Symmetry Feature and Construction for the 3-Band Tight Framelets with Prescribed Properties

A construction approach for the 3-band tight wavelet frames by factorization of paraunitary matrix is developed. Several necessary constraints on the filter lengths and symmetric features of wavelet frames are investigated starting at the constructed paraunitary matrix. The matrix is a symmetric extension of the polyphase matrix corresponding to 3-band tight wavelet frames. Further, the parameterizations of 3-band tight wavelet frames with 3N+1 filter lengths are established. Examples of framelets with symmetry/antisymmetry and Sobolev exponent are computed by appropriately choosing the parameters in the scheme