Hermite Interpolation in Loop Groups and Conjugate Quadrature Filter Approximation
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Hermite Interpolation in Loop Groups and Conjugate Quadrature Filter Approximation WAYNE M. LAWTON Department of Mathematics, National University of Singapore, 2 Science Drive 2, Singapore 117543, Republic of Singapore (Received: 29 September 2004) 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. Mathematics Subject Classifications (2000): 22E67(Primary), 41A29(Secondary), 42A10, 42A11, 42C40, 47H10, 47H11. Key words: loop groups, trigonometric, approximation, conjugate quadrature filter, jets, Brouwer topological degree, wavelets.
1. Introduction This paper addresses constrained Fourier approximation problems in which the range of functions is constrained to lie in a Lie group or homogeneous space (e.g., a sphere). Classical methods for approximating functions by functions that depend on a finite number of parameters include Taylor series, Fourier series, and more recently wavelet expansions that provide a multiscale representation of functions analogous to positional notation that was invented by Babylonian mathematicians four thousand years ago to represent numbers [44]. These methods assume that both the functions to be approximated and their approximants are real-valued or complex-valued, or possibly take values in a convex subset of a real or complex vector space. This assumption is critical in classical approximation because the set of approximants should be closed under linear combinations or convex combinations. Weierstrass’s proof that any continuous real-valued function on a compact subset of the real numbers can be uniformly approximated by a polynomial function uses four facts: the function can be extended to a continuous function on the real numbers having compact support, the extended function can be
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uniformly approximated by its convolution with a Gaussian function, the Gaussian function is analytic, hence it can be uniformly approximated by polynomials over compact subsets and the convolution of every function with a polynomial is a polynomial. Clearly this result extends to functions on a compact subset of real numbers having values in a convex subset of a finite-dimensional real or complex vector space. However, it does not extend to functions having values in a sphere, because every polynomial function with values in a sphere is constant. Furthermore, every trigonometric polynomial having values
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