# Strömberg wavelet

In mathematics, the **Strömberg wavelet** is a certain orthonormal wavelet discovered by Jan-Olov Strömberg and presented in a paper published in 1983.^{[1]} Even though the Haar wavelet was earlier known to be an orthonormal wavelet, Strömberg wavelet was the first smooth orthonormal wavelet to be discovered. The term *wavelet* had not been coined at the time of publishing the discovery of Strömberg wavelet and Strömberg's motivation was to find an orthonormal basis for the Hardy spaces.^{[1]}

## Definition

Let *m* be any non-negative integer. Let *V* be any discrete subset of the set *R* of real numbers. Then *V* splits *R* into non-overlapping intervals. For any *r* in *V*, let *I*_{r} denote the interval determined by *V* with *r* as the left endpoint. Let *P*^{(m)}(*V*) denote the set of all functions *f*(*t*) over *R* satisfying the following conditions:

*f*(*t*) is square integrable.*f*(*t*) has continuous derivatives of all orders up to*m*.*f*(*t*) is a polynomial of degree*m*+ 1 in each of the intervals*I*_{r}.

If *A*_{0} = {. . . , -2, -3/2, -1, -1/2} ∪ {0} ∪ {1, 2, 3, . . .} and *A*_{1} = *A*_{0} ∪ { 1/2 } then the **Strömberg wavelet** of order *m* is a function *S*^{m}(*t*) satisfying the following conditions:^{[1]}

- [math]\displaystyle{ S^m(t)\in P^{(m)} (A_1). }[/math]
- [math]\displaystyle{ \Vert S^m(t)\Vert=1 }[/math], that is, [math]\displaystyle{ \int_R\vert S^m(t)\vert^2\, dt = 1. }[/math]
- [math]\displaystyle{ S^m(t) }[/math] is orthogonal to [math]\displaystyle{ P^{(m)}(A_0) }[/math], that is, [math]\displaystyle{ \int_R S^m(t)\, f(t)\, dt=0 }[/math] for all [math]\displaystyle{ f(t)\in P^{(m)}(A_0). }[/math]

### Properties of the set *P*^{(m)}(*V*)

The following are some of the properties of the set *P*^{(m)}(*V*):

- Let the number of distinct elements in
*V*be two. Then*f*(*t*) ∈*P*^{(m)}(*V*) if and only if*f*(*t*) = 0 for all*t*. - If the number of elements in
*V*is three or more than*P*^{(m)}(*V*) contains nonzero functions. - If
*V*_{1}and*V*_{2}are discrete subsets of*R*such that*V*_{1}⊂*V*_{2}then*P*^{(m)}(*V*_{1}) ⊂*P*^{(m)}(*V*_{2}). In particular,*P*^{(m)}(*A*_{0}) ⊂*P*^{(m)}(*A*_{1}). - If
*f*(*t*) ∈*P*^{(m)}(*A*_{1}) then*f*(*t*) =*g*(*t*) + α λ(*t*) where α is constant and*g*(*t*) ∈*P*^{(m)}(*A*_{0}) is defined by*g*(*r*) =*f*(*r*) for*r*∈*A*_{0}.

### Strömberg wavelet as an orthonormal wavelet

The following result establishes the Strömberg wavelet as an orthonormal wavelet.^{[1]}

### Theorem

Let *S*^{m} be the Strömberg wavelet of order *m*. Then the following set

- [math]\displaystyle{ \left\{2^{j/2}S^m(2^jt-k):j,k \text{ integers }\right\} }[/math]

is a complete orthonormal system in the space of square integrable functions over *R*.

## Strömberg wavelets of order 0

In the special case of Strömberg wavelets of order 0, the following facts may be observed:

- If
*f*(*t*) ∈*P*^{0}(*V*) then*f*(*t*) is defined uniquely by the discrete subset {*f*(*r*) :*r*∈*V*} of*R*. - To each
*s*∈*A*_{0}, a special function λ_{s}in*A*_{0}is associated: It is defined by λ_{s}(*r*) = 1 if*r*=*s*and λ_{s}(*r*) = 0 if*s*≠*r*∈*A*_{0}. These special elements in*P*(*A*_{0}) are called*simple tents*. The special simple tent λ_{1/2}(*t*) is denoted by λ(*t*)

### Computation of the Strömberg wavelet of order 0

As already observed, the Strömberg wavelet *S*^{0}(*t*) is completely determined by the set { *S*^{0}(*r*) : *r* ∈ *A*_{1} }. Using the defining properties of the Strömbeg wavelet, exact expressions for elements of this set can be computed and they are given below.^{[2]}

- [math]\displaystyle{ S^0(k) = S^0(1)(\sqrt{3}-2)^{k-1} }[/math] for [math]\displaystyle{ k=1,2,3, \ldots }[/math]
- [math]\displaystyle{ S^0(\tfrac{1}{2}) = -S^0(1)\left(\sqrt{3}+\tfrac{1}{2}\right) }[/math]
- [math]\displaystyle{ S^0(0) = S^0(1)(2\sqrt{3}-2) }[/math]
- [math]\displaystyle{ S^0(-\tfrac{k}{2}) = S^0(1)(2\sqrt{3}-2)(\sqrt{3}-2)^k }[/math] for [math]\displaystyle{ k=1,2,3, \ldots }[/math]

Here *S*^{0}(1) is constant such that ||*S*^{0}(*t*)|| = 1.

### Some additional information about Strömberg wavelet of order 0

The Strömberg wavelet of order 0 has the following properties.^{[2]}

- The Strömberg wavelet
*S*^{0}(*t*) oscillates about*t*-axis. - The Strömberg wavelet
*S*^{0}(*t*) has exponential decay. - The values of
*S*^{0}(*t*) for positive integral values of*t*and for negative half-integral values of*t*are related as follows: [math]\displaystyle{ S^0(-k/2)=(10-6\sqrt{3})S^0(k) }[/math] for [math]\displaystyle{ k=1,2,3,\ldots\,. }[/math]

- The Strömberg wavelet

## References

- ↑
^{1.0}^{1.1}^{1.2}^{1.3}Janos-Olov Strömberg,*A modified Franklin system and higher order spline systems on R*, Conference on Harmonic Analysis in Honor of A. Zygmond, Vol. II, W. Beckner, et al (eds.) Wadsworth, 1983, pp.475-494^{n}as unconditional bases for Hardy spaces - ↑
^{2.0}^{2.1}P. Wojtaszczyk (1997).*A Mathematical Introduction to Wavelets*. Cambridge University Press. pp. 5–14. ISBN 0521570204. https://archive.org/details/mathematicalintr00wojt.

Original source: https://en.wikipedia.org/wiki/Strömberg wavelet.
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