Extension principles for tight wavelet frames of periodic functions

A unitary extension principle for constructing normalized tight wavelet frames of periodic functions of one or higher dimensions is established. While the wavelets are nonstationary, the method much simplifies their construction by reducing it to a matrix extension problem that involves finite rows of complex numbers. Further flexibility is achieved by reformulating the result as an oblique extension principle. In addition, with a constructive proof, necessary and sufficient conditions for a solution of the matrix extension problem are obtained. A complete characterization of all possible solutions is also provided. As illustration, parametric families of trigonometric polynomial tight wavelet frames are constructed.