Quantitative Approximation Theory 
Investigators: Thierry Blu, Michael Unser 

Summary: We have investigated the behavior of the approximation error of a signal in a splinelike 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. 

When approximating a signal in a splinelike 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. 

We made some refinements to the StrangFix 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 piecewisesmooth 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. 




