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Quantitative Approximation Theory

Investigators: Thierry Blu, Michael Unser

Summary: We have investigated the behavior of the approximation error of a signal in a spline-like basis as a function of the sampling step. We have proposed a general Fourier analysis method that yields quantitative error estimates as well as a whole series of bounds and asymptotic formulæ.

Introduction

When approximating a signal in a spline-like basis, the approximation error typically decreases and eventually vanishes as the sampling step T gets smaller. A standard result from approximation theory is that this error qualtitatively decays like the Nth power of T, where N is the order of approximation of the representation. In signal processing applications, it is important to have some more precise error estimates to select an appropriate sampling step and also in order to optimize the approximation algorithm.

Main Contributions

We made some refinements to the Strang-Fix theory of approximation and derived simple quantitative error bounds that are directly applicable to the selection of the appropriate sampling step and of the approximation procedure (interpolation or projection). In particular, we showed that splines are generally superior to other wavelet representations for approximating piecewise-smooth functions.

We have obtained exact asymptotic error formulæ that become valid as soon as the sampling step gets sufficiently small when compared to the smoothness scale of the signal.

Period: 1997-2002

Major Publications

[1] 

T. Blu, M. Unser, "Approximation Error for Quasi-Interpolators and (Multi-) Wavelet Expansions," Applied and Computational Harmonic Analysis, vol. 6, no. 2, pp. 219-251, March 1999.

[2] 

T. Blu, M. Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part I—Interpolators and Projectors," IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2783-2795, October 1999.

[3] 

T. Blu, M. Unser, "Quantitative Fourier Analysis of Approximation Techniques: Part II—Wavelets," IEEE Transactions on Signal Processing, vol. 47, no. 10, pp. 2796-2806, October 1999.

[4] 

M. Jacob, T. Blu, M. Unser, "Sampling of Periodic Signals: A Quantitative Error Analysis," IEEE Transactions on Signal Processing, vol. 50, no. 5, pp. 1153-1159, May 2002.

[5] 

M. Unser, I. Daubechies, "On the Approximation Power of Convolution-Based Least Squares Versus Interpolation," IEEE Transactions on Signal Processing, vol. 45, no. 7, pp. 1697-1711, July 1997.

[6] 

M. Unser, "Sampling—50 Years After Shannon," Proceedings of the IEEE, vol. 88, no. 4, pp. 569-587, April 2000.

[7] 

M. Unser, "Approximation Power of Biorthogonal Wavelet Expansions," IEEE Transactions on Signal Processing, vol. 44, no. 3, pp. 519-527, March 1996.

Related Topics

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