Squeezable Orthogonal Bases: Accuracy and Smoothness

We present a method for generating local orthogonal bases on arbitrary partitions of R from a given local orthogonal shift-invariant basis via what we call a squeeze map. We give necessary and sufficient conditions for a squeeze map to generate a nonuniform basis that preserves any smoothness and/or accuracy (polynomial reproduction) of the shift-invariant basis. When the shift-invariant basis has sufficient smoothness or accuracy, there is a unique squeeze map associated with a given partition that preserves this property and, in this case, the squeeze map may be calculated locally in terms of the ratios of adjacent intervals. If both the smoothness and accuracy are large enough, then the resulting nonuniform space contains the nonuniform spline space characterized by that smoothness and accuracy. Our examples include a multiresolution on nonuniform partitions such that each space has a local orthogonal basis consisting of continuous piecewise quadratic functions. We also construct a family of smooth, local, orthogonal, piecewise polynomial generators with arbitrary approximation order.