# Shannon Wavelets Theory

Authors: Carlo Cattani
Journal: Mathematical Problems in Engineering
 (2009)

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## Abstract

Shannon wavelets are studied together with their differential properties (known as connection
coefficients). It is shown that the Shannon sampling theorem can be considered in a more general
approach suitable for analyzing functions ranging in multifrequency bands. This generalization
coincides with the Shannon wavelet reconstruction ofL2(ℝ) functions. The differential properties
of Shannon wavelets are also studied through the connection coefficients. It is shown that Shannon
wavelets are C∞-functions and their any order derivatives can be analytically defined by some kind of
a finite hypergeometric series. These coefficients make it possible to define the wavelet reconstruction
of the derivatives of the Cℓ-functions.


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## Body

## 1. Introduction

Wavelets [1] are localized functions which are a very useful tool in many different applications: signal analysis, data compression, operator analysis, and PDE
solving (see, e.g., [2] and references therein). The main feature of wavelets
is their natural splitting of objects into different scale components [1, 3] according to the multiscale
resolution analysis. For the  L2(ℝ) functions, that is, functions with decay to infinity, wavelets give the best approximation. When the function is localized
in space, that is, the bottom length of the function is within a short interval
(function with a compact support), such as pulses, any other reconstruction,
but wavelets, leads towards undesirable problems such as the Gibbs phenomenon
when the approximation is made in the Fourier basis. In this paper, it is shown
that Shannon wavelets are the most expedient basis for the analysis of impulse
functions (pulses) [4]. The approximation can be simply performed and the
reconstruction by Shannon wavelets range in multifrequency bands. Comparing
with the Shannon sampling theorem where the frequency band is only one, the
reconstruction by Shannon wavelets can be done for functions ranging in
different frequency bands. Shannon sampling theorem [5] plays a fundamental role in
signal analysis and, in particular, for the reconstruction of a signal from a
digital sampling. Under suitable hypotheses (on a given signal function)
a few sets of values (samples) and a preliminary chosen basis (made
by the sinc function) enable us to completely reconstruct the continuous
signal. This reconstruction is alike the reconstruction of a function as a
series expansion (such as polynomial, i.e., Taylor series, or trigonometric
functions, i.e., Fourier series), but for the first time the reconstruction (in
the sampling theorem) makes use of the sinc function, that is a localized
function with decay to zero. Together with the Shannon sampling theorem (and
reconstruction), also the wavelets series become very popular, as well as the
bases with compact support. It has been recognized that on the sinc functions
one can settle the family of Shannon wavelets. The main properties of these
wavelets will be shown and discussed. Moreover, the connection coefficients
[6–9] (also called refinable
integrals) will be computed by giving some finite formulas for any order
derivatives (see also some preliminary results in
[2, 10–12]). These coefficients enable
us to define any order derivatives of the Shannon scaling and wavelet basis
and it is shown that also the derivatives are orthogonal.


## 2. Shannon Wavelets

Sinc function or Shannon scaling function is the
starting point for the definition of the Shannon wavelet family [11]. It can be shown that the
Shannon wavelets coincide with the real part of the harmonic wavelets [2, 10, 13, 14], which are the band-limited complex functions(2.1)ψkn(x)≡def2n/2e4πi(2nx−k)−e2πi(2nx−k)2πi(2nx−k),with n,k∈ℤ.
Harmonic wavelets form an orthonormal basis and give rise to a multiresolution
analysis [1–3, 14, 15]. In the frequency domain,
they are very well localized and defined on compact support intervals, but they
have a very slow decay in the space variable. However, in dealing with real
problems it is more expedient to make use of real basis. By focussing on the
real part of the harmonic family, we can take advantage of the basic properties
of harmonic wavelets together with a more direct physical interpretation of the
basis.Let us take, as scaling functionφ(x),
the sinc function (Figure 1) (2.2)φ(x)=sinc⁡x=defsin⁡πxπx=eπix−e−πix2πixand for the dilated and
translated instances(2.3)φkn(x)=2n/2φ(2nx−k)=2n/2sin⁡π(2nx−k)π(2nx−k)=2n/2eπi(2nx−k)−e−πi(2nx−k)2πi(2nx−k).The parameters n,k give, respectively, a compression (dilation)
of the basic function (2.2) and a translation along the x-axis. The family of translated instances {φ(x−k)} is an orthonormal basis for the banded
frequency functions [5] (Shannon theorem). For this reason, they can be used
to define the Shannon multiresolution analysis as follows. The scaling
functions do not represent a basis, in a functional space, therefore we need to
define a family of functions (based on scaling) which are a basis; they are
called the wavelet functions and the corresponding analysis the multiresolution
analysis.Figure 1
Shannon scaling functionφ(x) (thick line) and wavelet (dashed line) ψ(x).Let(2.4)f^(ω)=f(x)^=def12π∫−∞∞f(x)e−iωxdxbe the Fourier transform of the
function f(x)∈L2(ℝ) and(2.5)f(x)=2π∫−∞∞f^(ω)eiωxdωits inverse transform. The
Fourier transform of (2.2) gives us(2.6)φ^(ω)=12πχ(ω+3π)={1(2π),−π≤ω<π,0,elsewhere,with(2.7)χ(ω)={1,2π≤ω<4π,0,elsewhere.Analogously for the dilated and
translated instances of scaling function it is(2.8)φ^kn(ω)=2−n/22πe−iω(k+1)/2nχ(ω2n+3π).From the given scaling function,
it is possible to define the corresponding wavelet function [1, 15] according to the
following.Theorem 2.1.
The Shannon
wavelet, in the Fourier domain, is(2.9)ψ^(ω)=12πe−iω[χ(2ω)+χ(−2ω)].Proof.
It can be easily shown that the scaling function (2.6)
fulfills the condition(2.10)φ^(ω)=H(ω2)φ^(ω2),which characterizes the
multiresolution analysis [1] with(2.11)H(ω2)=χ(ω+3π).Thus the corresponding wavelet
function can be defined as [1, 15](2.12)ψ^(ω)=e−iωH(ω2±2π)¯φ^(ω2).
WithH(ω/2−2π) we have(2.13)ψ^(ω)=e−iωH(ω2−2π)¯φ^(ω2)=e−iωχ(ω+3π−2π)12πχ(ω2+3π)=12πe−iωχ(ω+π)χ(ω2+3π)=12πe−iωχ(2ω),then analogously with H(ω/2+2π) we obtain(2.14)ψ^(ω)=12πe−iωχ(−2ω),so that (2.9) follows.For the whole family of dilated-translated instances,
it is(2.15)ψ^kn(ω)=2−n/22πe−iω(k+1)/2n[χ(ω2n−1)+χ(−ω2n−1)].The Shannon wavelet function in
the real domain can be obtained from (2.9) by the inverse Fourier transform (Figure 1) (2.16)ψ(x)=sin⁡π(x−1/2)−sin⁡2π(x−1/2)π(x−1/2)=e−2iπx(−i+eiπx+e3iπx+ie4iπx)(π−2πx),and by the space shift and
compression we have the whole family of dilated and translated instances:(2.17)ψkn(x)=2n/2sin⁡π(2nx−k−1/2)−sin⁡2π(2nx−k−1/2)π(2nx−k−1/2).By summarizing (2.3) and (2.17), the Shannon wavelet
theory is based on the following functions [11]:(2.18)φkn(x)=2n/2sin⁡π(2nx−k)π(2nx−k),ψkn(x)=2n/2sin⁡π(2nx−k−1/2)−sin⁡2π(2nx−k−1/2)π(2nx−k−1/2)in the space domain, and
collecting (2.8) and (2.15), we have in the frequency domain(2.19)φ^kn(ω)=2−n/22πe−iωk/2nχ(ω2n+3π),ψ^kn(ω)=−2−n/22πe−iω(k+1/2)/2n[χ(ω2n−1)+χ(−ω2n−1)].The inner product is defined as(2.20)〈f,g〉≡def∫−∞∞f(x)g(x)¯dx,which, according to the Parseval
equality, can be expressed also as(2.21)〈f,g〉≡def∫−∞∞f(x)g(x)¯dx=2π∫−∞∞f^(ω)g^(ω)¯dω=2π〈f^,g^〉,where the bar stands for the
complex conjugate.With respect to the inner product (2.20), we can
show the following theorem [11].Theorem 2.2.
Shannon wavelets are orthonormal functions in the
sense that(2.22)〈ψkn(x),ψhm(x)〉=δnmδhk,with δnm,δhk being the Kroenecker symbols.Proof.
(2.23)〈ψkn(x),ψhm(x)〉=2π〈ψ^kn(ω),ψ^hm(ω)〉=2π∫−∞∞2−n/22πe−iω(k+1/2)/2n[χ(ω2n−1)+χ(−ω2n−1)]2−m/22πeiω(h+1/2)/2m[χ(ω2m−1)+χ(−ω2m−1)]dω=2−(n+m)/22π∫−∞∞e−iω(k+1/2)/2n+iω(h+1/2)/2m[χ(ω2n−1)+χ(−ω2n−1)][χ(ω2m−1)+χ(−ω2m−1)]dωwhich is zero for n≠m.
For n=m it is(2.24)〈ψkn(x),ψhn(x)〉=2−n2π∫−∞∞e−iω(h−k)/2n[χ(ω2n−1)+χ(−ω2n−1)]dωand, according to
(2.7), by the
change of variable ξ=ω/2n−1(2.25)〈ψkn(x),ψhn(x)〉=14π[∫−4π−2πe−2i(h−k)ξdξ+∫2π4πe−2i(h−k)ξdξ].For h=k (and n=m), it is trivially
(2.26)〈ψkn(x),ψkn(x)〉=1.For h≠k,
it is(2.27)∫2π4πe−2i(h−k)ξdξ=i2(h−k)(e−4iπ(h−k)−e−8iπ(h−k))=0,and analogously ∫−4π−2πe−2i(h−k)ξdξ=0.Moreover, we have the following theorem
[11].Theorem 2.3.
The translated instances of the Shannon scaling
functionsφkn(x),
at the level n=0,
are orthogonal in the sense that(2.28)〈φk0(x),φh0(x)〉=δkh,being φk0(x)=defφ(x−k).Proof.
It is(2.29)〈φkn(x),φhm(x)〉=2π〈φ^kn(ω),φ^hm(ω)〉=2π∫−∞∞2−n/22πe−iωk/2nχ(ω2n+3π)2−m/22πeiωh/2mχ(ω2m+3π)dω=2−(n+m)/22π∫−∞∞e−iω(k/2n−h/2m)χ(ω2n+3π)χ(ω2m+3π)dω.When m=n,
we have(2.30)〈φkn(x),φhn(x)〉=2−n2π∫−2nπ2nπe−iω(k−h)/2ndω=2nsin⁡[(h−k)π](h−k)π.Since h,k∈ℤ,
there follows that(2.31)sin⁡[(h−k)π](h−k)π={1,h=k0,h≠k}=δkh,that is,(2.32)〈φkn(x),φhn(x)〉=δkh.When m≠n,
let's say m<n,
we have(2.33)〈φkn(x),φhm(x)〉=2−(n+m)/22π∫−2mπ2mπe−iω(k/2n−h/2m)dω,that is,(2.34)〈φkn(x),φhm(x)〉=2(m+n)/2sin⁡[(h−2m−nk)π](h−2m−nk)π.
Whenm≠n,
the last expression is always different from zero, in fact (since m<n)(2.35)sin⁡[(h−k2|m−n|)π]=0⟹[h−k2|m−n|]π=sπ,s∈ℤthat is,(2.36)h=s+k2|m−n|,h,k,s∈ℤand h∈ℤ only if m=n.
Therefore, in order to have the orthogonality it must
be
m=n,
so that(2.37)〈φkn(x),φhn(x)〉=2nδkh.and, in particular, when n=0,(2.38)〈φk0(x),φh0(x)〉=δkh.As a consequence of this proof we have that(2.39)φk0(h)=δkh(h,k∈ℤ).The scalar product of the
(Shannon) scaling functions with the corresponding wavelets is
characterized by the following [11].Theorem 2.4.
The translated instances of the Shannon scaling
functionsφkn(x),
at the level n=0,
are orthogonal to the Shannon wavelets in the sense that(2.40)〈φk0(x),ψhm(x)〉=0,m≥0,being φk0(x)=defφ(x−k).Proof.
It is(2.41)〈φkn(x),ψhm(x)〉=2π〈φ^kn(ω),ψ^hm(ω)〉=2π∫−∞∞2−n/2e−iωk/2nχ(ω2n+3π)2−m/22πeiω(h+1/2)/2m[χ(ω2m−1)+χ(−ω2m−1)]dω=2−(n+m)/2∫−∞∞e−iωk/2n+iω(h+1/2)/2mχ(ω2n+3π)[χ(ω2m−1)+χ(−ω2m−1)]dωwhich is zero for m≥n≥0 (since, according to (2.7), the compact support
of the characteristic functions do not intersect).
On the contrary, it can be easily seen that, form<n,
it is(2.42)〈φkn(x),ψhm(x)〉=2−(n+m)/2∫2mπ2nπe−iωk/2n+iω(h+1/2)/2mdω=−21+(m+n)/2(ieiπ[2−m+n−1(1+2h)−k]+eiπ(h−2m−nk))2n(1+2h)−21+mkand this product,  in general,
does not vanish.


## 3. Reconstruction of a Function by Shannon Wavelets

Letf(x)∈L2(ℝ) be a function such that for any value of the
parameters n,k∈ℤ,
it is(3.1)|∫−∞∞f(x)φk0(x)dx|≤Akn<∞,|∫0∞f(x)ψkn(x)dx|≤Bkn<∞,and B⊂L2(ℝ) the Paley-Wiener space, that is, the space of
band-limited functions such that,(3.2)suppf^⊂[−b,b],b<∞.For the representation with
respect to the basis (2.18), it is b=π.
According to the sampling theorem (see, e.g., [5]) we have the
the following.Theorem 3.1 (Shannon).
Iff(x)∈L2(ℝ) and suppf^⊂[−π,π], the series(3.3)f(x)=∑k=−∞∞αkφk0(x)uniformly converges to f(x),
and(3.4)αk=f(k).Proof.
In order to compute the values of the coefficients,
we have to evaluate the series in correspondence of the integer:(3.5)f(h)=∑k=−∞∞αkφk0(h)=∑k=−∞∞αkδkh=αh,having taken into account (2.39).
The convergence follows from the hypotheses onf(x).
In particular, the importance of the band-limited frequency can be easily seen
by applying the Fourier transform to (3.3):(3.6)f^(ω)=∑k=−∞∞f(k)φ^k0(x)=(2.8)12π∑k=−∞∞f(k)e−iωkχ(ω+3π)=12πχ(ω+3π)∑k=−∞∞f(k)e−iωkso that(3.7)f^(ω)={12π∑k=−∞∞f(k)e−iωk,ω∈[−π,π],0,ω∉[−π,π].In other words, if the function
is band limited (i.e., with compact support in the frequency domain), it can be
completely reconstructed by a discrete Fourier series. The Fourier coefficients
are the values of the function f(x) sampled at the integers.As a generalization of the Paley-Wiener space, and in
order to generalize the Shannon theorem, we define the spaceBψ⊇B of functions f(x) such that the integrals(3.8)αk=〈f(x),φk0(x)〉=∫−∞∞f(x)φk0(x)dx,βkn=〈f(x),ψkn(x)〉=∫−∞∞f(x)ψkn(x)dxexist and are finite. According
to (2.20) and (2.21), it is in the Fourier domain that(3.9)αk=2π〈f(x)^,φk0(x)^〉=∫−∞∞f^(ω)φ^k0(ω)dω=∫02πf^(ω)eiωkdω,βkn=2π〈f(x)^,ψkn(x)^〉=⋯=2−n/2∫2n+1π2n+2πf^(ω)eiωk/2ndω.
Let us prove the
following.Theorem 3.2 (Shannon generalization).
Iff(x)∈Bψ⊂L2(ℝ) and suppf^⊆ℝ, the series(3.10)f(x)=∑h=−∞∞αhφh0(x)+∑n=0∞∑k=−∞∞βknψkn(x)converges to f(x),
with αh and βkn given by (3.8) and
(3.9).
In particular, when suppf^⊆[−2Nπ,2Nπ], it is(3.11)f(x)=∑h=−∞∞αhφh0(x)+∑n=0N∑k=−∞∞βknψkn(x).Proof.
The representation (3.10) follows from the
orthogonality of the scaling and Shannon wavelets (Theorems
2.2,
2.3,
2.4). The
coefficients, which exist and are finite, are given by (3.8). The convergence
of the series is a consequence of the wavelet axioms.
It should be noticed that(3.12)suppf^=[−π,π]⋃n=0,…,∞[−2n+1π,−2nπ]∪[2nπ,2n+1π]so that for a band-limited
frequency signal, that is, for a signal whose frequency belongs to the first
band [−π,π],
this theorem reduces to the Shannon. But, more in general, one has to deal with
a signal whose frequency range in different bands, even if
practically banded, since it is N<∞.
In this case, we have some nontrivial contributions to the series coefficients
from all the bands, ranging from [−2Nπ,2Nπ]:(3.13)suppf^=[−π,π]⋃n=0,…,N[−2n+1π,−2nπ]∪[2nπ,2n+1π].In the frequency domain, (3.10)
gives(3.14)f(x)=∑h=−∞∞αhφh0(x)+∑n=0∞∑k=−∞∞βknψkn(x),f^(ω)=∑h=−∞∞αhφ^h0(ω)+∑n=0∞∑k=−∞∞βknψ^kn(ω),f^(ω)=(2.19)12π∑h=−∞∞αhe−iωhχ(ω+3π)+12π∑n=0∞∑k=−∞∞2−n/2βkne−iω(k+1)/2n[χ(ω2n−1)+χ(−ω2n−1)],that is,(3.15)f^(ω)=12πχ(ω+3π)∑h=−∞∞αhe−iωh+12πχ(ω2n−1)∑n=0∞∑k=−∞∞2−n/2βkne−iω(k+1)/2n+12πχ(−ω2n−1)∑n=0∞∑k=−∞∞2−n/2βkne−iω(k+1)/2n.There follows that the Fourier
transform is made by the composition of coefficients at different frequency
bands. When βkn=0,
for all n,k∈ℤ,
we obtain the Shannon sampling theorem as a special case.Of course, if we limit the dilation factorn≤N<∞,
for a truncated series, we have the approximation of f(x),
given by(3.16)f(x)≅∑h=−SSαhφ(x−h)+∑n=0N∑k=−MMβknψkn(x).By rearranging the many terms of
the series with respect to the different scales, for a fixed N we have(3.17)f(x)≅∑h=−SSαhφ(x−h)+∑n=0Nfn(x),fn(x)=∑k=−MMβknψkn(x),where fn(x) represent the component of the function f(x) at the scale 0≤n≤N (i.e., in the band [2nπ,2n+1π]), and f(x) results from a multiscale approximation or
better from the multiband reconstruction.


### 3.1. Examples

Let us first compute the approximate wavelet
representation of the even function(3.18)f(x)=e−4x2cos⁡2πx.The bottom length (i.e., the main part) of the
functionf(x) is concentrated in the interval [−0.2,0.2].
With a low scale n=3, we can have a good approximation
(Figures 2,
4)
of the function even with a small number k of translation. In fact, with |k|≤3 the absolute value of the approximation error
is less than 7% (see Figure 4). The higher number of the
translation parameter k improves the approximation of the function on
its “tails,” in the sense that by increasing the number of translation
parametersk the oscillation on “tails” is reduced.
We can see that with |k|≤10 the approximation error is reduced up to 3%.
Moreover, the approximation error tends to zero with |x|→∞.Shannon wavelet reconstruction (dashed) of the
even functionf(x)=e−4x2cos2πx,
with nmax=3,−3≤k≤3  (bottom right). Scale approximation with
(a) n=0,−3≤k≤3, (b) n=1,−3≤k≤3, (c) n=2,−3≤k≤3,
(d) n=3,−3≤k≤3,
(e) 0≤n≤3,−3≤k≤3,
(f) n=3,−5≤k≤5.
(a)(b)(c)(d)(e)(f)The multiscale representation is given by(3.19)f(x)≅α0φ(x)+∑n=03fn(x),fn(x)=∑k=−33βknψkn(x),so that at the higher scales
there are the higher frequency oscillations (see Figure 2).
It should be also
noticed that the lower scale approximations f0(x),f1(x),f2(x) represent the major content of the
amplitude. In other words, f0(x)+f1(x)+f2(x) gives a good representation of
(3.18) in the origin, while f3(x),
with its higher oscillations, makes a good approximation of the tails of (3.18).
Therefore, if we are interested in the evolution of the peak in the origin, we
can restrict ourselves to the analysis of the lower scales. If we are
interested in the evolution either of the tails or the high frequency, we must
take into consideration the higher scales (in our case f3(x)).If we compare the Shannon wavelet reconstruction with
the Fourier integral approach, in the Fourier method the
following hold.(1)
It is impossible to have a series expansion
except for the periodic functions.(2)
It is impossible to focus, as it is done with
the Shannon series, on the contribution of each basis to the function. In other words, the
projection off(x) on each term {cosξx,sinξx} of the Fourier basis is not
evident. There follows that it is impossible to decompose the
profile with the components at different scales.(3)
The integral transform performs an integral
over the whole real axis for a function which is substantially zero (overℝ), except in the “small” interval (−ε,ε).As a second example, let us consider the approximate
wavelet representation of the odd function(3.20)f(x)=e−(16x)2/2+e−4x2sin⁡2πx.The bottom length (i.e., the main part) of the
functionf(x) is concentrated in the interval [−0.2,0.2].
Also in this case, for a localized function, with a low scale n=3 we can have a good approximation
(Figures 3,
4)
of the function even with a small number k of translation. However, in this case, the
error can be reduced (around the origin) by adding some translated instances,
but it remains nearly constant far from the  origin. In fact, with |k|≤3 the absolute value of the approximation error
is less than 10% (8% in the origin, Figure 4). The higher number of
the translation parameter k improves the approximation of the function on
its “tails,” in the sense that by increasing the number of
translation parameters
k the oscillation on “tails” is reduced
and becomes constant (around 10%). But we can see that with |k|≤10 the approximation error in the origin is
reduced up to 3%.Shannon wavelet reconstruction (dashed) of the
odd functionf(x)=e−(16x)2/2+e−4x2sin2πx,
with N=nmax=3,−3≤k≤3     (bottom right). Scale approximation with (a) n=0,−3≤k≤3,
(b) n=1,−3≤k≤3,
(c) n=2,−3≤k≤3,
(d) n=3,−3≤k≤3,
(e) 0≤n≤3,−3≤k≤3,
(f) n=3,−5≤k≤5.
(a)(b)(c)(d)(e)(f)Error of the Shannon wavelet reconstruction
of the even function (top)f(x)=e−4x2cos2πx,
with N=nmax=3 and the odd function f(x)=e−(16x)2/2+e−4x2sin2πx,
with N=nmax=3 (bottom right) with different values of kmax.
(a)(b)(c)(d)


## 3.1. Examples

Let us first compute the approximate wavelet
representation of the even function(3.18)f(x)=e−4x2cos⁡2πx.The bottom length (i.e., the main part) of the
functionf(x) is concentrated in the interval [−0.2,0.2].
With a low scale n=3, we can have a good approximation
(Figures 2,
4)
of the function even with a small number k of translation. In fact, with |k|≤3 the absolute value of the approximation error
is less than 7% (see Figure 4). The higher number of the
translation parameter k improves the approximation of the function on
its “tails,” in the sense that by increasing the number of translation
parametersk the oscillation on “tails” is reduced.
We can see that with |k|≤10 the approximation error is reduced up to 3%.
Moreover, the approximation error tends to zero with |x|→∞.Shannon wavelet reconstruction (dashed) of the
even functionf(x)=e−4x2cos2πx,
with nmax=3,−3≤k≤3  (bottom right). Scale approximation with
(a) n=0,−3≤k≤3, (b) n=1,−3≤k≤3, (c) n=2,−3≤k≤3,
(d) n=3,−3≤k≤3,
(e) 0≤n≤3,−3≤k≤3,
(f) n=3,−5≤k≤5.
(a)(b)(c)(d)(e)(f)The multiscale representation is given by(3.19)f(x)≅α0φ(x)+∑n=03fn(x),fn(x)=∑k=−33βknψkn(x),so that at the higher scales
there are the higher frequency oscillations (see Figure 2).
It should be also
noticed that the lower scale approximations f0(x),f1(x),f2(x) represent the major content of the
amplitude. In other words, f0(x)+f1(x)+f2(x) gives a good representation of
(3.18) in the origin, while f3(x),
with its higher oscillations, makes a good approximation of the tails of (3.18).
Therefore, if we are interested in the evolution of the peak in the origin, we
can restrict ourselves to the analysis of the lower scales. If we are
interested in the evolution either of the tails or the high frequency, we must
take into consideration the higher scales (in our case f3(x)).If we compare the Shannon wavelet reconstruction with
the Fourier integral approach, in the Fourier method the
following hold.(1)
It is impossible to have a series expansion
except for the periodic functions.(2)
It is impossible to focus, as it is done with
the Shannon series, on the contribution of each basis to the function. In other words, the
projection off(x) on each term {cosξx,sinξx} of the Fourier basis is not
evident. There follows that it is impossible to decompose the
profile with the components at different scales.(3)
The integral transform performs an integral
over the whole real axis for a function which is substantially zero (overℝ), except in the “small” interval (−ε,ε).As a second example, let us consider the approximate
wavelet representation of the odd function(3.20)f(x)=e−(16x)2/2+e−4x2sin⁡2πx.The bottom length (i.e., the main part) of the
functionf(x) is concentrated in the interval [−0.2,0.2].
Also in this case, for a localized function, with a low scale n=3 we can have a good approximation
(Figures 3,
4)
of the function even with a small number k of translation. However, in this case, the
error can be reduced (around the origin) by adding some translated instances,
but it remains nearly constant far from the  origin. In fact, with |k|≤3 the absolute value of the approximation error
is less than 10% (8% in the origin, Figure 4). The higher number of
the translation parameter k improves the approximation of the function on
its “tails,” in the sense that by increasing the number of
translation parameters
k the oscillation on “tails” is reduced
and becomes constant (around 10%). But we can see that with |k|≤10 the approximation error in the origin is
reduced up to 3%.Shannon wavelet reconstruction (dashed) of the
odd functionf(x)=e−(16x)2/2+e−4x2sin2πx,
with N=nmax=3,−3≤k≤3     (bottom right). Scale approximation with (a) n=0,−3≤k≤3,
(b) n=1,−3≤k≤3,
(c) n=2,−3≤k≤3,
(d) n=3,−3≤k≤3,
(e) 0≤n≤3,−3≤k≤3,
(f) n=3,−5≤k≤5.
(a)(b)(c)(d)(e)(f)Error of the Shannon wavelet reconstruction
of the even function (top)f(x)=e−4x2cos2πx,
with N=nmax=3 and the odd function f(x)=e−(16x)2/2+e−4x2sin2πx,
with N=nmax=3 (bottom right) with different values of kmax.
(a)(b)(c)(d)


## 4. Reconstruction of the Derivatives

Letf(x)∈L2(ℝ) and let f(x) be a differentiable function f(x)∈Cp with p sufficiently high. The reconstruction of a
function f(x) given by (3.10) enables us to compute also its
derivatives in terms of the wavelet decomposition(4.1)dℓdxℓf(x)=∑h=−∞∞αhdℓdxℓφh0(x)+∑n=0∞∑k=−∞∞βkndℓdxℓψkn(x),so that, according to (3.10),
the derivatives of f(x) are known when the derivatives(4.2)dℓdxℓφh0(x),dℓdxℓψkn(x)are given.By a direct computation, we can easily evaluate the
first and second derivatives of the scaling function(4.3)ddxφkn(x)=2n[−1+(2nx−k)πcot⁡((2nx−k)π)]2nx−kφkn(x),d2dx2φkn(x)=22n{2−[π(2nx−k)]2−2π(2nx−k)cot⁡((2nx−k)π)}(2nx−k)2φkn(x),respectively. However, on this
way, higher-order derivatives cannot be easily expressed. Indeed, according to
(3.10), we have to compute the wavelet decomposition of the derivatives:(4.4)dℓdxℓφh0(x)=∑k=−∞∞λhk(ℓ)φk0(x)+∑n=0∞∑k=−∞∞Λhk(ℓ)nψkn(x),dℓdxℓψhm(x)=∑k=−∞∞Γhk(ℓ)mφk0(x)+∑n=0∞∑k=−∞∞γhk(ℓ)mnψkn(x),
with
(4.5)λkh(ℓ)≡def〈dℓdxℓφk0(x),φh0(x)〉,γkh(ℓ)nm≡def〈dℓdxℓψkn(x),ψhm(x)〉,(4.6)Λkh(ℓ)n≡def〈dℓdxℓφk0(x),ψhn(x)〉,Γhk(ℓ)m≡def〈dℓdxℓψhn(x),φh0(x)〉,
being
the connection coefficients [6–9, 11] (or refinable integrals).Their computation can be easily performed in the Fourier
domain, thanks to equality (2.21). In fact, in the Fourier domain
the ℓ-order derivatives of the
(scaling) wavelet functions are(4.7)dℓdxℓφkn(x)^=(iω)ℓφ^kn(ω),dℓdxℓψkn(x)^=(iω)ℓψ^kn(ω)and according to
(2.19),(4.8)dℓdxℓφkn(x)^=(iω)ℓ2−n/22πe−iωk/2nχ(ω2n+3π),dℓdxℓψkn(x)^=−(iω)ℓ2−n/22πe−iω(k+1/2)/2n[χ(ω2n−1)+χ(−ω2n−1)].Taking into account (2.21), we can easily compute the
connection coefficients in the frequency domain(4.9)λkh(ℓ)=2π〈dℓdxℓφk0(x)^,φh0(x)^〉,γkh(ℓ)nm=2π〈dℓdxℓψkn(x)^,ψhm(x)^〉,with the derivatives given by
(4.8).For the explicit computation, we need some preliminary
theorems (for a sketch of the proof see also [11]).Theorem 4.1.
For givenm∈ℤ, ℓ∈ℕ,
it is(4.10)∫xℓemxdx=(1−|μ(m)|)xℓ+1ℓ+1+μ(m)emx|m|ℓ+1∑s=1ℓ+1(−1)[1+μ(m)](2ℓ−s+1)/2ℓ!(|m|x)ℓ−s+1(ℓ−s+1)!+Const,where(4.11)μ(m)=sign(m)={1,m>0,−1,m<0,0,m=0.Proof.
Whenm=0,
(4.10) trivially follows. When m≠0,
by a partial integration we get the iterative formula(4.12)∫xℓemxdx={μ(m)1|m|emx,ℓ=0,μ(m)1|m|[xℓemx−ℓ∫xℓ−1emxdx],ℓ>0,from where by the explicit
computation of iterative terms and rearranging the many terms, (4.10) holds.The following corollary
follows. From Theorem4.1, after a substitution x→iξ, we have the following corollary.Corollary 4.2.
For
givenm∈ℤ, ℓ∈ℕ,
it is(4.13)∫(iξ)ℓeimξdξ=iℓ(1−|μ(m)|)ξℓ+1ℓ+1−iμ(m)eimξ∑s=1ℓ+1(−1)[1+μ(m)](2ℓ−s+1)/2ℓ!(iξ)ℓ−s+1(ℓ−s+1)!|m|s+Const.In
particular, taking into account that(4.14)eikπ=(−1)k={1,k=±2s,−1,k=±(2s+1),s∈ℕ,
we have the
following corollary.Corollary 4.3.
For givenm∈ℤ∪{0}, ℓ∈ℕ, and n∈ℕ,
it is(4.15)∫−nπnπ(iξ)ℓeimξdξ=iℓ(1−|μ(m)|)(nπ)ℓ+1[1+(−1)ℓ]ℓ+1+iμ(m)(−1)mn+1∑s=1ℓ+1(−1)[1+μ(m)](2ℓ−s+1)/2ℓ!(inπ)ℓ−s+1(ℓ−s+1)!|m|s[1−(−1)ℓ−s+1].More in general, the following corollary
holds.Corollary 4.4.
For givenm∈ℤ, ℓ∈ℕ, and a,b∈ℤ(a<b), it is(4.16)∫aπbπ(iξ)ℓeimξdξ=iℓ(1−|μ(m)|)πℓ+1(bℓ+1−aℓ+1)ℓ+1−iμ(m)∑s=1ℓ+1(−1)[1+μ(m)](2ℓ−s+1)/2ℓ!(iπ)ℓ−s+1(ℓ−s+1)!|m|s[(−1)mbbℓ−s+1−(−1)maaℓ−s+1].As a particular case, the following
corollaries hold.Corollary 4.5.
For givenm∈ℤ, ℓ∈ℕ, and b∈ℤ(0<b), it is(4.17)∫0bπ(iξ)ℓeimξdξ=iℓ(1−|μ(m)|)πℓ+1bℓ+1ℓ+1−iμ(m)[∑s=1ℓ(−1)[1+μ(m)](2ℓ−s+1)/2ℓ!(iπ)ℓ−s+1(−1)mbbℓ−s+1(ℓ−s+1)!|m|s+(−1)(1+μ(m))ℓ/2ℓ![(−1)mb−1]|m|ℓ+1].Corollary 4.6.
For
givenm∈ℤ, ℓ∈ℕ, it is(4.18)∫02π(iξ)ℓeimξdξ=iℓ(1−|μ(m)|)(2π)ℓ+1ℓ+1−iμ(m)∑s=1ℓ(−1)[1+μ(m)](2ℓ−s+1)/2ℓ!(2iπ)ℓ−s+1(ℓ−s+1)!|m|s.Thus we can show that the following
theorem holds.Theorem 4.7.
The any order connection
coefficients (4.5)1of the scaling functions φk0(x) are(4.19)λkh(ℓ)={(−1)k−hiℓ2π∑s=1ℓℓ!πss![i(k−h)]ℓ−s+1[(−1)s−1],k≠h,iℓπℓ+12π(ℓ+1)[1+(−1)ℓ],k=h,
or, shortly,(4.20)λkh(ℓ)=iℓπℓ2(ℓ+1)[1+(−1)ℓ](1−|μ(k−h)|)+(−1)k−h|μ(k−h)|iℓ2π∑s=1ℓℓ!πss![i(k−h)]ℓ−s+1[(−1)s−1].Proof.
From (4.9), (4.8),
(4.7), (2.21), (2.19), it is(4.21)λkh(ℓ)=12π∫−∞∞(iω)ℓe−i(k−h)ωχ(ω+3π)χ(ω+3π)dω,that is,(4.22)λkh(ℓ)=12π∫−∞∞(iω)ℓe−i(k−h)ωχ(ω+3π)χ(ω+3π)dω=12π∫−ππ(iω)ℓe−i(k−h)ωdω=iℓ2π∫−ππωℓe−i(k−h)ωdω.The last integral, according to
(4.15)
(with n=1), gives (4.20).Thus we have at the lower-order derivativesℓ≤5(4.23)λkh(1)=−(−1)k−hk−h,λ00(1)=0,λkh(2)=−2(−1)k−h(k−h)2,λ00(2)=−π23,λkh(3)=(−1)k−h(k−h)2π2−6(k−h)3,λ00(3)=0,λkh(4)=4(−1)k−h(k−h)2π2−6(k−h)4,λ00(4)=π45,λkh(5)=(−1)k−h(k−h)4π4−20(k−h)2π2+120(k−h)5,λ00(5)=0.Analogously for the connection coefficients (4.5)2,
we have the following theorem.Theorem 4.8.
The any
order connection coefficients (4.5)2 of the Shannon wavelets (2.18)2 are (4.24)γkh(ℓ)nm=δnm{iℓ(1−|μ(h−k)|)πℓ2nℓ−1ℓ+1(2ℓ+1−1)(1+(−1)ℓ)+μ(h−k)∑s=1ℓ+1(−1)[1+μ(h−k)](2ℓ−s+1)/2ℓ!iℓ−sπℓ−s(ℓ−s+1)!|h−k|s(−1)−s−2(h+k)2nℓ−s−1×{2ℓ+1[(−1)4h+s+(−1)4k+ℓ]−2s[(−1)3k+h+ℓ+(−1)3h+k+s]}},respectively, for ℓ≥1,
and γkh(0)nm=δkhδnm.Proof.
From (4.9), (4.8),
(4.7), (2.21), (2.19), it is(4.25)γkh(ℓ)nm=def〈dℓdxℓψkn(x),ψhm(x)〉=(4.9)2π〈dℓdxℓψkn(x)^,ψhm(x)^〉=(4.7)2π〈(iω)ℓψ^kn(ω),ψ^hm(ω)〉=(2.21)2π∫−∞∞(iω)ℓψkn(ω)ψ^hm(ω)¯dω=(2.19)2π∫−∞∞(iω)ℓ2−n/22πe−iω(k+1/2)/2n[χ(ω2n−1)+χ(−ω2n−1)]×2−m/22πeiω(h+1/2)/2m[χ(ω2m−1)+χ(−ω2m−1)]dω,from where, according to the
definition (2.7), it is(4.26)γkh(ℓ)nm=0,n≠m,and (for n=m)(4.27)γkh(ℓ)nn=2−n2π∫−∞∞(iω)ℓe−iω(k−h)/2n[χ(ω2n−1)+χ(−ω2n−1)]dω=2−n2π[∫−2n+1π−2nπ(iω)ℓe−iω(k−h)/2ndω+∫2nπ2n+1π(iω)ℓe−iω(k−h)/2ndω].By taking into account (4.16),
(4.24) is proven.Theorem 4.9.
The connection coefficients are
recursively given by the matrix at the lowest scale level:(4.28)γkh(ℓ)nn=2ℓ(n−1)γkh(ℓ)11.Moreover,
it is(4.29)γkh(2ℓ+1)nn=−γhk(2ℓ+1)nn,γkh(2ℓ)nn=γhk(2ℓ)nn.Let us now prove that the mixed connection
coefficients (4.6) are zero. It is enough to show the following
theorem.Theorem 4.10.
The mixed coefficients (4.6)1 of the Shannon wavelets are(4.30)Λkh(ℓ)n=0.Proof.
From (4.9), (4.8),
(4.7), (2.21), (2.19), it is(4.31)Λkh(ℓ)n=def〈dℓdxℓφk0(x),ψhm(x)〉=2π〈dℓdxℓφk0(x)^,ψhm(x)^〉=(4.7)2π〈(iω)ℓφ^k0(ω),ψ^hm(ω)〉=(2.21)2π∫−∞∞(iω)ℓφk0(ω)ψ^hm(ω)¯dω=(2.19)2π∫−∞∞(iω)ℓ2−n/22πe−iωk/2nχ(ω2n+3π)×2−m/22πeiω(h+1/2)/2m[χ(ω2m−1)+χ(−ω2m−1)]dω,from where, since(4.32)χ(ω2n+3π)[χ(ω2m−1)+χ(−ω2m−1)]=0,the theorem is proven.As a consequence, we have that theℓ-order derivatives of the Shannon scaling and
wavelets are(4.33)dℓdxℓφh0(x)=∑k=−∞∞λhk(ℓ)φk0(x),dℓdxℓψhm(x)=∑n=0∞∑k=−∞∞γhk(ℓ)mnψkn(x).In other words, the following
theorem holds.Theorem 4.11.
The derivatives of the Shannon
scaling function are orthogonal to the derivatives of the Shannon wavelets(4.34)〈dℓdxℓφh0(x),dpdxpψhm(x)〉=0.Proof.
It follows directly from (4.33) and
the orthogonality of the Shannon functions according  to Theorem 2.4.


### 4.1. First- and Second-Order Connection Coefficients

For the first and second derivatives of the Shannon
wavelets, we have (see [11])(4.35)ddxψkn(x)=∑h=−∞∞γkh′nnψhn(x),d2dx2ψkn(x)=∑h=−∞∞γkh′′nnψhn(x),with (4.24)(4.36)γkh′nn=μ(h−k)∑s=12(−1)[1+μ(h−k)](2−s+1)/2i1−sπ1−s(2−s)!|h−k|s(−1)−s−2(h+k)2n−s−1×{4[(−1)4h+s+(−1)4k+1]−2s[(−1)3k+h+1+(−1)3h+k+s]},γkh′′nn=−(1−|μ(h−k)|)π222n+μ(h−k)∑s=13(−1)[1+μ(h−k)](5−s)/22i2−sπ2−s(3−s)!|h−k|s(−1)−s−2(h+k)22n−s−1×{8[(−1)4h+s+(−1)4k+2]−2s[(−1)3k+h+2+(−1)3h+k+s]},
respectively.A disadvantage in
(4.33) is that derivatives are expressed as infinite sum. However, since the
wavelets are mainly localized in a short range interval, a good approximation
can be obtained with a very few terms of the series. The main advantage of (4.33)
is that the derivatives are expressed in terms of the wavelet basis.Analogously, we obtain for the first and second
derivative of the scaling function(4.37)ddxφk0(x)=∑h=−∞∞λkh′φh0(x),d2dx2φk0(x)=∑h=−∞∞λkh′′φh0(x),with (4.20)(4.38)λkh′=(−1)h−kμ(h−k)∑s=12(−1)[1+μ(h−k)](3−s)/2i1−sπ1−s2(2−s)!|h−k|s[1+(−1)1−s],λkh′′=−(1−|μ(h−k)|)π22+(−1)h−kμ(h−k)∑s=13(−1)[1+μ(h−k)](5−s)/2i2−sπ2−s(3−s)!|h−k|s[1+(−1)2−s].f The coefficients of derivatives are real values as can
be shown by a direct computation(4.39)γkh′11k=−2k=−1k=0k=1k=2h=−20−12−14−16−18h=−1120−12−14−16h=014120−12−14h=11614120−12h=2181614120(4.40)γkh′22k=−2k=−1k=0k=1k=2h=−20−1−12−13−14h=−110−1−12−13h=01210−1−12h=1131210−1h=214131210 If we consider a dyadic discretization of the x-axis such that(4.41)xk=2−n(k+12),k∈ℤ,that is,(4.42)k=−2k=−1k=0k=1k=2n=0−1.5−0.50.51.52.5n=1−0.75−0.250.250.751.25n=2−0.375−0.1250.1250.3750.625there results(4.43)ψkn(2−n(k+12))=−2n/2,k∈ℤ.Thus (4.33) at dyadic points xk=2−n(k+1/2)
becomes
(4.44)[ddxψkn(x)]x=xk=−2n/2∑h=−∞∞γkhnn,[d2dx2ψkn(x)]x=xk=−2n/2∑h=−∞∞Γkhnn.
For instance, (see the above tables) in x1=2−1(1+1/2),
(4.45)[ddxψ11(x)]x=x1=3/4=−21/2∑h=−∞∞γ1h11≅−21/2∑h=−22γ1h11=−21/2(16+14)=−5212.Analogously, it is(4.46)φkn(2−n(k+12))=21+n/2π,k∈ℤ,from where, in xk=(k+1/2),
it is(4.47)[ddxφk0(x)]x=xk=2π∑h=−∞∞λkh,[d2dx2φk0(x)]x=xk=2π∑h=−∞∞Λkh.Outside the dyadic points, the approximation is quite
good even with low values of the parametersn,k.
For instance, we have (Figure 5) the approximation(4.48)ddxφ00(x)=cos⁢⁡πxx−sin⁡πxπx2≅∑h=−55λ0hφh0(x).Figure 5
Approximation (plain) of the first derivative of the functionφ00(x) (bold) by using the connection coefficients.


## 4.1. First- and Second-Order Connection Coefficients

For the first and second derivatives of the Shannon
wavelets, we have (see [11])(4.35)ddxψkn(x)=∑h=−∞∞γkh′nnψhn(x),d2dx2ψkn(x)=∑h=−∞∞γkh′′nnψhn(x),with (4.24)(4.36)γkh′nn=μ(h−k)∑s=12(−1)[1+μ(h−k)](2−s+1)/2i1−sπ1−s(2−s)!|h−k|s(−1)−s−2(h+k)2n−s−1×{4[(−1)4h+s+(−1)4k+1]−2s[(−1)3k+h+1+(−1)3h+k+s]},γkh′′nn=−(1−|μ(h−k)|)π222n+μ(h−k)∑s=13(−1)[1+μ(h−k)](5−s)/22i2−sπ2−s(3−s)!|h−k|s(−1)−s−2(h+k)22n−s−1×{8[(−1)4h+s+(−1)4k+2]−2s[(−1)3k+h+2+(−1)3h+k+s]},
respectively.A disadvantage in
(4.33) is that derivatives are expressed as infinite sum. However, since the
wavelets are mainly localized in a short range interval, a good approximation
can be obtained with a very few terms of the series. The main advantage of (4.33)
is that the derivatives are expressed in terms of the wavelet basis.Analogously, we obtain for the first and second
derivative of the scaling function(4.37)ddxφk0(x)=∑h=−∞∞λkh′φh0(x),d2dx2φk0(x)=∑h=−∞∞λkh′′φh0(x),with (4.20)(4.38)λkh′=(−1)h−kμ(h−k)∑s=12(−1)[1+μ(h−k)](3−s)/2i1−sπ1−s2(2−s)!|h−k|s[1+(−1)1−s],λkh′′=−(1−|μ(h−k)|)π22+(−1)h−kμ(h−k)∑s=13(−1)[1+μ(h−k)](5−s)/2i2−sπ2−s(3−s)!|h−k|s[1+(−1)2−s].f The coefficients of derivatives are real values as can
be shown by a direct computation(4.39)γkh′11k=−2k=−1k=0k=1k=2h=−20−12−14−16−18h=−1120−12−14−16h=014120−12−14h=11614120−12h=2181614120(4.40)γkh′22k=−2k=−1k=0k=1k=2h=−20−1−12−13−14h=−110−1−12−13h=01210−1−12h=1131210−1h=214131210 If we consider a dyadic discretization of the x-axis such that(4.41)xk=2−n(k+12),k∈ℤ,that is,(4.42)k=−2k=−1k=0k=1k=2n=0−1.5−0.50.51.52.5n=1−0.75−0.250.250.751.25n=2−0.375−0.1250.1250.3750.625there results(4.43)ψkn(2−n(k+12))=−2n/2,k∈ℤ.Thus (4.33) at dyadic points xk=2−n(k+1/2)
becomes
(4.44)[ddxψkn(x)]x=xk=−2n/2∑h=−∞∞γkhnn,[d2dx2ψkn(x)]x=xk=−2n/2∑h=−∞∞Γkhnn.
For instance, (see the above tables) in x1=2−1(1+1/2),
(4.45)[ddxψ11(x)]x=x1=3/4=−21/2∑h=−∞∞γ1h11≅−21/2∑h=−22γ1h11=−21/2(16+14)=−5212.Analogously, it is(4.46)φkn(2−n(k+12))=21+n/2π,k∈ℤ,from where, in xk=(k+1/2),
it is(4.47)[ddxφk0(x)]x=xk=2π∑h=−∞∞λkh,[d2dx2φk0(x)]x=xk=2π∑h=−∞∞Λkh.Outside the dyadic points, the approximation is quite
good even with low values of the parametersn,k.
For instance, we have (Figure 5) the approximation(4.48)ddxφ00(x)=cos⁢⁡πxx−sin⁡πxπx2≅∑h=−55λ0hφh0(x).Figure 5
Approximation (plain) of the first derivative of the functionφ00(x) (bold) by using the connection coefficients.


## 5. Conclusion

In this paper,
the theory of Shannon wavelets has been analyzed showing the main properties of
these functions sharply localized in frequency. The reconstruction formula for
theL2(ℝ) functions has been given not only for the
function but also for its derivatives. The derivative of the Shannon wavelets
has been computed by a finite formula (both for the scaling and for the
wavelet) for any order derivative. Indeed, to achieve this task, it was enough
to compute connection coefficients, that is, the wavelet coefficients of the
basis derivatives. These coefficients were obtained as a finite series (for any
order derivatives). In Latto's method [6, 8, 9],
instead, these coefficients were obtained only (for the Daubechies wavelets) by
using the inclusion axiom but in approximated form and only for the first two
order derivatives. The knowledge of the derivatives of the basis enables us to
approximate a function and its derivatives and it is an expedient tool for the projection
of differential operators in the numerical computation of the solution of both
partial and ordinary differential equations [2, 3, 10, 13].



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*Source: 164808-2008-09-29.xml*