De Boor-Fix dual functionals for transformation from polynomial basis to convolution basis

Abstract

Basis transformation plays a central role in data representations and in many problem solving techniques such as Fourier transform and wavelet transform. A pertinent basis gives pertinent information. Dual functionals are a mathematical tool for basis transformation. This work presents, from the perspective of algebraic computing, a generalization of the de Boor-Fix dual functionals for the Bernstein basis functions to the case of the convolution basis functions. The convolution basis functions can be characterized as polar forms of the corresponding Bernstein basis functions. These new dual functionals can be used for computing algebraically the expansion coefficients of any polynomial over the convolution basis which has applications in computer-aided geometric design and secret information encoding.