Construction of smooth compactly supported windows generating dual pairs of gabor frames

摘要

Let g be any real-valued, bounded and compactly supported function, whose integer-translates {T_kg}_(k∈□) form a partition of unity. Based on a new construction of dual windows associated with Gabor frames generated by g, we present a method to explicitly construct dual pairs of Gabor frames. This new method of construction is based on a family of polynomials which is closely related to the Daubechies polynomials, used in the construction of compactly supported wavelets. For any k ∈ □ ∪ {∞} we consider the Meyer scaling functions and use these to construct compactly supported windows g ∈ C~k(□) associated with a family of smooth compactly supported dual windows {h_n}_(n=1) ∞. For any n ∈ □ the pair of dual windows g, h n ∈ C~k(□) have compact support in the interval [-2/3, 2/3] and share the property of being constant on half the length of their support. We therefore obtain arbitrary smoothness of the dual pair of windows g, h_n without increasing their support.