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Wavelets, Splines, and the Reproduction of Polynomials

Module by: Rebecca Willett. E-mail the author

Summary: On well-known property of valid scaling functions in wavelet analysis is that the scaling functions can perfectly reproduce a polynomial of degree no more than the wavelet smoothness. This property can be easily derived using the Unser-Blu Scaling Function / Spline Factorization Theorem.

Recall from spline theory that fractional B-splines reproduce polynomials of order less than or equal to the ceiling of the spline order. This means that any polynomial of a certain order can be expressed as a linear combination of B-splines; that is, the B-splines form a basis for polynomials. Specifically, for some pp which is a nonnegative integer no greater than α+1α1, we have

∑kk∈Zkp⁢ β + α ⁢x−k=xp+ a p , 1 ⁢xp−1+…+ a p , p - 1 ⁢x+ a p , p k k k p β + α x k x p a p , 1 x p 1 … a p , p - 1 x a p , p (1)
The collection of all these polynomials for all pp no greater than α+1α1 forms a basis for polynomials of order no greater than α+1α1.

This combined with the Unser-Blu Scaling Function/Spline Factorization Theorem leads to a straighforward proof of the fact that scaling functions reproduce polynomials up to a degree proportional to their smoothness.

Proposition 1

Let φ 0 φ 0 be any distribution such that ∫ φ 0 ⁢xdx=1 x φ 0 x 1 adn ∫xi⁢ φ 0 ⁢xdx<∞ x x i φ 0 x , for all i∈1…n i 1 … n , where n=⌈α⌉ n α . Then φ⁢x= β + α * φ 0 ⁢x φ x β + α φ 0 x reproduces polynomials of degree lesser or equal to n=⌈α⌉ n α .

Proof

  1. First note that
    ∑kk∈Zkp⁢φ⁢x−k=∑kk∈Zkp⁢ β + γ - 1 * φ 0 ⁢(x−k) k k k p φ x k k k k p β + γ - 1 φ 0 x k (2)
    ∑kk∈Zkp⁢φ⁢x−k= φ 0 *∑k=0p a p , k ⁢xp−k k k k p φ x k φ 0 k 0 p a p , k x p k (3)
  2. φ 0 ⁢x*xp=xp+ b 1 ⁢xp−1+…+ b p φ 0 x x p x p b 1 x p 1 … b p

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