuous. However, in a-band case, there is no such scaling function except Haar scaling function. G.Walter's sampling theorem for wavelet subspaces corresponding to this scaling function has the interpolant with compact support. Therefore, the signals in multiresolution subspaces can be reconstructed exactly and fast without truncated errors.