# Connexions

You are here: Home » Content » Wavelets, Splines, and the Reproduction of Polynomials

### Recently Viewed

This feature requires Javascript to be enabled.

# 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

kkZkp β + α xk=xp+ a p , 1 xp1++ 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 i1n i 1 n , where n=α n α . Then φx= 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
kkZkpφxk=kkZkp β + γ - 1 * φ 0 (xk) k k k p φ x k k k k p β + γ - 1 φ 0 x k
(2)
kkZkpφxk= φ 0 *k=0p a p , k xpk 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 xp1++ b p φ 0 x x p x p b 1 x p 1 b p

## Content actions

PDF | EPUB (?)

### What is an EPUB file?

EPUB is an electronic book format that can be read on a variety of mobile devices.

My Favorites (?)

'My Favorites' is a special kind of lens which you can use to bookmark modules and collections. 'My Favorites' can only be seen by you, and collections saved in 'My Favorites' can remember the last module you were on. You need an account to use 'My Favorites'.

| A lens I own (?)

#### Definition of a lens

##### Lenses

A lens is a custom view of the content in the repository. You can think of it as a fancy kind of list that will let you see content through the eyes of organizations and people you trust.

##### What is in a lens?

Lens makers point to materials (modules and collections), creating a guide that includes their own comments and descriptive tags about the content.

##### Who can create a lens?

Any individual member, a community, or a respected organization.

##### What are tags?

Tags are descriptors added by lens makers to help label content, attaching a vocabulary that is meaningful in the context of the lens.

| External bookmarks