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B-splines

Module by: Alena Scott

Summary: This module explains B-splines, which form a basis for splines.

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B-splines are symmetrical, bell-shaped functions constructed from the ( n+1 n 1 ) fold convolution of a rectangular pulse β 0 x β 0 x : β 0 x=1if|x|<0.50.5if|x|=0.50otherwise β 0 x 1 x 0.5 0.5 x 0.5 0 β n x= β + , 1 0 * β + , 2 0 *…* β + , n 0 * β + , n + 1 0 x β n x β + , 1 0 β + , 2 0 … β + , n 0 β + , n + 1 0 x

Theorem 1: Schoenberg

Polynomial splines with equally spaced knots can be uniquely characterized in terms of a B-spline expansion.

Example

That is, for a spline sx s x of degree nn, there exist unique coefficients ck c k such that sx=∑k=-∞∞ck β n x−k=c* β n x s x k c k β n x k c β n x

Understanding B-splines can help us better understand splines, since B-splines are the building blocks.

Some properties of B-splines

  1. The derivative of a B-spline of order nn with respect to xx is equal to the sum of two B-splines of order n−1 n 1 evaluated at shifts of xx ddx β n x= β n - 1 x+0.5− β n - 1 x−0.5 x β n x β n - 1 x 0.5 β n - 1 x 0.5
  2. The integral of a B-spline of order nn is equal to the infinite sum of B-splines of order n+1 n 1 . ∫-∞x β n tdt=∑k=0∞ β n + 1 x−0.5−k t x β n t k 0 β n + 1 x 0.5 k
  3. Notice that this Fourier transform is related to the n+1 n 1 fold convolution construction of the B-splines. β ^ n ω=sinω2ω2n+1 β ^ n ω ω 2 ω 2 n 1
  4. B-splines are compactly supported.
  5. They are the shortest possible polynomial splines.
  6. β n x β n x is a piecewise polynomial of degree nn: β n x=1n!∑k=0n+1n+1k-1kx−k+n+12n β n x 1 n k 0 n 1 n 1 k 1 k x k n 1 2 n where x + n=xnifx≥00ifx<0 x + n x n x 0 0 x 0

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