Approximation Order: Why the Asymptotic Constant Matters

T. Blu, M. Sühling, P. Thévenaz, M. Unser

Proceedings of the Second Pacific Rim Conference on Mathematics, Taipei, Taiwan (People's Republic of China), January 4-8, 2001, pp. II.3–II.4.

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We consider the approximation (either interpolation, or least-squares) of L² functions in the shift-invariant space V_T = span_k∈Z { φ(t ⁄ T − k) } that is generated by the single shifted function φ. We measure the approximation error in an L² sense and evaluate the asymptotic equivalent of this error as the sampling step T tends to zero. Let ƒ ∈ L² and ƒ_T be its approximation in V_T. It is well-known that, if φ satisfies the Strang-Fix conditions of order L, and under mild technical constraints, || ƒ − ƒ_T || _L2 = O(T^L).

In this presentation however, we want to be more accurate and concentrate on the constant C_φ which is such that || ƒ − ƒ_T || _L2 = C_φ || ƒ^(L) || _L2 T^L + o(T^L).

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