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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.



We consider the approximation (either interpolation, or least-squares) of L2 functions in the shift-invariant space VT = spankZ { φ(tTk) } 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 ƒT be its approximation in VT. It is well-known that, if φ satisfies the Strang-Fix conditions of order L, and under mild technical constraints, || ƒ − ƒT || L2 = O(TL).

In this presentation however, we want to be more accurate and concentrate on the constant Cφ which is such that || ƒ − ƒT || L2 = Cφ || ƒ(L) || L2 TL + o(TL).


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