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Title Linear Combinations of B-Splines as Generating Functions for Signal Approximation
Author Manuel J. C. S. Reis, Paulo J S G Ferreira, Salviano F. S. P. Soares
Journal Digital Signal Processing
Volume 15
Number 3
Pages 226-236
Month May
Year 2005
DOI [1]
Group (before 2015) Signal Processing Laboratory
Indexed by ISI Yes


The advantages of B-splines for signal representation are well known. This paper stresses a fact that seems to be less well known, namely, the possibility of using linear combinations of B-splines to obtain representations that are more stable than the usual ones. We give the best possible Riesz bounds for these linear combinations and calculate their duals, in a generalized sampling context.

To put this into perspective, consider an approximation scheme based on a certain kernel function. Often, it would lead to a Riesz sequence or a Riesz basis. Riesz bases are a generalization of orthonormal bases, and can be regarded as the result of applying a bounded invertible operator to the elements of an orthonormal basis. A Riesz sequence, on the other hand is an incomplete basis, as it is merely a Riesz basis for the closed linear span of the set of functions under consideration.

The numerical stability of such representations is important in practice, as it dictates the magnitude of the adverse effects due to noisy data. The stability of Riesz bases and Riesz sequences can be measured by looking at the size of their Riesz bounds. Orthonormal bases are perfect from the viewpoint of numerical stability, and their Riesz bounds are both equal to unity. As the ratio of the upper to the lower bound increases, the numerical stability of the representation decreases. It is well known that the tighter (closer to each other) these bounds are the less any small perturbations in the input data will be felt at the output. This paper points out that the Riesz bounds associated with bases built using certain linear combinations of B-splines are better from the stability point of view than bases directly based on B-splines.