Reproducing Polynomials with B-Splines: Unterschied zwischen den Versionen

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A B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math><ref>I.J. Schoenberg: Cardinal interpolation and spline functions</ref>:
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A B-Spline of order <math>N</math> is known to be able to reproduce any polynomial up to order <math>N</math><ref>I.J. Schoenberg: "Cardinal interpolation and spline functions", ''J. Approx. Theory volume 2'', pp. 167-206, 1969</ref>:
  
 
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Version vom 19. Juli 2010, 14:14 Uhr

A B-Spline of order is known to be able to reproduce any polynomial up to order [1]:

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.

  1. I.J. Schoenberg: "Cardinal interpolation and spline functions", J. Approx. Theory volume 2, pp. 167-206, 1969