3. Wavelets

The idea of multiresolution analysis is to approximate vectors of L<sup>2</sup> ð Þ IR with variable degrees of resolution. This is achieved through a multiresolution analysis scheme defined by means of some axioms. The first axiom is the existence of a sequence f g Vn <sup>n</sup><sup>∈</sup> <sup>Z</sup> of subspaces of <sup>L</sup><sup>2</sup> ð Þ IR nested one inside the other, that is,

$$
\cdots \subset V\_{-2} \subset V\_{-1} \subset V\_0 \subset V\_1 \subset V\_2 \subset \cdots \tag{23}
$$

The idea is that if one approximates (in a least square sense) a function f with vectors belonging to Vn, the approximation error gets smaller as n increases since every vector of Vn also belongs to Vnþ1. Note, however, that (23) does not grant that we will be able to approximate f with an error as small as desired; in order to grant this, we need another axiom

$$\overline{\bigcup\_{n \in \mathbb{Z}} V\_n} = L^2(\mathbb{R}) \tag{24}$$

where the overline denotes set closure (in the topology induced by the norm on L2 ð Þ IR ). Axiom (24) requires that every vector of <sup>L</sup><sup>2</sup> ð Þ IR is in the closure of the union in the left hand; this means that given any ϵ . 0 and f ∈ L<sup>2</sup> ð Þ IR , it is possible to find an element of the union whose distance from f is less than ϵ. In other words, (24) means that whatever f ∈ L<sup>2</sup> ð Þ IR and whatever the chosen maximum approximation error allowed ϵ, one can find a space Vn that approximates f with the required precision.

An axiom dual to (24) is

$$\bigcap\_{n \in \mathbb{Z}} V\_n = \{\mathbf{O}\} \tag{25}$$

that requires that there is only one "lowest resolution vector," that is, the null vector.

### Remark 3.1.

In order to see that axiom (24) is not obvious, it is more convenient to work with Hilbert space L<sup>2</sup> ð Þ ½ � 0; 1 . Recall that functions x↦ cos 2ð Þ πnx , x↦ sin 2ð Þ πnx , and n∈ IN and the constant 1 are an orthogonal basis of L<sup>2</sup> ð Þ ½ � 0; 1 .

Define S<sup>0</sup> ¼ f g sin 2ð Þ πð Þ 2k t ; k∈ IN as the set of all the even-numbered sines, and define V<sup>0</sup> as the space generated by S0, that is,

$$V\_0 \coloneqq \mathbf{span}\, \mathbf{S}\_0 \tag{26}$$

Now define spaces Vn, n , 0 by removing one vector at time from the basis of V0, and define spaces Vn, n , 0 by adding one odd harmonic at time. More precisely, define

$$V\_n = \text{span}\,\mathcal{S}\_n\tag{27}$$

ϕ ¼ ∑ i∈ Z

where the left-hand side scalar product is the usual scalar product in ℓ<sup>2</sup>

Note that starting from a ϕ that satisfies a two-scale equation like (37), it is possible to recover a full multiresolution analysis. Indeed, one defines V<sup>0</sup> according to (36) and Vn by repeated applications of (35). Two-scale Eq. (37) grants that the

central to wavelet theory. Function ϕ is known as scaling function.

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DOI: http://dx.doi.org/10.5772/intechopen.82820

Note that from the orthonormality of τi=<sup>2</sup>Sϕi∈ Z follows

Note also that (37) shows that ϕ is the fixed point of operator

O ≔ ∑ i∈ Z gi τ<sup>i</sup>=<sup>2</sup>

ð Þ IR in order to obtain ϕ. This is indeed possible, but the

<sup>ψ</sup> <sup>¼</sup> <sup>δ</sup><sup>i</sup> orthonormal basis (42)

<sup>ϕ</sup> <sup>¼</sup> <sup>0</sup> <sup>∀</sup>i, j<sup>∈</sup> <sup>Z</sup> <sup>W</sup><sup>0</sup> orthogonal to <sup>V</sup><sup>0</sup> (43)

Vnþ<sup>1</sup> ¼ Vn⊕Wn (40)

<sup>¼</sup> <sup>δ</sup>k,<sup>0</sup> (44)

<sup>¼</sup> <sup>0</sup> (45)

<sup>g</sup>�nþ<sup>1</sup> (46)

ð Þ IR with its translations and

, a possible hi that satisfies (44) and (45) is

Sϕ ψ ∈ V<sup>1</sup> (41)

This suggests that maybe one could start from a sequence gi and apply repeat-

Since Vnþ<sup>1</sup> ⊃Vn one can consider the orthogonal complement of Vn in Vnþ1; call

It is possible to find, starting from the two-scale Eq. (37), a function ψ such that

By using (41) and the orthonormality of τ<sup>i</sup>=<sup>2</sup>Sϕ, it is possible to rewrite (42) and

<sup>i</sup><sup>∈</sup> <sup>Z</sup> is an orthonormal basis of W0. This implies that it must be for all

hi τ<sup>i</sup>=<sup>2</sup>

ψ ¼ ∑ i∈ Z

> τ2k hi; hi

> > τ2k hi; gi

hi ¼ �ð Þ<sup>1</sup> <sup>i</sup>

orthogonal filter bank. Moreover, if gi and ϕ are known, one can obtain ψ by choosing hi according to (46) and computing ψ according to (41). Function ψ is

This shows that sequences hi and gi are the impulse responses of a two-channel

ψ; τ<sup>i</sup>

τj ψ; τ<sup>i</sup>

(43) as conditions on hi, namely,

It is easy to verify that, given gi

known as wavelet, and it generates the whole L<sup>2</sup>

gi ; τ2<sup>k</sup> gi <sup>¼</sup> <sup>φ</sup>; <sup>τ</sup>�<sup>k</sup>

Remark 3.2.

Remark 3.3.

nesting axiom (32) is satisfied.

edly O to a vector of L<sup>2</sup>

it Wn, that is,

τi ψ

dilations.

43

theoretical details are out of scope here; see [5].

gi τi=<sup>2</sup>

for some sequence gi : Z ! IR. Eq. (37) is known as two-scale equation and it is

Sϕ (37)

<sup>φ</sup> <sup>¼</sup> <sup>δ</sup>k,<sup>0</sup> (38)

S (39)

ð Þ Z .

where

$$\mathcal{S}\_{\mathfrak{n}} = \begin{pmatrix} \mathcal{S}\_{\mathfrak{n}+1} \left\{ \sin \left( 2\pi (-2\mathfrak{n})t \right) \right\} & \text{if } \mathfrak{n} \le \mathbf{0} \\ \mathcal{S}\_{\mathfrak{n}-1} \cup \left\{ \sin \left( 2\pi (2\mathfrak{n}-1)t \right) \right\} & \text{if } \mathfrak{n} > \mathbf{0} \end{pmatrix} \tag{28}$$

It is clear that the sequence of spaces defined in this way satisfies axiom (23), but not (24), since, for example, function cos 2ð Þ πt is orthogonal to every Vn. Note that this construction can be repeated for any Hilbert space using an orthonormal basis of the space instead of sines and cosines. Another axiom makes more precise the idea of "increasing resolution" by asking that vectors in Vn vary twice as faster than the vectors in Vn�1. In order to make this more precise, define operator S : L<sup>2</sup> ð Þ! IR <sup>L</sup><sup>2</sup> ð Þ IR as the rescaling operator ½ � <sup>S</sup><sup>f</sup> ð Þ¼ <sup>x</sup> ffiffi 2 <sup>p</sup> <sup>f</sup>ð Þ <sup>2</sup><sup>x</sup> . Note that because of the multiplication by ffiffi 2 <sup>p</sup> , <sup>S</sup> is unitary, that is, k k <sup>S</sup><sup>f</sup> <sup>¼</sup> k k<sup>f</sup> . The new axiom is

$$f \in V\_n \Leftrightarrow \mathcal{S}f \in V\_{n+1} \tag{29}$$

It follows that Vn <sup>¼</sup> <sup>S</sup>nV0. With this position one can interpret (25) by saying that the slowest function is the constant (and the only constant in L<sup>2</sup> ð Þ IR is the zero).

The last axiom puts a constraint on the structure of V<sup>0</sup> by asking that is generated by a function ϕ and its translations. In order to make this more precise, define the operator τ<sup>t</sup> : L<sup>2</sup> ð Þ! IR <sup>L</sup><sup>2</sup> ð Þ IR associated to a translation of <sup>t</sup> as <sup>τ</sup><sup>t</sup> ½ � <sup>f</sup> ð Þ¼ <sup>x</sup> f xð Þ � <sup>t</sup> . Note that also τ<sup>t</sup> is unitary and that the exponential notation is convenient since <sup>τ</sup><sup>a</sup>τ<sup>b</sup> <sup>¼</sup> <sup>τ</sup><sup>a</sup>þ<sup>b</sup>. Observe also the commutation relation <sup>S</sup>τ<sup>t</sup> <sup>¼</sup> <sup>τ</sup><sup>t</sup>=<sup>2</sup>S. The last axiom can be written as

$$\exists \phi \in L^2(\mathbb{R}) : V\_0 = \text{span}\{\tau^i \phi, \quad i \in \mathbb{Z}\} \tag{30}$$

Often as part of axioms, it is required that ϕ is orthogonal to its translations, that is,

$$
\langle \mathfrak{r}^i \mathfrak{g}, \mathfrak{r}^j \mathfrak{g} \rangle = \delta\_{i,j} \tag{31}
$$

However, it is not necessary to include (31) explicitly in the axioms since, given a ϕ that satisfies (30), it is possible to orthonormalize it, so that it satisfies (31), with a well-known "Fourier trick" [3]. Therefore, we will suppose (31) satisfied.

It is worth to summarize here the axioms

$$
\cdots \subset V\_{-2} \subset V\_{-1} \subset V\_0 \subset V\_1 \subset V\_2 \subset \cdots \tag{32}
$$

$$\overline{\bigcup\_{n \in \mathbb{Z}} V\_n} = L^2(\mathbb{R}) \tag{33}$$

$$\bigcap\_{n \in \mathbb{Z}} V\_n = \{0\} \tag{34}$$

$$V\_{n+1} = \mathcal{S} V\_n \tag{35}$$

$$W\_0 = \text{span}\{\tau^i \phi, \quad i \in \mathbb{Z}\} \quad \exists \phi \in L^2(\mathbb{R}) \tag{36}$$

The axioms above allow us to determine a property of ϕ. Note that since V<sup>1</sup> ⊃V0, ϕ∈ V1. Note also that set Sτ<sup>i</sup> <sup>ϕ</sup> <sup>¼</sup> <sup>τ</sup><sup>i</sup>=<sup>2</sup>Sϕi<sup>∈</sup> <sup>Z</sup> � � is an orthonormal basis of V1. It follows that one can write ϕ as linear combination of Sϕ and its half-integer translations, that is,

Wavelets for Differential Equations and Numerical Operator Calculus DOI: http://dx.doi.org/10.5772/intechopen.82820

$$\phi = \sum\_{i \in \mathbb{Z}} \mathbf{g}\_i \,\,\, \pi^{i/2} \mathcal{S} \phi \,\,\, \tag{37}$$

for some sequence gi : Z ! IR. Eq. (37) is known as two-scale equation and it is central to wavelet theory. Function ϕ is known as scaling function.

### Remark 3.2.

Vn ¼ spanSn (27)

2

f ∈ Vn ⇔ Sf ∈ Vnþ<sup>1</sup> (29)

ð Þ IR associated to a translation of <sup>t</sup> as <sup>τ</sup><sup>t</sup> ½ � <sup>f</sup> ð Þ¼ <sup>x</sup> f xð Þ � <sup>t</sup> .

<sup>ϕ</sup> � � <sup>¼</sup> <sup>δ</sup>i,j (31)

⋯ ⊂V�<sup>2</sup> ⊂V�<sup>1</sup> ⊂V<sup>0</sup> ⊂V<sup>1</sup> ⊂V<sup>2</sup> ⊂ ⋯ (32)

ϕ; i∈ Z � � (30)

ð Þ IR (33)

ð Þ IR (36)

Vn ¼ f g0 (34)

Vnþ<sup>1</sup> ¼ SVn (35)

<sup>ϕ</sup> <sup>¼</sup> <sup>τ</sup><sup>i</sup>=<sup>2</sup>Sϕi<sup>∈</sup> <sup>Z</sup> � � is an orthonormal basis of

<sup>p</sup> , <sup>S</sup> is unitary, that is, k k <sup>S</sup><sup>f</sup> <sup>¼</sup> k k<sup>f</sup> . The new axiom is

<sup>p</sup> <sup>f</sup>ð Þ <sup>2</sup><sup>x</sup> . Note that because

ð Þ IR is the

(28)

Sn <sup>¼</sup> Snþ<sup>1</sup> f g sin 2ð Þ <sup>π</sup>ð Þ �2<sup>n</sup> <sup>t</sup> if n , <sup>0</sup> Sn�<sup>1</sup> ∪ f g sin 2ð Þ πð Þ 2n � 1 t if n . 0

the vectors in Vn�1. In order to make this more precise, define operator

that the slowest function is the constant (and the only constant in L<sup>2</sup>

ð Þ IR as the rescaling operator ½ � <sup>S</sup><sup>f</sup> ð Þ¼ <sup>x</sup> ffiffi

It is clear that the sequence of spaces defined in this way satisfies axiom (23), but not (24), since, for example, function cos 2ð Þ πt is orthogonal to every Vn. Note that this construction can be repeated for any Hilbert space using an orthonormal basis of the space instead of sines and cosines. Another axiom makes more precise the idea of "increasing resolution" by asking that vectors in Vn vary twice as faster than

It follows that Vn <sup>¼</sup> <sup>S</sup>nV0. With this position one can interpret (25) by saying

The last axiom puts a constraint on the structure of V<sup>0</sup> by asking that is generated by a function ϕ and its translations. In order to make this more precise, define

Note that also τ<sup>t</sup> is unitary and that the exponential notation is convenient since <sup>τ</sup><sup>a</sup>τ<sup>b</sup> <sup>¼</sup> <sup>τ</sup><sup>a</sup>þ<sup>b</sup>. Observe also the commutation relation <sup>S</sup>τ<sup>t</sup> <sup>¼</sup> <sup>τ</sup><sup>t</sup>=<sup>2</sup>S. The last axiom can

ð Þ IR : <sup>V</sup><sup>0</sup> <sup>¼</sup> span <sup>τ</sup><sup>i</sup>

τi ϕ; τ<sup>j</sup>

a well-known "Fourier trick" [3]. Therefore, we will suppose (31) satisfied.

⋃ n ∈ Z

<sup>V</sup><sup>0</sup> <sup>¼</sup> span <sup>τ</sup><sup>i</sup>

⋂ n∈ Z

The axioms above allow us to determine a property of ϕ. Note that since

V1. It follows that one can write ϕ as linear combination of Sϕ and its half-integer

Often as part of axioms, it is required that ϕ is orthogonal to its translations,

However, it is not necessary to include (31) explicitly in the axioms since, given a ϕ that satisfies (30), it is possible to orthonormalize it, so that it satisfies (31), with

Vn <sup>¼</sup> <sup>L</sup><sup>2</sup>

ϕ; i∈ Z � � ∃ϕ∈ L<sup>2</sup>

�

2

ð Þ! IR <sup>L</sup><sup>2</sup>

∃ϕ∈ L<sup>2</sup>

It is worth to summarize here the axioms

V<sup>1</sup> ⊃V0, ϕ∈ V1. Note also that set Sτ<sup>i</sup>

translations, that is,

42

where

S : L<sup>2</sup>

zero).

ð Þ! IR <sup>L</sup><sup>2</sup>

the operator τ<sup>t</sup> : L<sup>2</sup>

be written as

that is,

of the multiplication by ffiffi

Wavelet Transform and Complexity

Note that from the orthonormality of τi=<sup>2</sup>Sϕi∈ Z follows

$$
\langle \mathbf{g}\_i, \pi^{2k} \mathbf{g}\_i \rangle = \langle \boldsymbol{\varrho}, \pi^{-k} \boldsymbol{\varrho} \rangle = \delta\_{k,0} \tag{38}
$$

where the left-hand side scalar product is the usual scalar product in ℓ<sup>2</sup> ð Þ Z . Remark 3.3.

Note that starting from a ϕ that satisfies a two-scale equation like (37), it is possible to recover a full multiresolution analysis. Indeed, one defines V<sup>0</sup> according to (36) and Vn by repeated applications of (35). Two-scale Eq. (37) grants that the nesting axiom (32) is satisfied.

Note also that (37) shows that ϕ is the fixed point of operator

$$\mathcal{O} \coloneqq \sum\_{i \in \mathbb{Z}} \mathbf{g}\_i \quad \mathfrak{r}^{i/2}\mathcal{S} \tag{39}$$

This suggests that maybe one could start from a sequence gi and apply repeatedly O to a vector of L<sup>2</sup> ð Þ IR in order to obtain ϕ. This is indeed possible, but the theoretical details are out of scope here; see [5].

Since Vnþ<sup>1</sup> ⊃Vn one can consider the orthogonal complement of Vn in Vnþ1; call it Wn, that is,

$$V\_{n+1} = V\_n \oplus W\_n \tag{40}$$

It is possible to find, starting from the two-scale Eq. (37), a function ψ such that τi ψ <sup>i</sup><sup>∈</sup> <sup>Z</sup> is an orthonormal basis of W0. This implies that it must be for all

$$\Psi = \sum\_{i \in \mathbb{Z}} h\_i \quad \mathfrak{r}^{i/2} \mathfrak{S} \phi \quad \psi \in V\_1 \tag{41}$$

$$
\langle \boldsymbol{\Psi}, \boldsymbol{\tau}^i \boldsymbol{\Psi} \rangle = \delta\_i \qquad \text{orthogonal basis} \tag{42}
$$

$$\langle \pi^i \boldsymbol{\mu}, \pi^i \boldsymbol{\phi} \rangle = \mathbf{0} \qquad \forall i, j \in \mathbb{Z} \qquad \text{ $W\_0$  orthogonal to  $V\_0$ } \tag{43}$$

By using (41) and the orthonormality of τ<sup>i</sup>=<sup>2</sup>Sϕ, it is possible to rewrite (42) and (43) as conditions on hi, namely,

$$
\langle \pi^{2k} h\_i, h\_i \rangle = \delta\_{k,0} \tag{44}
$$

$$
\langle \pi^{2k} h\_i, \mathbf{g}\_i \rangle = \mathbf{0} \tag{45}
$$

It is easy to verify that, given gi , a possible hi that satisfies (44) and (45) is

$$h\_i = (-\mathbf{1})^i \mathbf{g}\_{-n+1} \tag{46}$$

This shows that sequences hi and gi are the impulse responses of a two-channel orthogonal filter bank. Moreover, if gi and ϕ are known, one can obtain ψ by choosing hi according to (46) and computing ψ according to (41). Function ψ is known as wavelet, and it generates the whole L<sup>2</sup> ð Þ IR with its translations and dilations.

This has also another interesting consequence. Suppose f ∈ L<sup>2</sup> ð Þ IR and that

$$\left\langle \gamma\_i^{(1)} = \left\langle \mathfrak{r}^{i/2} \mathcal{S} \phi, f \right\rangle \right\rangle = \left\langle \mathcal{S} \mathfrak{r}^i \phi, f \right\rangle \tag{47}$$

are the coefficients of its projection on V1. Suppose we need the coefficients γ ð Þ 0 <sup>k</sup> <sup>¼</sup> <sup>τ</sup>kϕ; <sup>f</sup> � � of the projection on <sup>V</sup>0. It is possible to exploit the two-scale equation

$$\begin{aligned} \left< \mathbf{y}\_k^{(0)} = \left< \mathbf{r}^k \boldsymbol{\phi}, f \right> &= \left< \mathbf{r}^k \sum\_{i \in \mathbb{Z}} \mathbf{g}\_i \mathcal{S} \mathbf{r}^i \boldsymbol{\phi}, f \right> \\ &= \sum\_{i \in \mathbb{Z}} \mathbf{g}\_i \left< \mathcal{S} \mathbf{r}^{i+2k} \boldsymbol{\phi}, f \right> \\ &= \sum\_{i \in \mathbb{Z}} \mathbf{g}\_i \boldsymbol{\upchi}\_{i+2k}^{(1)} = \left[ \mathbf{g}\_- \, ^\* \boldsymbol{\upchi}\_i^{(1)} \right]\_{2k} \end{aligned} \tag{48}$$

where <sup>g</sup>� is the time-reversed version of gi . Eq. (48) shows that it is possible to go to the space at lower resolution by means of a filtering by <sup>g</sup>� and a decimation by a factor of two. Similarly, by calling η ð Þ 0 <sup>k</sup> <sup>¼</sup> <sup>τ</sup><sup>k</sup>ψ; <sup>f</sup> � � the coefficients relative to the projection of f on W0, one can obtain

$$\eta\_k^{(0)} = \sum\_{i \in \mathbb{Z}} h\_i \boldsymbol{\gamma}\_{i+2k}^{(1)} = \left[ h\_- \, ^\* \boldsymbol{\gamma}\_i^{(1)} \right]\_{2k} \tag{49}$$

P xð Þ¼ ∑

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DOI: http://dx.doi.org/10.5772/intechopen.82820

In other words, V<sup>0</sup> contains all the polynomials of degree less than ℓ.

ϕHð Þ¼ x

It is immediate to verify that ϕ<sup>H</sup> satisfies a two-scale equation

2

<sup>ϕ</sup><sup>H</sup> <sup>¼</sup> <sup>1</sup> ffiffi 2

to create the corresponding wavelet, use prescription hi ¼ �ð Þ<sup>1</sup> <sup>i</sup>

<sup>ϕ</sup><sup>H</sup> <sup>¼</sup> <sup>1</sup> ffiffi 2

makes it not well suited to approximate smooth functions.

Example 3.1. (Haar wavelet).

Figure 1.

with coefficients <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>g</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffi

Example 3.2 (Sinc wavelet).

Sinc function

45

i ∈ Z ciτ<sup>i</sup> ϕ � �

(a) Splitting coefficient sequence into a low-resolution and a high-resolution one using a two-channel filter bank. (b) Iteration of structure (a) makes a fast algorithm for computing the wavelet coefficients.

The simplest example of wavelet is the Haar wavelet whose scaling function is

<sup>p</sup> <sup>S</sup>ϕ<sup>H</sup> <sup>þ</sup> <sup>τ</sup><sup>1</sup>=<sup>2</sup>

<sup>p</sup> <sup>S</sup>ϕ<sup>H</sup> � <sup>τ</sup><sup>1</sup>=<sup>2</sup>

Note that the Haar wavelet is compactly supported, but it is discontinuous. This

An example in some sense opposite to the Haar wavelet is the Sinc wavelet. In this case V<sup>0</sup> is the space of "low-pass" functions, that is, functions whose Fourier transform is zero outside interval ½ � �π; π . As well known, V<sup>0</sup> is generated by the

1 if x∈ ½ � 0; 1 0 else �

Sϕ<sup>H</sup>

Sϕ<sup>H</sup>

<sup>p</sup> . Note that trivially <sup>τ</sup><sup>2</sup><sup>k</sup>ϕH; <sup>ϕ</sup><sup>H</sup>

� � (53)

� � (54)

� � <sup>¼</sup> <sup>δ</sup>k. In order

<sup>g</sup>�iþ<sup>1</sup> to get

ð Þ x (51)

(52)

Figure 1a shows this idea: the sequence of high-resolution coefficients are processed with a two-channel orthogonal filter bank, and the coefficients relative to the lower resolution space Vn exit from one branch, and the coefficients relative to the "missing details" space Wn exit from the other. The idea can be iterated several times; see Figure 1b. This is the basis of the well-known fast algorithm to compute wavelet coefficients and also the origin of the minor, and very common, misnomer in calling Figure 1b a "discrete-time wavelet transform."

An interesting characteristic of wavelets is that they can be used to detect the local regularity of a function. This is similar to what happened with Fourier transform where a function that is discontinuous has a Fourier transform that decays as 1=ω; if the function is continuous but not derivable, its Fourier transform decays as 1=ω<sup>2</sup> and so on. With the wavelet transform happens something similar, with the scale playing the role of frequency. The interesting difference is that while a Fourier transform that decays as 1=ω tells us that there is at least one discontinuity, but not where, with the wavelet transform the slow decay with the scale is localized around the discontinuity. The precise claim of this property requires the introduction of the concept of Lipschitz regularity and would take us too far; see [6]. This suggests that when approximating the unknown function in a PDE, we can keep high-resolution coefficients only in the neighborhood of singularities, saving on computational effort.

We will say that wavelet ψ has ℓ vanishing moments if

$$\int\_{\mathbb{R}} x^k \mu(\mathbf{x}) d\mathbf{x} = \mathbf{0} \qquad k = 0, 1, \dots, \ell - 1 \tag{50}$$

An interesting property of compactly supported wavelets with ℓ vanishing moments is that the corresponding scaling function (not the wavelet itself) can reproduce polynomials of degree at most ℓ � 1 in the sense that if P xð Þ is a polynomial with degree less than ℓ, there exist coefficients ci such that

Wavelets for Differential Equations and Numerical Operator Calculus DOI: http://dx.doi.org/10.5772/intechopen.82820

Figure 1. (a) Splitting coefficient sequence into a low-resolution and a high-resolution one using a two-channel filter bank. (b) Iteration of structure (a) makes a fast algorithm for computing the wavelet coefficients.

$$P(\mathbf{x}) = \left[\sum\_{i \in \mathbb{Z}} c\_i \pi^i \phi\right](\mathbf{x}) \tag{51}$$

In other words, V<sup>0</sup> contains all the polynomials of degree less than ℓ. Example 3.1. (Haar wavelet).

The simplest example of wavelet is the Haar wavelet whose scaling function is

$$\phi\_H(\mathbf{x}) = \begin{pmatrix} \mathbf{1} & \text{if } \mathbf{x} \in [0, \mathbf{1}] \\ \mathbf{0} & \text{else} \end{pmatrix} \tag{52}$$

It is immediate to verify that ϕ<sup>H</sup> satisfies a two-scale equation

$$\phi\_H = \frac{1}{\sqrt{2}} \left( \mathcal{S} \phi\_H + \pi^{1/2} \mathcal{S} \phi\_H \right) \tag{53}$$

with coefficients <sup>g</sup><sup>0</sup> <sup>¼</sup> <sup>g</sup><sup>1</sup> <sup>¼</sup> <sup>1</sup><sup>=</sup> ffiffi 2 <sup>p</sup> . Note that trivially <sup>τ</sup><sup>2</sup><sup>k</sup>ϕH; <sup>ϕ</sup><sup>H</sup> � � <sup>¼</sup> <sup>δ</sup>k. In order to create the corresponding wavelet, use prescription hi ¼ �ð Þ<sup>1</sup> <sup>i</sup> <sup>g</sup>�iþ<sup>1</sup> to get

$$\phi\_H = \frac{1}{\sqrt{2}} \left( \mathcal{S} \phi\_H - \pi^{1/2} \mathcal{S} \phi\_H \right) \tag{54}$$

Note that the Haar wavelet is compactly supported, but it is discontinuous. This makes it not well suited to approximate smooth functions.

Example 3.2 (Sinc wavelet).

An example in some sense opposite to the Haar wavelet is the Sinc wavelet. In this case V<sup>0</sup> is the space of "low-pass" functions, that is, functions whose Fourier transform is zero outside interval ½ � �π; π . As well known, V<sup>0</sup> is generated by the Sinc function

This has also another interesting consequence. Suppose f ∈ L<sup>2</sup>

Sϕ; f D E

are the coefficients of its projection on V1. Suppose we need the coefficients

i∈ Z gi Sτ<sup>i</sup> ϕ; f

� �

gi Sτiþ2<sup>k</sup> ϕ; f � �

> ∗ γ ð Þ1 i h i

xkψð Þ <sup>x</sup> dx <sup>¼</sup> <sup>0</sup> <sup>k</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, …, <sup>ℓ</sup> � <sup>1</sup> (50)

∗γ ð Þ1 i h i

2k

<sup>k</sup> <sup>¼</sup> <sup>τ</sup><sup>k</sup>ψ; <sup>f</sup> � � the coefficients relative to the

. Eq. (48) shows that it is possible to

<sup>2</sup><sup>k</sup> (49)

<sup>k</sup> <sup>¼</sup> <sup>τ</sup>kϕ; <sup>f</sup> � � of the projection on <sup>V</sup>0. It is possible to exploit the two-scale

<sup>ϕ</sup>; <sup>f</sup> � � <sup>¼</sup> <sup>τ</sup><sup>k</sup> <sup>∑</sup>

¼ ∑ i∈ Z

¼ ∑ i∈ Z gi γ ð Þ1 <sup>i</sup>þ2<sup>k</sup> <sup>¼</sup> <sup>g</sup>�

ð Þ 0

Figure 1a shows this idea: the sequence of high-resolution coefficients are processed with a two-channel orthogonal filter bank, and the coefficients relative to the lower resolution space Vn exit from one branch, and the coefficients relative to the "missing details" space Wn exit from the other. The idea can be iterated several times; see Figure 1b. This is the basis of the well-known fast algorithm to compute wavelet coefficients and also the origin of the minor, and very common, misnomer

An interesting characteristic of wavelets is that they can be used to detect the local regularity of a function. This is similar to what happened with Fourier transform where a function that is discontinuous has a Fourier transform that decays as 1=ω; if the function is continuous but not derivable, its Fourier transform decays as 1=ω<sup>2</sup> and so on. With the wavelet transform happens something similar, with the scale playing the role of frequency. The interesting difference is that while a Fourier transform that decays as 1=ω tells us that there is at least one discontinuity, but not where, with the wavelet transform the slow decay with the scale is localized around the discontinuity. The precise claim of this property requires the introduction of the concept of Lipschitz regularity and would take us too far; see [6]. This suggests that when approximating the unknown function in a PDE, we can keep high-resolution coefficients only in the neighborhood of singularities, saving on computational

An interesting property of compactly supported wavelets with ℓ vanishing moments is that the corresponding scaling function (not the wavelet itself) can reproduce polynomials of degree at most ℓ � 1 in the sense that if P xð Þ is a polynomial

go to the space at lower resolution by means of a filtering by <sup>g</sup>� and a decimation by

<sup>¼</sup> <sup>S</sup>τ<sup>i</sup>

γ ð Þ1 <sup>i</sup> <sup>¼</sup> <sup>τ</sup>i=<sup>2</sup>

γ ð Þ 0 <sup>k</sup> <sup>¼</sup> <sup>τ</sup><sup>k</sup>

Wavelet Transform and Complexity

where <sup>g</sup>� is the time-reversed version of gi

η ð Þ 0 <sup>k</sup> ¼ ∑ i∈ Z hiγ ð Þ1 <sup>i</sup>þ2<sup>k</sup> <sup>¼</sup> <sup>h</sup>�

in calling Figure 1b a "discrete-time wavelet transform."

We will say that wavelet ψ has ℓ vanishing moments if

with degree less than ℓ, there exist coefficients ci such that

Z IR

a factor of two. Similarly, by calling η

projection of f on W0, one can obtain

γ ð Þ 0

equation

effort.

44

ð Þ IR and that

(48)

ϕ; f � � (47)

$$\text{sinc}(\mathfrak{x}) = \frac{\sin \left( \mathfrak{x} \mathfrak{x} \right)}{\mathfrak{x} \mathfrak{x}} \tag{55}$$

these three characteristics? The answer was given by Daubechies. It turns out that imposing all the three characteristics is very demanding and only a small family of

gi <sup>¼</sup> <sup>ϕ</sup>; <sup>τ</sup>i=<sup>2</sup>

According to (60) if ϕ has compact support, then gi has a finite number of coefficients that are different from zero. Since gi needs to be orthogonal to its even translations, its length (i.e., the number of nonzero coefficients) must be necessar-

Moreover, if the iteration of operator O in (39) converges, it is easy to see that gi has a finite number of coefficients, and then the limit function has compact support. This suggests that it "suffices" to find a finite length sequence gi that is orthogonal to its own even translation and iterates operator O to obtain the desired scaling functions. It actually turns out that this can be done, although there are lots of technical details to be taken care of (e.g., about convergence of O<sup>k</sup> and smoothness

Every member of the Daubechies family is identified by the length 2N of the sequence gi (remember that the length of gi is necessarily even). It can be proven that the resulting scaling function has N vanishing moments and its smoothness grows with N; see Table 1. See also Figure 2 that shows the results of the first three iterations of O (first row), the final scaling function (second row), and the wavelet

The construction given above is the original idea of multiresolution analysis. Since the early 1990s, many researchers worked in this field, and many variations and extensions have been introduced. Here we briefly recall those that have more

Multiwavelets are a generalization of standard multiresolution analysis in the sense that now scaling functions and wavelets are vectors of functions. This means that Vn is not generated by the translations of a single function but from the translations of many functions. Every idea of standard multiresolution analysis can be reformulated without much difficulty in this case, with the most notable difference that the two-scale equation now has vector function and coefficients that are matrices. Multiwavelets can accommodate scaling factors different from two and there is a larger choice for compact support wavelets. See [7] for more details.

N 2 3 4 5 6 7 8 9 10 β<sup>S</sup> 1 1.415 1.775 2.096 2.388 2.658 2.914 3.161 3.402 β<sup>H</sup> 0.550 0.915 1.275 1.596 1.888 2.158 2.415 2.661 2.902

Hölder β<sup>H</sup> and Sobolev β<sup>S</sup> regularity exponent of Daubechies' wavelets as function of length 2 N of gi

.

cients gi in the two-scale equation can be obtained as

DOI: http://dx.doi.org/10.5772/intechopen.82820

Wavelets for Differential Equations and Numerical Operator Calculus

of the resulting ϕ); see [5] for details.

(third row) of three different Daubechies wavelets.

interest in the field of differential equation solutions.

An easy observation is that if ϕ is orthogonal to its own translations, the coeffi-

Sϕ

D E (60)

wavelets exists.

ily even [3].

3.2 Extensions

3.2.1 Multiwavelets

Table 1.

47

and its translations, that is,

$$V\_0 = \text{span}\{\tau^k \text{sinc}\}\_{k \in \mathbb{Z}} \tag{56}$$

This suggests to use ϕ<sup>S</sup> ¼ sinc as scaling function. The fact that a two-scale equation is satisfied is easily checked in frequency since V<sup>1</sup> is the space of functions whose Fourier transform is zero outside ½ � �2π; 2π ; therefore, every function of V<sup>0</sup> is contained in V1, as desired.

The corresponding wavelet ψ<sup>S</sup> is easily characterized in frequency as the function whose Fourier transform is

$$\Psi\_{\mathbb{S}}(\boldsymbol{\alpha}) = \begin{pmatrix} \mathbf{1} & \text{if } \boldsymbol{\pi} \le |\boldsymbol{\alpha}| \le 2\pi \\ \mathbf{0} & \text{otherwise} \end{pmatrix} \tag{57}$$

It is easy to verify that ψ<sup>S</sup> ∈ V1, ψS⊥V1, and τ<sup>2</sup><sup>k</sup>ψS; ψ<sup>S</sup> � � <sup>¼</sup> <sup>δ</sup>k.

As said above, the Sinc example is somehow the opposite of Haar wavelet: it is arbitrarily differentiable, but it has infinite support; actually, it decays very slowly (as Oð Þ 1=x ), and this introduces several practical issues. Moreover, sequences gi and hi are of infinite length, and they decay slowly too (they do not even have a ztransform), making it difficult to implement it.

Example 3.3 (spline wavelet).

An example intermediate between Haar and Sinc wavelet is represented by spline spaces of degree d. In this case Vð Þ <sup>d</sup> <sup>0</sup> is defined as the space of piecewise polynomial functions that are d times differentiable (with continuous derivative), with the "breaking points" on the integer, more precisely

$$\mathcal{V}\_0^{(d)} = \{ f \in L^2(\mathbb{R}) \cap \mathcal{C}\_{d-1}, f \big|\_{[k,k+1]} = \text{polynomial degree } d \forall k \in \mathbb{Z} \} \tag{58}$$

It is easy to see that Vð Þ <sup>d</sup> <sup>1</sup> <sup>¼</sup> <sup>S</sup>Vð Þ <sup>d</sup> <sup>0</sup> is a similar space of piecewise polynomial functions but with the breaking points in half integers. It follows that every function in Vð Þ <sup>d</sup> <sup>0</sup> also belongs to <sup>V</sup>ð Þ <sup>d</sup> <sup>1</sup> , giving rise to a multiresolution analysis.

A generator for Vð Þ <sup>d</sup> <sup>0</sup> can easily be obtained as a suitable translation (necessary to align the breaking points) of

$$\mathbf{rect}^{\*(d)}(\mathbf{x}) = \begin{cases} \mathbf{rect}(\mathbf{x}) = \begin{cases} 1 & \text{if } |\mathbf{x}| < 1/2 \\ 0 & \text{otherwise} \end{cases} & \text{if } d = 0 \\ \mathbf{rect}^{\*(d-1)\*}\mathbf{rect}(\mathbf{x}) & \text{if } d > 0 \end{cases} \tag{59}$$

that is, the rect convolved with itself d times. Function rect<sup>∗</sup>ð Þ <sup>d</sup> has compact support but it is not, however, orthogonal to its own translations. It can be orthogonalized with the Fourier trick, but the result has no compact support. For more details about this case, see [3].

### 3.1 Compactly supported wavelets: Daubechies' wavelets

In the examples above, we found multiresolution analysis whose scaling function had at most two out of the following three desirable characteristics: orthogonality, smoothness, and compact support. Is it possible to find a wavelet that has all Wavelets for Differential Equations and Numerical Operator Calculus DOI: http://dx.doi.org/10.5772/intechopen.82820

these three characteristics? The answer was given by Daubechies. It turns out that imposing all the three characteristics is very demanding and only a small family of wavelets exists.

An easy observation is that if ϕ is orthogonal to its own translations, the coefficients gi in the two-scale equation can be obtained as

$$\mathbf{g}\_i = \left\langle \phi, \pi^{i/2} \mathcal{S} \phi \right\rangle \tag{60}$$

According to (60) if ϕ has compact support, then gi has a finite number of coefficients that are different from zero. Since gi needs to be orthogonal to its even translations, its length (i.e., the number of nonzero coefficients) must be necessarily even [3].

Moreover, if the iteration of operator O in (39) converges, it is easy to see that gi has a finite number of coefficients, and then the limit function has compact support. This suggests that it "suffices" to find a finite length sequence gi that is orthogonal to its own even translation and iterates operator O to obtain the desired scaling functions. It actually turns out that this can be done, although there are lots of technical details to be taken care of (e.g., about convergence of O<sup>k</sup> and smoothness of the resulting ϕ); see [5] for details.

Every member of the Daubechies family is identified by the length 2N of the sequence gi (remember that the length of gi is necessarily even). It can be proven that the resulting scaling function has N vanishing moments and its smoothness grows with N; see Table 1. See also Figure 2 that shows the results of the first three iterations of O (first row), the final scaling function (second row), and the wavelet (third row) of three different Daubechies wavelets.

## 3.2 Extensions

sincð Þ¼ x

<sup>V</sup><sup>0</sup> <sup>¼</sup> span <sup>τ</sup><sup>k</sup>

This suggests to use ϕ<sup>S</sup> ¼ sinc as scaling function. The fact that a two-scale equation is satisfied is easily checked in frequency since V<sup>1</sup> is the space of functions whose Fourier transform is zero outside ½ � �2π; 2π ; therefore, every function of V<sup>0</sup> is

The corresponding wavelet ψ<sup>S</sup> is easily characterized in frequency as the func-

<sup>Ψ</sup>Sð Þ¼ <sup>ω</sup> 1 if <sup>π</sup> , <sup>∣</sup>ω<sup>∣</sup> , <sup>2</sup><sup>π</sup> 0 otherwise

As said above, the Sinc example is somehow the opposite of Haar wavelet: it is arbitrarily differentiable, but it has infinite support; actually, it decays very slowly (as Oð Þ 1=x ), and this introduces several practical issues. Moreover, sequences gi and hi are of infinite length, and they decay slowly too (they do not even have a z-

An example intermediate between Haar and Sinc wavelet is represented by

polynomial functions that are d times differentiable (with continuous derivative),

functions but with the breaking points in half integers. It follows that every func-

(

that is, the rect convolved with itself d times. Function rect<sup>∗</sup>ð Þ <sup>d</sup> has compact support but it is not, however, orthogonal to its own translations. It can be orthogonalized with the Fourier trick, but the result has no compact support. For more

In the examples above, we found multiresolution analysis whose scaling function had at most two out of the following three desirable characteristics: orthogonality, smoothness, and compact support. Is it possible to find a wavelet that has all

�

It is easy to verify that ψ<sup>S</sup> ∈ V1, ψS⊥V1, and τ<sup>2</sup><sup>k</sup>ψS; ψ<sup>S</sup>

with the "breaking points" on the integer, more precisely

�

rectð Þ¼ x

<sup>1</sup> <sup>¼</sup> <sup>S</sup>Vð Þ <sup>d</sup>

8 ><

3.1 Compactly supported wavelets: Daubechies' wavelets

transform), making it difficult to implement it.

ð Þ IR <sup>∩</sup>Cd�<sup>1</sup>; <sup>f</sup> � �

Example 3.3 (spline wavelet).

Vð Þ <sup>d</sup>

tion in Vð Þ <sup>d</sup>

46

<sup>0</sup> <sup>¼</sup> <sup>f</sup> <sup>∈</sup> <sup>L</sup><sup>2</sup>

It is easy to see that Vð Þ <sup>d</sup>

A generator for Vð Þ <sup>d</sup>

align the breaking points) of

details about this case, see [3].

spline spaces of degree d. In this case Vð Þ <sup>d</sup>

<sup>0</sup> also belongs to <sup>V</sup>ð Þ <sup>d</sup>

rect<sup>∗</sup>ð Þ <sup>d</sup> ð Þ¼ <sup>x</sup>

and its translations, that is,

Wavelet Transform and Complexity

contained in V1, as desired.

tion whose Fourier transform is

sin ð Þ πx πx

sinc � �

(55)

(57)

(58)

<sup>k</sup> <sup>∈</sup> <sup>Z</sup> (56)

� � <sup>¼</sup> <sup>δ</sup>k.

<sup>0</sup> is defined as the space of piecewise

o

½ � <sup>k</sup>;kþ<sup>1</sup> <sup>¼</sup> polynomial degree <sup>d</sup>∀k<sup>∈</sup> <sup>Z</sup>

<sup>1</sup> , giving rise to a multiresolution analysis.

1 if∣x∣ , 1=2 0 otherwise

rect<sup>∗</sup>ð Þ <sup>d</sup>�<sup>1</sup> <sup>∗</sup>rectð Þ <sup>x</sup> if <sup>d</sup> . <sup>0</sup>

<sup>0</sup> can easily be obtained as a suitable translation (necessary to

>: , (59)

<sup>0</sup> is a similar space of piecewise polynomial

if d ¼ 0

The construction given above is the original idea of multiresolution analysis. Since the early 1990s, many researchers worked in this field, and many variations and extensions have been introduced. Here we briefly recall those that have more interest in the field of differential equation solutions.

### 3.2.1 Multiwavelets

Multiwavelets are a generalization of standard multiresolution analysis in the sense that now scaling functions and wavelets are vectors of functions. This means that Vn is not generated by the translations of a single function but from the translations of many functions. Every idea of standard multiresolution analysis can be reformulated without much difficulty in this case, with the most notable difference that the two-scale equation now has vector function and coefficients that are matrices. Multiwavelets can accommodate scaling factors different from two and there is a larger choice for compact support wavelets. See [7] for more details.


Table 1.

Hölder β<sup>H</sup> and Sobolev β<sup>S</sup> regularity exponent of Daubechies' wavelets as function of length 2 N of gi .

sometimes called the prediction step, and it is interpreted as a filter that predicts the

The advantage of this form is that, being similar to the Feistel structure used in cryptography [8], it is exactly invertible even if operations are implemented in fixed-point arithmetic. Actually, the invertibility does not depend on the detail of Pi and Ui; that can be anything, even nonlinear. Another interesting advantage of this predict/update idea is that it does not require a regular domain, allowing to bring the wavelet concept to more general contexts. For example, [9] uses this idea to

By now it should be clear how multiresolution analysis can be applied to differential equation solution: by using scaling functions and/or wavelets as basis functions in approximation (6). All the approaches described in Section 2 can be used with wavelet: collocation, Galerkin PDE method, weighted residual method, meshless methods, etc. Before describing some details of few approaches described

What makes wavelet interesting is their multiresolution property and the fact that a wisely chosen wavelet (smooth and/or with many vanishing gradients) has interesting "singularity sensing" properties: in the neighborhood of a singularity (discontinuity, nondifferentiability, etc.), the coefficients decay as a function of scale with a speed that depends on the singularity involved (similar to what Fourier transform does, only on a local level), but away from the singularity, they decade fast [3]. This implies that good approximations can be obtained with few coefficients, using high-resolution decomposition only where it is necessary, reducing the size of the matrices involved in the solution of the PDE. A similar effect can be obtained, for example, in FEM by using a finer mesh around points of large variation. However, using this approach in an adaptive way would require to adjust at running time the mesh, a potentially heavy operation. Wavelets have the potential of employing an adaptive resolution in an easier way. See, for example, [10] for few

While orthogonality is considered an important feature in many theoretically wavelet papers, in the context of differential equation solution, it plays a smaller role. The reason is that basis functions enter in the scalar products associated with the various methods via the differential operator D, and it is not guaranteed that D will preserve orthogonality (that would give rise to many zero entries, that is,

Actually, orthogonality is preserved if the two basis functions have disjoint support in space (since differential operators do not extend the support) or in frequency (since differential operators are translation-invariant and in frequency they become a product). This suggests that in the context of differential equations, compact support and well-localization in frequency are more important than just orthogonality. In a sense, they represent a "robust" orthogonality condition.

It is true that true compact support in frequency is less common than compact support in space. With the exception of few very special and theoretical cases (e.g., Sinc), the best we can get is a rapid decay in frequency. This means that the scalar product of two basis functions separated in frequency will be maybe very small, but not zero. Nevertheless, even this kind of "almost sparseness" can be exploited. A general issue with wavelets is that it can be difficult to impose boundary conditions since they have no natural interpolation property that would make

odd samples from the even ones; filter U is sometimes called update.

Wavelets for Differential Equations and Numerical Operator Calculus

4. Some examples of application of wavelets to PDE

in the literature, it is worth to do some general remarks.

examples of adaptive techniques employing wavelets.

sparser matrices).

Remark 4.1.

49

solve differential equations on the sphere.

DOI: http://dx.doi.org/10.5772/intechopen.82820

Figure 2. Daubechies' wavelets. First three iterations of O (first row), the final scaling function (second row), and the wavelet (third row) of three different Daubechies' wavelets.

### 3.2.2 Second-generation wavelets

Two-scale Eq. (37) and the resulting filter bank-based procedure work well when the data are sampled on a regular grid and/or the functions of interest are defined on IR<sup>d</sup>. Since there are many applications that do not satisfy this requirement (e.g., differential equations on general manifolds), the idea of secondgeneration wavelet has been introduced.

The starting point is the so-called lifting form of filter bank (Figure 1). It is possible to show that any two-channel filter bank (Figure 1) can be implemented as shown in Figure 3. In the lifting approach, the input signal is split into odd and even samples by a serial-to-parallel converter. The first branch is filtered, and the result combined with the other branch; the result of this operation is filtered again and combined with the first branch, and this iterated as long as necessary. Filter P is

Figure 3. Lifting implementation of the two-channel filter banks associated with a wavelet analysis.

## Wavelets for Differential Equations and Numerical Operator Calculus DOI: http://dx.doi.org/10.5772/intechopen.82820

sometimes called the prediction step, and it is interpreted as a filter that predicts the odd samples from the even ones; filter U is sometimes called update.

The advantage of this form is that, being similar to the Feistel structure used in cryptography [8], it is exactly invertible even if operations are implemented in fixed-point arithmetic. Actually, the invertibility does not depend on the detail of Pi and Ui; that can be anything, even nonlinear. Another interesting advantage of this predict/update idea is that it does not require a regular domain, allowing to bring the wavelet concept to more general contexts. For example, [9] uses this idea to solve differential equations on the sphere.
