Hermite Interpolation in Loop Groups and Conjugate Quadrature Filter Approximation

Abstract A classical result of Weierstrass ensures that any continuous finite length trajectory in a vector space can be uniformly approximated by one whose coordinates are trigonometric functions. We derive an analogous result for trajectories in spheres and apply it to show that a continuous frequency response of a conjugate quadrature filter can be uniformly approximated by the frequency response of a finitely supported conjugate quadrature filter. We also extend this result, so as to preserve specified roots of the frequency response, and derive an approximation result for refinable functions whose integer translates are orthonormal. Our methods utilize properties of loop groups, jets, and the Brouwer topological degree.