Tight periodic wavelet frames and approximation orders

A systematic study on tight periodic wavelet frames and their approximation orders is conducted. We identify a necessary and sufficient condition, in terms of refinement masks, for applying the unitary extension principle for periodic functions to construct tight wavelet frames. Then a theory on the approximation orders of truncated tight frame series is established, which facilitates the construction of tight periodic wavelet frames with desirable approximation orders. Finally, a notion of vanishing moments for periodic wavelets, which is missing in the current literature, is introduced and related to frame approximation orders and sparse representations of locally smooth functions. As illustrations, the results are applied to two classes of examples: one is band-limited and the other is time-localized.