for all s with Re(s) > Re(¯). Moreover, let E be the set of the real parts

s

of s such that the Laplace integral exists and denote by » the in¬mum of

E. If » happens to be ¬nite, the Laplace integral exists in the half-plane

Re(s) > ». If » = ’∞ then it exists for all s ∈ C; » is called the abscissa

of convergence.

We recall that the Laplace transform enjoys properties completely analo-

gous to those of the Fourier transform. The inverse Laplace transform is

denoted formally as L’1 and is such that

f (t) = L’1 [L(s)].

10.11 Transforms and Their Applications 457

Example 10.9 Let us consider the ordinary di¬erential equation y (t) + ay(t) =

g(t) with y(0) = y0 . Multiplying by est , integrating between 0 and ∞ and passing

to the Laplace transform, yields

sY (s) ’ y0 + aY (s) = G(s). (10.83)

Should G(s) be easily computable, (10.83) would furnish Y (s) and then, by ap-

plying the inverse Laplace transform, the generating function y(t). For instance,

if g(t) is the unit step function, we obtain

11 1 y0 1

y(t) = L’1 (1 ’ e’at ) + y0 e’at .

’ + =

as s+a s+a a

•

For an extensive presentation and analysis of the Laplace transform see,

e.g., [Tit37]. In the next section we describe a discrete version of the Laplace

transform, known as the Z-transform.

10.11.4 The Z-Transform

De¬nition 10.3 Let f be a given function, de¬ned for any t ≥ 0, and

∆t > 0 be a given time step. The function

∞

f (n∆t)z ’n , z∈C

Z(z) = (10.84)

n=0

is called the Z-transform of the sequence {f (n∆t)} and is denoted by

Z[f (n∆t)].

The parameter ∆t is the sampling time step of the sequence of samples

f (n∆t). The in¬nite sum (10.84) converges if

|z| > R = lim sup n |f (n∆t)|.

n’∞

It is possible to deduce the Z-transform from the Laplace transform as

follows. Denoting by f0 (t) the piecewise constant function such that f0 (t) =

f (n∆t) for t ∈ (n∆t, (n + 1)∆t), the Laplace transform L[f0 ] of f0 is the

function

∞ (n+1)∆t

∞

f0 (t)e’st dt = e’st f (n∆t) dt

L(s) =

n=0 n∆t

0

∞ ∞

e’ns∆t ’ e’(n+1)s∆t 1 ’ e’s∆t

f (n∆t)e’ns∆t .

= f (n∆t) =

s s

n=0 n=0

458 10. Orthogonal Polynomials in Approximation Theory

The discrete Laplace transform Z d [f0 ] of f0 is the function

∞

f (n∆t)e’ns∆t .

d

Z (s) =

n=0

Then, the Z-transform of the sequence {f (n∆t), n = 0, . . . , ∞} coincides

with the discrete Laplace transform of f0 up to the change of variable

z = e’s∆t . The Z-transform enjoys similar properties (linearity, scaling,

convolution and product) to those already seen in the continuous case.

The inverse Z-transform is denoted by Z ’1 and is de¬ned as

f (n∆t) = Z ’1 [Z(z)].

The practical computation of Z ’1 can be carried out by resorting to classi-

cal techniques of complex analysis (for example, using the Laurent formula

or the Cauchy theorem for residual integral evaluation) coupled with an

extensive use of tables (see, e.g., [Pou96]).

10.12 The Wavelet Transform

This technique, originally developed in the area of signal processing, has

successively been extended to many di¬erent branches of approximation

theory, including the solution of di¬erential equations. It is based on the

so-called wavelets, which are functions generated by an elementary wavelet

through traslations and dilations. We shall limit ourselves to a brief intro-

duction of univariate wavelets and their transform in both the continuous

and discrete cases referring to [DL92], [Dau88] and to the references cited

therein for a detailed presentation and analysis.

10.12.1 The Continuous Wavelet Transform

Any function

t’„

1

hs,„ (t) = √ h t∈R

, (10.85)

s

s

that is obtained from a reference function h ∈ L2 (R) by means of traslations

by a traslation factor „ and dilations by a positive scaling factor s is called

a wavelet. The function h is called an elementary wavelet.

Its Fourier transform, written in terms of ω = 2πν, is

√

Hs,„ (ω) = sH(sω)e’iω„ , (10.86)

where i denotes the imaginary unit and H(ω) is the Fourier transform of

the elementary wavelet. A dilation t/s (s > 1) in the real domain produces

10.12 The Wavelet Transform 459

therefore a contraction sω in the frequency domain. Therefore, the factor

1/s plays the role of the frequency ν in the Fourier transform (see Section

10.11.1). In wavelets theory s is usually referred to as the scale. Formula

(10.86) is known as the ¬lter of the wavelet transform.

De¬nition 10.4 Given a function f ∈ L2 (R), its continuous wavelet trans-

form Wf = W[f ] is a decomposition of f (t) onto a wavelet basis {hs,„ (t)},

that is

∞

¯

Wf (s, „ ) = f (t)hs,„ (t) dt, (10.87)

’∞

where the overline bar denotes complex conjugate.

When t denotes the time-variable, the wavelet transform of f (t) is a func-

tion of the two variables s (scale) and „ (time shift); as such, it is a repre-

sentation of f in the time-scale space and is usually referred to as time-scale

joint representation of f . The time-scale representation is the analogue of

the time-frequency representation introduced in the Fourier analysis. This

latter representation has an intrinsic limitation: the product of the res-

olution in time ∆t and the resolution in frequency ∆ω must satisfy the

following constraint (Heisenberg inequality)

1

∆t∆ω ≥ (10.88)

2

which is the counterpart of the Heisenberg uncertainty principle in quantum

mechanics. This inequality states that a signal cannot be represented as

a point in the time-frequency space. We can only determine its position

within a rectangle of area ∆t∆ω in the time-frequency space.

The wavelet transform (10.87) can be rewritten in terms of the Fourier

transform F (ω) of f as

√ ∞

s ¯

F (ω)H(sω)eiω„ dω,

Wf (s, „ ) =

2π

’∞

which shows that the wavelets transform is a bank of wavelet ¬lters char-

acterized by di¬erent scales. More precisely, if the scale is small the wavelet

is concentrated in time and the wavelet transform provides a detailed de-

scription of f (t) (which is the signal). Conversely, if the scale is large, the

wavelet transform is able to resolve only the large-scale details of f . Thus,

the wavelet transform can be regarded as a bank of multiresolution ¬lters.

The theoretical properties of this transform do not depend on the partic-

ular elementary wavelet that is considered. Hence, speci¬c bases of wavelets

can be derived for speci¬c applications. Some examples of elementary wave-

lets are reported below.

460 10. Orthogonal Polynomials in Approximation Theory

Example 10.10 (Haar wavelets) These functions can be obtained by choos-

ing as the elementary wavelet the Haar function de¬ned as

±

if x ∈ (0, 1 ),

1 2

’1 if x ∈ ( 1 , 1),

h(x) =

2

0 otherwise.

Its Fourier transform is the complex-valued function

ω

H(ω) = 4ie’iω/2 1 ’ cos( ) /ω,

2

which has symmetric module with respect to the origin (see Figure 10.8). The

bases that are obtained from this wavelet are not used in practice due to their

•

ine¬ective localization properties in the frequency domain.

1.5

1.5

1

1

0.5

0

0.5

’0.5

’1

0

’1.5

’80 ’60 ’40 ’20 0 20 40 60 80

’0.5 0 0.5 1 1.5

FIGURE 10.8. The Haar wavelet (left) and the module of its Fourier transform

(right)

Example 10.11 (Morlet wavelets) The Morlet wavelet is de¬ned as follows

(see [MMG87])

2

h(x) = eiω0 x e’x /2

.

Thus, it is a complex-valued function whose real part has a real positive Fourier

transform, symmetric with respect to the origin, given by

√ 2 2

π e’(ω’ω0 ) /2 + e’(ω+ω0 ) /2 .

H(ω) =

•

We point out that the presence of the dilation factor allows for the wavelets

to easily handle possible discontinuities or singularities in f . Indeed, using

the multi-resolution analysis, the signal, properly divided into frequency

bandwidths, can be processed at each frequency by suitably tuning up the

scale factor of the wavelets.

10.12 The Wavelet Transform 461

1.6

1

0.8 1.4

0.6

1.2

0.4

1

0.2

0.8

0

’0.2

0.6

’0.4

0.4

’0.6

0.2

’0.8

0

’1

’10 ’8 ’6 ’4 ’2 0 2 4 6 8 10

’10 ’8 ’6 ’4 ’2 0 2 4 6 8 10

FIGURE 10.9. The real part of the Morlet wavelet (left) and the real part of the

corresponding Fourier transforms (right) for ω0 = 1 (solid line), ω0 = 2.5 (dashed

line) and ω0 = 5 (dotted line)

Recalling what was already pointed out in Section 10.11.1, the time lo-

calization of the wavelet gives rise to a ¬lter with in¬nite bandwidth. In

particular, de¬ning the bandwidth ∆ω of the wavelet ¬lter as

«∞ 2

∞

∆ω = |H(ω)|2 dω ,

ω 2 |H(ω)|2 dω/

’∞ ’∞

then the bandwidth of the wavelet ¬lter with scale equal to s is

«∞ 2

∞

1

∆ωs = ω 2 |H(sω)|2 dω/ |H(sω)|2 dω = ∆ω.

s

’∞ ’∞

Consequently, the quality factor Q of the wavelet ¬lter, de¬ned as the in-

verse of the bandwidth of the ¬lter, is independent of s since

1/s

Q= = ∆ω

∆ωs

provided that (10.88) holds. At low frequencies, corresponding to large

values of s, the wavelet ¬lter has a small bandwidth and a large temporal

width (called window) with a low resolution. Conversely, at high frequencies

the ¬lter has a large bandwidth and a small temporal window with a high

resolution. Thus, the resolution furnished by the wavelet analysis increases

with the frequency of the signal. This property of adaptivity makes the

wavelets a crucial tool in the analysis of unsteady signals or signals with

fast transients for which the standard Fourier analysis turns out to be

ine¬ective.

10.12.2 Discrete and Orthonormal Wavelets

The continuous wavelet transform maps a function of one variable into a bi-

dimensional representation in the time-scale domain. In many applications

462 10. Orthogonal Polynomials in Approximation Theory

this description is excessively rich. Resorting to the discrete wavelets is

an attempt to represent a function using a ¬nite (and small) number of

parameters.

A discrete wavelet is a continuous wavelet that is generated by using

discrete scale and translation factors. For s0 > 1, denote by s = sj the

0

scale factors; the dilation factors usually depend on the scale factors by

setting „ = k„0 sj , „0 ∈ R. The corresponding discrete wavelet is

0

’j/2 ’j/2

h(s’j (t ’ k„0 sj )) = s0 h(s’j t ’ k„0 ).

hj,k (t) = s0 0 0 0

The scale factor sj corresponds to the magni¬cation or the resolution of

0

the observation, while the translation factor „0 is the location where the

observations are made. If one looks at very small details, the magni¬cation

must be large, which corresponds to large negative index j. In this case the

step of translation is small and the wavelet is very concentrated around the

observation point. For large and positive j, the wavelet is spread out and

large translation steps are used.

The behavior of the discrete wavelets depends on the steps s0 and „0 .

When s0 is close to 1 and „0 is small, the discrete wavelets are close to the

continuous ones. For a ¬xed scale s0 the localization points of the discrete

wavelets along the scale axis are logarithmic as log s = j log s0 . The choice

s0 = 2 corresponds to the dyadic sampling in frequency. The discrete time-

step is „0 sj and, typically, „0 = 1. Hence, the time-sampling step is a

0

function of the scale and along the time axis the localization points of the

wavelet depend on the scale.

For a given function f ∈ L1 (R), the corresponding discrete wavelet trans-

form is

∞

¯

Wf (j, k) = f (t)hj,k (t) dt.

’∞

It is possible to introduce an orthonormal wavelet basis using discrete di-

lation and traslation factors, i.e.

∞

¯ ∀i, j, k, l ∈ Z.

hi,j hk,l (t) dt = δik δjl,

’∞

With an orthogonal wavelet basis, an arbitrary function f can be recon-

structed by the expansion

f (t) = A Wf (j, k)hj,k (t),

j,k∈Z

where A is a constant that does not depend on f .

As of the computational standpoint, the wavelet discrete transform can

be implemented at even a cheaper cost than the FFT algorithm for com-

puting the Fourier transform.

10.13 Applications 463

10.13 Applications

In this section we apply the theory of orthogonal polynomials to solve two

problems arising in quantum physics. In the ¬rst example we deal with

Gauss-Laguerre quadratures, while in the second case the Fourier analysis

and the FFT are considered.

10.13.1 Numerical Computation of Blackbody Radiation

The monochromatic energy density E(ν) of blackbody radiation as a func-

tion of frequency ν is expressed by the following law

ν3

8πh

E(ν) = 3 hν/K T ,

’1

ce B

where h is the Planck constant, c is the speed of light, KB is the Boltz-

mann constant and T is the absolute temperature of the blackbody (see,

for instance, [AF83]).

To compute the total density of monochromatic energy that is emitted

by the blackbody (that is, the emitted energy per unit volume) we must

evaluate the integral

∞ ∞

x3

E(ν)dν = ±T 4