Affine dual frames and Extension Principles

In this work we provide three new characterizations of affine dual frames constructed from refinable functions. The first one is similar to Daubechies et al. (2003) [10, Proposition 5.2], but without any decay assumptions on the generators of a pair of affine systems. The second one reveals the geometric significance of the Mixed Fundamental function and the third one shows that the Mixed Oblique Extension Principle actually characterizes dual framelets. We also extend recent results on the characterization of affine Parseval frames obtained in Stavropoulos (2012) [27, Theorem 2.3].