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 Ir 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 Ir.

If A0 = {. . . , -2, -3/2, -1, -1/2} ∪ {0} ∪ {1, 2, 3, . . .} and A1 = A0 ∪ { 1/2 } then the Strömberg wavelet of order m is a function Sm(t) satisfying the following conditions:[1]

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

Properties of the set P(m)(V)

The following are some of the properties of the set P(m)(V):

1. 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.
2. If the number of elements in V is three or more than P(m)(V) contains nonzero functions.
3. If V1 and V2 are discrete subsets of R such that V1V2 then P(m)(V1) ⊂ P(m)(V2). In particular, P(m)(A0) ⊂ P(m)(A1).
4. If f(t) ∈ P(m)(A1) then f(t) = g(t) + α λ(t) where α is constant and g(t) ∈ P(m)(A0) is defined by g(r) = f(r) for rA0.

Strömberg wavelet as an orthonormal wavelet

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

Theorem

Let Sm be the Strömberg wavelet of order m. Then the following set

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

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

Strömberg wavelets of order 0

The graph of the Strömberg wavelet of order 0. The graph is scaled such that the value of the wavelet function at 1 is 1.

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

1. If f(t) ∈ P0(V) then f(t) is defined uniquely by the discrete subset {f(r) : rV} of R.
2. To each sA0, a special function λs in A0 is associated: It is defined by λs(r) = 1 if r = s and λs(r) = 0 if srA0. These special elements in P(A0) are called simple tents. The special simple tent λ1/2(t) is denoted by λ(t)

Computation of the Strömberg wavelet of order 0

As already observed, the Strömberg wavelet S0(t) is completely determined by the set { S0(r) : rA1 }. Using the defining properties of the Strömbeg wavelet, exact expressions for elements of this set can be computed and they are given below.[2]

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

Here S0(1) is constant such that ||S0(t)|| = 1.

• The values of S0(t) for positive integral values of t and for negative half-integral values of t are related as follows: $\displaystyle{ S^0(-k/2)=(10-6\sqrt{3})S^0(k) }$ for $\displaystyle{ k=1,2,3,\ldots\,. }$