Reproducing Polynomials with B-Splines: Unterschied zwischen den Versionen

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In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order <math>N</math>. This is needed for certain applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework where where any kernel <math>\varphi</math> reproducing polynomials, that is, satisfying the Strang-Fix conditions, can be used. However, among all possible kernels, the B-Splines have the smallest possible support.
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In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order <math>N</math>. This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel <math>\varphi</math> reproducing polynomials (that is, satisfying the Strang-Fix conditions) can be used. However, among all possible kernels, the B-Splines have the smallest possible support.
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An important question is how to obtain the coefficients <math>c_{m,n}</math> for the reproduction-formula. In this small article, I describe one way.

Version vom 19. Juli 2010, 14:06 Uhr

A B-Spline of order is known to be able to reproduce any polynomial up to order :

In words, a proper linear combination of shifted versions of a B-Spline can reproduce any polynomial up to order . This is needed for different applications, for example, for the Sampling at Finite Rate of Innovation (FRI) framework. In this case any kernel reproducing polynomials (that is, satisfying the Strang-Fix conditions) can be used. However, among all possible kernels, the B-Splines have the smallest possible support.

An important question is how to obtain the coefficients for the reproduction-formula. In this small article, I describe one way.