Localization of Dual Periodic Scaling and Wavelet Functions

In 1996, we constructed periodic interpolatory scaling functions φj , wavelet functions Lj and their dual basis φ̃j and L̃j with properties such as symmetry, biorthogonality, any order of smoothness, real-valuedness, explicit expressions and interpolatory. We proved the localization of φj in 1997, and in 1998 with Li proved the localization of Lj . In this paper we shall give a detailed proof of the localization for the dual functions φ̃j and L̃j .