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fachbereich mathematik funktionalanalysis und
SCHRIFTEN ZUR
FUNKTIONALANALYSIS
UND GEOMATHEMATIK
M. J. Fengler, W. Freeden, M. Gutting
The Spherical Bernstein Wavelet
Bericht 20 – Mai 2005
FACHBEREICH MATHEMATIK
THE SPHERICAL BERNSTEIN WAVELET
M. J. FENGLER∗ , W. FREEDEN∗ , M. GUTTING∗
Abstract. In this work we introduce a new bandlimited spherical wavelet: The Bernstein
wavelet. It possesses a couple of interesting properties. To be specific, we are able to construct
bandlimited wavelets free of oscillations. The scaling function of this wavelet is investigated with
regard to the spherical uncertainty principle, i.e., its localization in the space domain as well as
in the momentum domain is calculated and compared to the well-known Shannon scaling function.
Surprisingly, they possess the same localization in space although one is highly oscillating whereas
the other one shows no oscillatory behavior. Moreover, the Bernstein scaling function turns out to
be the first bandlimited scaling function known to the literature whose uncertainty product tends to
the minimal value 1.
Key words. Spherical Wavelets, Bernstein Polynomials, Constructive Approximation on the
Sphere, Uncertainty Principle
AMS subject classifications. 33C55, 42C10, 42C15, 42C40, 65T99
1. Motivation. Over the last decade wavelets have found important applications in numerous areas of mathematics, physics, engineering and computer science.
Wavelets form versatile tools for representing general functions or data sets. They
especially become more and more important in Earth sciences since most recent satellite missions deliver millions of data scattered around the globe. Meanwhile spherical
wavelets introduced by [5, 7, 8, 9] play the fundamental role in the analysis of regional,
high-frequent phenomena observed in geophysical, geodetic, meteorological and seismological applications (see, e.g., [2, 4, 5, 6] and the many references therein).
The spherical wavelets discussed here are based on expansions of Legendre polynomials. Hence, they form radial basis functions on the sphere whose argument depends
only on the spherical distance between the localization of the wavelet and its evaluation point. Moreover, they are usually designed to fulfill the partial differential
equations corresponding to the considered geophysical context (see [9]). In view of
constructive approximation on the sphere the wavelet techniques have shown their
strength in detecting local and regional (also time-dependent) effects (see [5, 6]) for
more than one decade. For example, studies of local changes of mass in the gravitational field observed from satellite data or crustal field modeling from Earth’s magnetic data represent recent examples of application. Also vectorial data are analyzed
by an intrinsic extension of scalar wavelets to vectorial wavelets, e.g., for modeling
atmospherical flows and oceanographic streams. Along this experience the wavelet
techniques have been extended to (non-linear) Galerkin methods for solving spherical
partial differential equations, such as the Navier-Stokes equation (see [2, 3]).
In the literature bandlimited and non-bandlimited wavelets are distinguished. Bandlimited wavelets are generated by a finite series of Legendre polynomials whereas for
non-bandlimited wavelets this Legendre expansion is infinite. Examples for both types
can be found in [5, 6, 9]. However, due to the construction as series of finitely many
Legendre polynomials in frequency domain, it has been common belief that it is not
possible to build spherical bandlimited wavelets that show no oscillations (except for
one that is required to obtain a vanishing zero moment). Just at this point this work
comes into play. We introduce a smooth, bandlimited, spherical wavelet: The spheri∗ University
of Kaiserslautern, Department of
({fengler/freeden/gutting}@mathematik.uni-kl.de).
1
Mathematics,
Geomathematics
Group
2
M. J. FENGLER, W. FREEDEN, M. GUTTING
cal Bernstein wavelet. Its great advantage is that it possesses the appealing property
of fast and stable numerical evaluation because it can be formulated in closed form
by an elementary function. In fact, it has been constructed via its explicit formula,
i.e., in the space domain whereas usually spherical wavelets are generated in the momentum domain via its Legendre coefficients. This different type of construction is
also reflected by the proofs in this article – many properties of the Bernstein wavelet
(especially its localization properties) are shown differently as for other well-known
wavelets. As the Bernstein wavelet shows no oscillatory behavior in space domain it
is also much easier to derive error estimates for spatial truncation as required in many
numerical procedures, e.g., spherical panel clustering [10].
The paper is organized as follows: First, we briefly introduce the mathematical
tools and notations that are needed later on. Second, we define the Bernstein wavelet
in the space domain in closed form and prove that all essential properties of spherical
wavelets are fulfilled. Surprisingly, when comparing the Bernstein scaling function
with kernels known to the literature, we realize that its space uncertainty is exactly
the same as in the Shannon case despite the huge oscillations of the latter. Finally,
we show that the uncertainty product (known as Heisenberg’s uncertainty product)
converges to 1 in case of the bandlimited Bernstein wavelet.
2. Preliminaries. In the following we adopt the notation from the monograph
[9]. The letters N, N0 and R denote the set of positive integers, non-negative integers
and real numbers. We write x, y to represent the elements of the three-dimensional
space R3 endowed with the Euclidian canonical basis {ε1 , ε2 , ε3 }. Then the inner
P3
product
is
denoted
by
x
·
y
=
i=1 xi yi . The corresponding norm of x is given by
√
|x| = x · x. The unit sphere is denoted by Ω, elements of it are usually written
by Greek letters, e.g., ξ or η. As customary the space of all real, square-integrable
functions F on Ω is called L2 (Ω). L2 (Ω) is a Hilbert space endowed with the inner
product
Z
hF, GiL2 (Ω) =
F (ξ)G(ξ) dω(ξ),
Ω
and associated norm
kF k
L2 (Ω)
=
Z
Ω
1/2
F (ξ) dω(ξ)
.
2
2
It is well known that L (Ω) possesses an orthonormal basis consisting of the realvalued spherical harmonics {Yn,k }n=0,...;k=−n,...,n of degree n and order k as defined,
for example, in [14, 15]. Then the spherical addition theorem
n
X
2n + 1
Pn (ξ · η) =
Yn,k (ξ)Yn,k (η)
4π
k=−n
connects the spherical harmonics Yn,k of degree n and the Legendre polynomial Pn
(for more details see [1, 9, 13]).
Scaling Functions and Wavelets. First of all we recapitulate the definition of
a so-called generator of a scaling function. As a matter of fact, the choice of this generator determines all properties of the spherical scaling function and its corresponding
wavelet.
THE SPHERICAL BERNSTEIN WAVELET
3
Definition 2.1 (Generator of a Scaling Function). A family {{Φ∧
J (n)}n∈N0 }J∈N0
is called a generator of a scaling function, if it satisfies the following requirements:
1. For all J ∈ N0
Φ∧
J (0) = 1,
(2.1)
2. for all J, J 0 ∈ N0 with J ≤ J 0 and all n ∈ N
∧
0 ≤ Φ∧
J (n) ≤ ΦJ 0 (n),
(2.2)
lim Φ∧
J (n) = 1.
(2.3)
3. for all n ∈ N
J→∞
For fixed J ∈ N0 the sequence {Φ∧
J (n)}n∈N0 is called the symbol of the corresponding scaling function ΦJ of scale J. According to [9] this scaling function of scale
J is defined by
ΦJ (ξ, η) =
∞
X
Φ∧
J (n)
n=0
2n + 1
Pn (ξ · η),
4π
while the (linear) wavelet of scale J reads as follows
ΨJ (ξ, η) =
∞
X
Ψ∧
J (n)
n=0
2n + 1
Pn (ξ · η),
4π
∧
∧
∧
where Ψ∧
J (n) is given via the linear refinement equation ΨJ (n) = ΦJ+1 (n) − ΦJ (n),
n ∈ N0 . The whole set {ΦJ }J∈N0 is called a scaling function, the set {ΨJ }J∈N0 is called
∧
a wavelet. If for all scales J the symbol {Φ∧
J (n)}n∈N0 , respectively {ΨJ (n)}n∈N0 , is
different from zero only for finitely many values of n, we call the corresponding scaling
function {ΦJ }J∈N0 , respectively the wavelet {ΨJ }J∈N0 , bandlimited. The most simple
representant of a scaling function is the well-known Shannon scaling function (see [5]).
Definition 2.2 (Shannon Scaling Function). Let J ∈ N0 and η ∈ Ω be fixed.
Then the Shannon scaling function of scale J is given by
ΦSJ (ξ, η)
=
J
2X
−1
ΦS∧
J (n)
n=0
2n + 1
Pn (ξ · η),
4π
where
(
1
ΦS∧
J (n) =
0
if n ∈ [0, 2J ),
else.
Obviously, its generator {{ΦS∧
J (n)}n∈N0 }J∈N0 fulfills all three conditions of Definition
2.1 and the Shannon scaling function is bandlimited.
Space Localization. We start by formulating the space localization property
mathematically: Suppose that F is real-valued and of class L2 (Ω). Moreover, assume
without loss of generality, that kF kL2 (Ω) = 1. Then, we follow [5, 9, 16] and associate
4
M. J. FENGLER, W. FREEDEN, M. GUTTING
to F the normal (radial) field η 7→ ηF (η), η ∈ Ω. This allows us to introduce the
expectation value of F in the space domain by
Z
eF =
ηF (η)2 dω(η),
(2.4)
Ω
3
such that eF ∈ R . In that sense, F (η)2 can be interpreted as probability density
function with respect to the measure dω(η). Immediately, we observe that |eF | ≤ 1.
Additionally, the variance in the space domain σF2 is given by
Z
2
σF = (η − eF )2 F (η)2 dω(η).
(2.5)
Ω
Hereby, σF can be regarded as the standard deviation of F . Let us clarify its geometrical meaning: Assume that eF 6= 0, then F localizes in the spherical cap
C = {η ∈ Ω|1 − η · eF /|eF | ≤ 1 − |eF |},
with radius σF . Clearly, we say F is “space localized” if the ratio
σF
∆F =
|eF |
(2.6)
between the diameter σF of the spherical cap C and the absolute value of the expectation value |eF | is small (a table with examples can be found in [2]). For a concise
description of the space localization as well as the uncertainty principle on the sphere
we refer the reader to [5, 9, 16]. According to [9] we can simplify the expression for the
expectation value eF if F is a radial basis function. In this case, we assume without
loss of generality that F is localized at ε3 = (0, 0, 1)T . Then (2.4) reads as
Z 1
eF = 2π
tF (t)2 dt ε3 .
(2.7)
−1
and following [9] the variance in (2.5) can be rewritten as
σF2 = 1 − |eF |2 .
(2.8)
Localization in Momentum. In order to define the localization in momentum
we follow the concepts of [5] and [16]. Thus, we consider the expectation value in the
momentum domain which is in fact the expectation value of the surface curl operator
L∗ on the sphere Ω. Suppose again that F is real valued with kF kL2 (Ω) = 1, but this
time F is an element of a space of Sobolev type H2 (Ω), i.e., for F ∈ H2l (Ω) with
l ∈ N there exists a function G ∈ L2 (Ω) whose Fourier coefficients G∧ (n, k) fulfill
G∧ (n, k) = (−n(n + 1))l F ∧ (n, k) for all n = 0, 1, . . . ; k = −n, . . . , n. For a concise
introduction to spherical Sobolev spaces see, e.g., [5, 6, 9].
The expectation value in the momentum domain of the function F is given by
Z
∗
(2.9)
eL
=
(L∗η F (η))F (η) dω(η),
F
Ω
∗
3
which by definition leads to the value eL
F = 0 ∈ R . Its variance in the momentum
L∗ 2
domain σF
is defined as
2
∗ 2 Z ∗
∗
)F
(η)
=
σFL
dω(η)
(Lη − eL
F
Ω
Z
(2.10)
= (−∆∗η F (η))F (η) dω(η),
Ω
THE SPHERICAL BERNSTEIN WAVELET
5
where the last equation easily results from the surface theorem of Stokes. Again we
can simplify this expression in the case of radial basis functions assuming without loss
of generality that F is localized in ε3
Z 1
∗ 2
F (t)Lt F (t) dt,
(2.11)
= −2π
σFL
−1
d
2 d
dt (1−t ) dt
where Lt =
denotes the Legendre operator. The localization in momentum
∗
∗
L∗
∆L
is
measured
by
the
square root of the variance, i.e., ∆L
F
F = σF . In consequence,
we are able to state the uncertainty principle for spherical functions (cf. [5, 9, 16])
which says that sharp localization in the space domain as well as in the momentum
domain is not possible.
Theorem 2.3 (Uncertainty Principle). Let F ∈ H2 (Ω) with kF kL2 (Ω) = 1. Then
∗ 2
2
(σF ) σFL
≥ |eF |2 and (if eF 6= 0)
∗
∆ F ∆L
F ≥1
(2.12)
The left hand side in (2.12) is called uncertainty product.
To avoid that the expectation value in momentum domain is vector valued and
equal to 0 for all functions we start over considering the negative Beltrami operator
−∆∗ instead of L∗ . Therefore, we have to assume that F ∈ H4 (Ω) with kF kL2 (Ω) = 1.
This leads to the expectation value of −∆∗
Z
∗ 2
∗
e−∆
−∆∗η F (η) F (η) dω(η) = σFL
=
(2.13)
F
Ω
which turns out to be the same as the variance of L∗ (see (2.10)). The variance with
respect to −∆∗ is defined as
Z 2
2
∗
∗ 2
(−∆∗ )2
−∆∗
F
(η)
−
e
(2.14)
dω(η)
=
e
=
σF−∆
(−∆∗η ) − e−∆
F
F
F
Ω
and the localization in momentum in terms of these expressions reads as follows (if
−∆∗
eF
6= 0)
σF−∆
∗
∆−∆
=
F
(−∆∗ )2
eF
∗
−(e−∆
F
−∆
eF
∗
∗ 2
)
12
12
∗
∗
= ∆L
= e−∆
F .
F
(2.15)
Note that this quantity does not change. Thus, the uncertainty principle (2.12) is left
unmodified.
3. The Bernstein Wavelet. In this section we introduce the Bernstein scaling
function and the corresponding Bernstein wavelet. The name Bernstein is motivated
J
−1
by the fact that it is proportional to the Bernstein polynomial B22J −1
on the interval
[−1, 1] (see, for instance, [17]). Later on, we also derive certain properties of this
kernel.
Definition 3.1 (Bernstein Scaling Function). Let η ∈ Ω be fixed. The Bernstein
scaling function of scale J ∈ N0 is given by
2J −1
2J−2 1 + ξ · η
B
ΦJ (ξ, η) =
.
(3.1)
π
2
6
M. J. FENGLER, W. FREEDEN, M. GUTTING
As usual in linear wavelet theory, we introduce the Bernstein wavelet of a certain
scale by differences of scaling functions of two consecutive scales.
Definition 3.2 (Bernstein Wavelet). Let η ∈ Ω be fixed. The Bernstein wavelet
of scale J ∈ N0 is given by
B
B
ΨB
J (ξ, η) = ΦJ+1 (ξ, η) − ΦJ (ξ, η).
Clearly, we are able to understand ΦB
J as bandlimited radial basis function with
,
such
that
symbol {ΦB∧
(n)}
n∈N0
J
ΦB
J (ξ, η)
=
ΦB
J (t)
=
J
2X
−1
ΦB∧
J (n)
n=0
2n + 1
Pn (t),
4π
t = ξ · η.
Interestingly, we find an explicit representation of the symbol by the following theorem.
Theorem 3.3 (Symbol of the Bernstein Scaling Function). Suppose that J ∈ N0 .
Then the scaling function ΦB
J (·, η) of scale J possesses the symbol given by
(
(2J )!(2J −1)!
if n ∈ [0, 2J ),
J −n−1)!(2J +n)!
B∧
(2
(3.2)
ΦJ (n) =
0
else.
Before going into the details of the proof of Theorem 3.3, we borrow the following
recursion formulae for the Legendre polynomials from [9, 13]: First, suppose that
n ≥ 0 then
(n + 1)Pn+1 (t) + nPn−1 (t) − (2n + 1)tPn (t) = 0,
(3.3)
for t ∈ [−1, 1] with P−1 (t) = 0. Second, let n ≥ 0, then
0
Pn0 (t) − Pn−2
(t) = (2n − 1)Pn−1 (t),
(3.4)
for t ∈ [−1, 1] with P−1 (t) = P−2 (t) = 0.
Based on these recursion formulae we are able to verify Theorem 3.3.
Proof. Formally, the symbol of ΦB
J can be computed by solving the integral
expression given by
ΦB∧
J (n) = 2π
Z
1
−1
2J−2
π
1+t
2
2J −1
Pn (t) dt.
For the following considerations we use the following abbreviation
Z 1
In,k =
(1 + t)k Pn (t) dt.
−1
Straightforward integration yields
I0,k =
2k+1
,
k+1
I1,k = I0,k
k
.
k+2
From the recurrence formula (3.3) we obtain (n ≥ 1)
(n + 1)In+1,k + nIn−1,k − (2n + 1)In,k+1 + (2n + 1)In,k = 0.
(3.5)
7
THE SPHERICAL BERNSTEIN WAVELET
6
0
10
−5
10
5
−10
10
4
−15
10
3
−20
10
−25
10
2
−30
10
1
−35
10
−40
10
0
10
20
30
n
40
50
60
70
0
−4
Fig. 3.1. Non-zero elements of the
B
B
symbol of ΦB
4 (stars), Φ5 (dots), and Φ6
(pluses).
−3
−2
−1
0
θ
1
2
3
4
Fig. 3.2. Bernstein scaling funcB
tion ΦB
4 (cos θ) (dashed), Φ5 (cos θ) (dotted), and ΦB
(cos
θ)
(solid).
6
6
0
10
−10
5
10
−20
10
4
−30
10
3
−40
10
2
−50
10
1
−60
10
0
−70
10
−80
10
0
20
40
60
n
80
100
120
140
Fig. 3.3. Non-zero elements of the
B
B
symbol of ΨB
4 (stars), Ψ5 (dots), and Ψ6
(pluses).
−1
−4
−3
−2
−1
0
θ
1
2
3
4
Fig. 3.4.
Bernstein wavelet
B
ΨB
4 (cos θ) (dashed), Ψ5 (cos θ) (dotted),
B
and Ψ6 (cos θ) (solid).
Furthermore, we find by partial integration and (3.4) that
(2n + 1)In,k+1 = −(k + 1)(In+1,k − In−1,k ).
Combining these results we arrive at the following recursion formula:
(n + k + 2)In+1,k = −(2n + 1)In,k + (k + 1 − n)In−1,k .
By letting k = 2J − 1 this gives us the following result.
ΦB∧
J (0) = 1,
2J − 1
,
2J + 1
B∧
J
B∧
(n + 2J + 1)ΦB∧
J (n + 1) = −(2n + 1)ΦJ (n) + (2 − n)ΦJ (n − 1).
ΦB∧
J (1) =
By induction over n we obtain the desired result.
These considerations also help us to compute an explicit representation of the L2 (Ω)norm of the Bernstein scaling function.
8
M. J. FENGLER, W. FREEDEN, M. GUTTING
Lemma 3.4 (Norm of the Bernstein Scaling Function). Let J ∈ N0 and η ∈ Ω be
fixed. Then the L2 (Ω)-norm of the Bernstein scaling function ΦB
J of scale J is given
by
2J−1
kΦB
.
J (·, η)kL2 (Ω) = p
π(2J+1 − 1)
(3.6)
Proof. The squared norm of ΦB
J can be evaluated by aid of
2
kΦB
J (·, η)kL2 (Ω)
1
= 2π
Z
1
= 2π
Z
−1
−1
2J −1 !2
2J−2 1 + t
dt
π
2
2J+1 −2
22J−4 1 + t
P0 (t) dt.
π2
2
Clearly, the latter integral is of the same type as in (3.5). Thus, we find by analogous
considerations and k = 2J+1 − 2 that
2
kΦB
J (·, η)kL2 (Ω) =
22J−2
.
π(2J+1 − 1)
We denote the L2 (Ω)-normalized Bernstein scaling function of scale J by
Φ̃B
J (·, η) =
ΦB
J (·, η)
.
B
kΦJ (·, η)kL2 (Ω)
Based on the results obtained so far, we are now able to prove that the Bernstein
scaling function forms an approximate identity.
Approximate Identity. Loosely spoken an approximate identity is a family of
kernels depending on a parameter which converges to the Dirac distribution as this
parameter tends to a limit value. In our case of scaling functions this parameter is
the scale J which tends to infinity. The third property of the generator, i.e., (2.3)
in Definition 2.1, ensures that any scaling function naturally forms an approximate
identity.
In conclusion, we have to show in the following that this limit holds true for the
Bernstein scaling function.
Theorem 3.5 (Approximate Identity). The Bernstein scaling function {ΦB
J }J∈N0
forms an approximate identity.
Proof. It is sufficient to prove that limJ→∞ ΦB∧
J (n) = 1 for fixed n ∈ N0 . To
accomplish this we take the logarithm of (3.2). Then
J
log ΦB∧
J (n)
=
2
X
log k +
k=1
=
=
J
2X
−1
k=1
J
2
X
log k −
k=2J −n
J
n
X
log
k=1
log k −
J
2X
+n
2J X
−n−1
k=1
log k −
J
2X
+n
log k
k=1
log k
k=2J
2 −k
2J + k
.
(3.7)
THE SPHERICAL BERNSTEIN WAVELET
9
Since n ∈ N0 is fixed, the series in (3.7) is finite. Then we obtain
lim
J→∞
log ΦB∧
J (n)
= lim
J→∞
n
X
log
k=1
2J − k
2J + k
= 0,
and, therefore, limJ→∞ ΦB∧
J (n) = 1.
Monotonicity. Since the first property (2.1) constituting a scaling function is
obviously fulfilled by ΦB∧
J (0), we are left with the monotonicity condition (2.2).
Theorem 3.6 (Monotonicity). The generator of the Bernstein scaling function
0
0
{ΦB
J }J∈N0 is monotonically increasing, i.e., for all J, J ∈ N0 with J ≤ J and all
n ∈ N we have
B∧
0 ≤ ΦB∧
J (n) ≤ ΦJ 0 (n).
Proof. From Theorem 3.3 it is clear that 0 ≤ ΦB∧
J (n) for all J ∈ N0 and all n ∈ N.
Therefore, it remains to prove the inequality between two different scales. It suffices
to show that, for fixed n ∈ N0 and arbitrary J, J 0 ∈ N0 with J ≤ J 0 , the following
ratio holds
ΦB∧
J (n)
≤ 1.
ΦB∧
J 0 (n)
Similar to Theorem 3.5 we consider the logarithms and obtain
!
0
n
X
(2J − k)(2J + k)
ΦB∧
J (n)
=
log
log B∧
(2J + k)(2J 0 − k)
ΦJ 0 (n)
k=1
which we want to be less or equal to 0. Thus, it is sufficient to show that the argument
of each logarithm in this sum is smaller than 1. This can be easily seen since from
0
J ≤ J 0 follows 2J ≤ 2J and therefore,
0
0
0
2J+J − k 2 + k(2J − 2J )
(2J − k)(2J + k)
≤ 1.
=
0
(2J + k)(2J − k)
2J+J 0 − k 2 + k(2J 0 − 2J )
Now all conditions of Definition 2.1 are guaranteed and we can rightly speak of the
Bernstein scaling function.
4. Oscillations. Throughout this section we are concerned with the oscillations
of the Bernstein and Shannon scaling function, respectively. Before going into the
mathematical details we like to motivate our considerations by comparing in Fig.
S
B
S
4.1 and Fig. 4.2 both normalized kernels Φ̃B
4 and Φ̃4 , and Φ̃6 and Φ̃6 , respectively.
Although both kernels are bandlimited to the same degree in each plot, we notice how
smoothly the Bernstein kernel decays to zero whereas the Shannon kernel oscillates
heavily.
Now, mathematically spoken, we understand oscillations as sign changes of the
slope.
In the Shannon case this consideration can be simplified if we prove that this kernel
10
M. J. FENGLER, W. FREEDEN, M. GUTTING
5
20
4
15
3
10
2
5
1
0
0
−1
−4
−3
−2
−1
0
θ
1
2
3
4
−5
−4
Fig. 4.1.
The L2 (Ω)-normalized
Shannon scaling function Φ̃S
4 (t) (solid)
and the L2 (Ω)-normalized Bernstein scaling function Φ̃B
4 (t) (dashed).
−3
−2
−1
0
θ
1
2
3
4
Fig. 4.2.
The L2 (Ω)-normalized
Shannon scaling function Φ̃S
6 (t) (solid)
and the L2 (Ω)-normalized Bernstein scaling function Φ̃B
6 (t) (dashed).
possesses only simple zeros which imply slope changes. To be more specific, we borrow
from [12] an alternative representation of ΦSJ (t), i.e.,
ΦSJ (t)
=
J
2X
−1
n=0
=
(
2n + 1
Pn (t)
4π
2J P2J (t)−P2J −1 (t)
4π
t−1
22J
4π
(4.1)
for t ∈ [−1, 1)
for t = 1.
(4.2)
Theorem 4.1 (Zeros of the Shannon Kernel). Let J ∈ N0 . Then the Shannon
scaling function ΦSJ (t) possesses exactly 2J − 1 distinct, simple zeros on the interval
[-1,1].
Proof. From (4.1) we see that ΦSJ (t) possesses at most 2J − 1 zeros. The case
J ≤ 1 is trivial. For J ≥ 2 it is sufficient to show that the numerator NJ (t) =
P2J (t) − P2J −1 (t) in (4.2) possesses 2J − 1 zeros in (-1,1). It is known that the
Legendre polynomial Pn possesses n distinct (simple) zeros on (−1, 1) (see [1]). Thus,
we denote by t1 , . . . , t2J the zeros of P2J in increasing order, and with s1 , . . . , s2J −1
the zeros of P2J −1 , respectively. Furthermore, the zeros of P2J −1 divide the zeros of
P2J (see [1]). In detail, we have si ∈ (ti , ti+1 ). Clearly, the expression
NJ (si ) = P2J (si ) − P2J −1 (si ) = P2J (si )
has the same alternating sign as P2J (si ), i = 1, . . . , 2J − 1. Therefore, we know from
the Intermediate Value Theorem of classical analysis that NJ (t) possesses at least
2J − 2 zeros in (t1 , t2J ). Finally, we show that the remaining zero is placed in (−1, t1 ).
This can be seen as follows: Keeping in mind that Pn (−1) = (−1)n we know that P2J
has a negative sign in (t1 , t2 ) since P2J (−1) = 1 and t1 denotes the first zero. Thus,
we obtain that NJ (s1 ) is negative and NJ (−1) = 2 is positive. Again, we can apply
the Intermediate Value Theorem and arrive at the desired result.
As an immediate consequence we find that the Shannon scaling function of scale
J ≥ 2 possesses 2J −2 sign changes of its slope on (−1, 1), i.e., oscillations. Obviously,
11
THE SPHERICAL BERNSTEIN WAVELET
the Bernstein scaling function as introduced in Definition 3.1 features no oscillations
since -1 is a zero of multiplicity 2J − 1.
5. Space Uncertainty. In this section we derive an explicit representation of
the space uncertainty of the Bernstein scaling function. In fact, we are led to the surprising result that the Shannon scaling function possesses the same space uncertainty.
Theorem 5.1 (Space Uncertainty of the Bernstein Scaling Function). Suppose
that J ∈ N and η ∈ Ω is fixed. Then the space uncertainty of the L2 (Ω)-normalized
Bernstein scaling function Φ̃B
J (·, η) of scale J is given by
√
∆Φ̃B (·,η) =
J
2J+1 − 1
,
2J − 1
where the expectation value and the variance in the space domain read as follows
2J − 1
η,
2J
2J+1 − 1
=
.
22J
eΦ̃B (·,η) =
J
2
σΦ̃
B (·,η)
J
Proof. Without loss of generality we set η = ε3 . From (2.7) we obtain
eΦ̃B (·,ε3 ) =
J
=
2π
Z
1
−1
2 t Φ̃B
(t)
dt ε3
J
2π
B
kΦJ (·, ε3 )k2L2 (Ω)
Z
1
!
2
ΦB
J (t)
t
−1
dt ε3 .
B
3
Substituting ΦB
J by (3.1) and kΦJ (·, ε )kL2 (Ω) by (3.6) we deduce

eΦ̃B (·,ε3 ) = 
J
=
=
J+2
(2
− 2)π
22J−2
2
2J+2 − 2
22J−2
Z
1
2J+1 − 1
2
Z
1
1
Z
2
t
π
−1
t 22J−4
−1
−1
J−2
1+t
2
1+t
2
1+t
2
2J −1 !2
2J+1 −2
2J+1 −2
!

dt ε3
dt ε3
!
P1 (t) dt ε3 .
Again, we observe that the latter term is of the same type as studied in (3.5). By
letting n = 1 and k = 2J+1 − 2 we find the desired result
eΦ̃B (·,ε3 ) =
J
2J − 1 3
ε .
2J
As an immediate consequence we deduce from (2.8) for the variance
2
σΦ̃
B (·,ε3 ) = 1 −
J
2J+1 − 1
(2J − 1)2
=
,
22J
22J
(5.1)
12
M. J. FENGLER, W. FREEDEN, M. GUTTING
such that the space uncertainty is given by
√
∆Φ̃B (·,ε3 ) =
J
2J+1 − 1
.
2J − 1
(5.2)
Next we deal with the amazing result that the L2 (Ω)-normalized Shannon scaling
function possesses the same space uncertainty as the normalized Bernstein scaling
function. First, we need a preliminary result on the norm of the Shannon scaling
function of an arbitrary scale.
Lemma 5.2 (Norm of the Shannon Scaling Function of scale J). Suppose that
J ∈ N0 and η ∈ Ω is fixed. Then the L2 (Ω)-norm of Shannon scaling function ΦB
J (·, η)
of scale J is given by
2J−1
kΦSJ (·, η)kL2 (Ω) = √ .
π
(5.3)
Proof. This result follows immediately from the definition of the Shannon generator (see Definition (2.2)) and the following norm identity
kΦSJ (·, η)k2L2 (Ω) =
∞
X
2n + 1 S∧ 2
ΦJ (n) .
4π
n=0
Based on the latter result we are now in the position to investigate the space uncertainty of the Shannon scaling function.
Theorem 5.3 (Space Uncertainty of the Shannon Scaling Function). Assume
that J ∈ N and η ∈ Ω is fixed. Then the space uncertainty of the L2 (Ω)-normalized
Shannon scaling function Φ̃SJ (·, η) of scale J is given by
√
2J+1 − 1
,
∆Φ̃S (·,η) =
J
2J − 1
where the expectation value and the variance in the space domain read as follows
2J − 1
η,
2J
J+1
2
−1
=
.
2J
2
eΦ̃S (·,η) =
J
2
σΦ̃
S (·,η)
J
Proof. We start with the explicit formula for the expectation value (see also [11])
!
∞
X
1
∧
3
eΦ̃J (·,ε3 ) =
(n + 1) Φ∧
J (n)ΦJ (n + 1) ε
2πkΦJ (·, ε3 )k2L2 (Ω) n=0

 ∞
P
∧
2
(n + 1)Φ∧
(n)Φ
(n
+
1)
J
J
 n=0
 3
ε .
=
∞


P
2
∧
(2n + 1) (ΦJ (n))
n=0
13
THE SPHERICAL BERNSTEIN WAVELET
2
Without loss of generality we set η = ε3 . The variance σΦ̃
can then be computed
J (·,η)
as in (2.8) and for the space uncertainty we have
∆Φ̃J (·,η)


σΦ̃J (·,η)

=
= 
|eΦ̃J (·,η) | 
∞
P
2
1) (Φ∧
J (n))
(2n +
n=0
∞
P
2
n=0
∧
(n + 1)Φ∧
J (n)ΦJ (n + 1)
2
 12


 − 1
 .


The results of the theorem are obtained by combining Definition 2.2 with these formulae.
Obviously, the last result indicates together with Theorem 4.1 that the quantity
∆F for the space localization (see (2.6)) does not reflect the amount of oscillations.
This is nicely illustrated in Fig. 4.1 and Fig. 4.2.
6. Uncertainty in Momentum and the Uncertainty Product. We now
derive the explicit formulae for the uncertainty in momentum and for the uncertainty
product of the Bernstein scaling function. We again compare these quantities with
their corresponding values in the Shannon case.
Theorem 6.1 (Uncertainty in Momentum of the Bernstein Scaling Function).
Let J ∈ N and η ∈ Ω be fixed. Then the uncertainty in momentum of the L2 (Ω)normalized Bernstein scaling function Φ̃B
J (·, η) of scale J is given by
−∆∗
∆Φ̃
B (·,η)
J
∗
∆L
Φ̃B
J (·,η)
=
√
=
2J − 1
√
2
where the expectation value (which coincides with the variance with respect to the L∗
operator) and the variance in the momentum domain (with respect to the operator
−∆∗ ) read as follows
2
2J − 1 L∗
= σΦ̃B (·,η) ,
J
2
J
2
J
2
J+1
∗
(2
−
1)
3
·
2
−1
−∆
=
σΦ̃
.
B (·,η)
J+1
4
2
−3
J
∗
−∆
eΦ̃
B (·,η) =
∗
2
L
Proof. Without loss of generality we set η = ε3 . For the calculation of σΦ̃
B (·,ε3 )
J
we use formula (2.11)
L∗
σΦ̃
B (·,ε3 )
J
2
= −2π
=
Z
1
B
Φ̃B
J (t)Lt Φ̃J (t) dt
−1
−2π
B
kΦJ (·, ε3 )k2L2 (Ω)
Z
1
−1
B
ΦB
J (t)Lt ΦJ (t) dt.
14
M. J. FENGLER, W. FREEDEN, M. GUTTING
Using (3.1) and (3.6) we get
L∗
σΦ̃
B (·,ε3 )
J
2J −1
2J −1
2J−2 1 + t
2J−2 1 + t
Lt
dt
π
2
π
2
−1
2J −1 2J −1
Z 1+t
−2π 2 (2J+1 − 1) 22J−4 1 1 + t
Lt
dt
=
22J−2
π2
2
2
−1
2J −1 2J −1
Z 1+t
−(2J+1 − 1) 1 1 + t
Lt
dt.
(6.1)
=
2
2
2
−1
−2π 2 (2J+1 − 1)
=
22J−2
2
Z
1
Simple differentiation yields the result of the application of the Legendre operator Lt
Lt
1+t
2
2J −1
2J −1
d
1+t
2 d
= (1 − t )
dt
dt
2
J
2
−2
2J − 1 1 + t
=
2J − 2 − 2J t .
2
2
Substituting this in (6.1) we obtain by aid of the Legendre polynomials P0 (t) = 1 and
P1 (t) = t
∗
L
σΦ̃
B (·,ε3 )
J
2
2J −1 J
2J −2
1+t
2 −1 1+t
2J − 2 − 2J t dt
2
2
2
−1
J+1
Z
2
−3
−(2J+1 − 1)(2J − 1) 1 1 + t
2J − 2 − 2J t dt
=
22
2
−1
2J+1 −3
Z 1
J+1
J
−(2
− 1)(2 − 1)
1+t
J
P0 (t)
=
(2 − 2)
dt
22
2
−1
2J+1 −3 !
Z 1
1+t
J
dt .
−2
P1 (t)
2
−1
=
−(2J+1 − 1)
2
Z
1
Once again we apply (3.5) to come up with
L∗
σΦ̃
B (·,ε3 )
J
2
−(2J+1 − 1)(2J − 1)
=
22
=
2J − 1
.
2
2(2J+1 − 3)
−2
(2 − 2) J+1
2
−2
(2J+1 − 2)(2J+1 − 1)
J
2
J
∗
From (2.13) we know that this also corresponds to the expectation value e−∆
.
3
Φ̃B
J (·,ε )
The uncertainty in momentum is given by
−∆∗
∆Φ̃
B (·,ε3 )
J
=
∗
∆L
3
Φ̃B
J (·,ε )
=
L∗
σΦ̃
B (·,ε3 )
J
=
√
2J − 1
√
.
2
15
THE SPHERICAL BERNSTEIN WAVELET
2
−∆∗
Finally, the variance σΦ̃
is computed via (2.14)
B (·,η)
J
∗
−∆
σΦ̃
B (·,η)
J
2
2
(−∆∗ )2
−∆∗
= eΦ̃B (·,η) − eΦ̃
B (·,η)
J
J
2
J
Z 2 −1
∗ 2 B
B
=
−∆ξ Φ̃J (ξ, η) Φ̃J (ξ, η) dω(ξ) −
2
Ω
J
2
Z 2 −1
∗ B
(ξ,
η)
∆
Φ̃
(ξ,
η)
dω(ξ)
−
=
∆∗ξ Φ̃B
.
J
ξ J
2
Ω
Again, this integral can be simplified as we are considering the structure as radial
basis function. Thus, without loss of generality we set η = ε3 .
Z Z 1
2
∗ B
3
∗ B
3
Lt Φ̃B
dt
∆ξ Φ̃J (ξ, ε ) ∆ξ Φ̃J (ξ, ε ) dω(ξ) = 2π
J (t)
Ω
−1
=
2π
B
kΦJ (·, ε3 )k2L2 (Ω)
Z
1
−1
2
Lt ΦB
dt.
J (t)
Inserting our known expressions for the norm (3.6) and for Lt ΦB
J (t)
Z Ω
∆∗ξ Φ̃B
J
2
2π 2 (2J+1 − 1)
(ξ, ε3 ) dω(ξ) =
22J−2
2J−2
π
2 Z
1
−1
Lt
1+t
2
!2
2J −1 !2
dt
2J −2
2J − 1 1 + t
(2J − 2 − 2J t)
dt
2
2
−1
2J+1 −4
Z 2
(2J+1 − 1)(2J − 1)2 1 1 + t
=
2J − 2 − 2J t dt.
23
2
−1
2J+1 − 1
=
2
Z
1
The last integral can be decomposed into 3 integrals of type (3.5) which compute to
(−∆∗ )2
eΦ̃B (·,ε3 ) =
J
23 (2J − 1)
(2J − 1)3
(2J+1 − 1)(2J − 1)2
= J+1
.
3
J+1
J+1
2
(2
− 3)(2
− 1)
2
−3
Therefore, we obtain
−∆∗
σΦ̃
B (·,ε3 )
J
2
(2J − 1)3
= J+1
−
2
−3
2J − 1
2
2
=
(2J − 1)2 3 · 2J+1 − 1
.
4
2J+1 − 3
For comparison we also calculate the uncertainty in momentum for the Shannon
scaling function.
Theorem 6.2 (Uncertainty in Momentum of the Shannon Scaling Function).
Let J ∈ N and η ∈ Ω be fixed. Then the uncertainty in momentum of the L2 (Ω)normalized Shannon scaling function Φ̃SJ (·, η) of scale J is given by
√
22J − 1
−∆∗
L∗
√
∆Φ̃
=
∆
=
S (·,η)
S (·,η)
Φ̃
J
J
2
where the expectation value (which coincides with the variance with respect to the L∗
operator) and the variance in the momentum domain (with respect to the operator
16
M. J. FENGLER, W. FREEDEN, M. GUTTING
−∆∗ ) read as follows
2
22J − 1 L∗
= σΦ̃S (·,η) ,
J
2
J
2
J
2 J
∗
(2
−
1)
(2
+
1)2
−∆
=
σΦ̃
.
S (·,η)
12
J
∗
e−∆
=
Φ̃S (·,η)
∗
−∆
Proof. We start with the following representation of eΦ̃
(see also [11])
(·,η)
J
∗
L
σΦ̃
J (·,η)
2
=
=
∞
X
1
2
(2n + 1)n(n + 1)(Φ∧
J (n))
4πkΦJ (·, η)k2L2 (Ω) n=0
∞
P
n=0
2
(2n + 1)n(n + 1)(Φ∧
J (n))
∞
P
n=0
.
(6.2)
2
(2n + 1)(Φ∧
J (n))
The uncertainty in momentum is then given by the square root of (6.2). Finally, the
2
−∆∗
variance σΦ̃
is deduced from
(·,η)
J
∗
−∆
σΦ̃
(·,η)
J
2
=
(−∆∗ )2
J (·,η)
= eΦ̃
∞
P
n=0
2
∗
− e−∆
Φ̃ (·,η)
J
2
(2n + 1)n2 (n + 1)2 (Φ∧
J (n))
∞
P
n=0
2
(2n + 1)(Φ∧
J (n))

∞
P
2
(2n + 1)n(n + 1)(Φ∧
J (n))
 n=0
−
∞

P
n=0
2
(2n + 1)(Φ∧
J (n))
2

 .

The results of the theorem are obtained by combining Definition 2.2 with these formulae.
Now we have all the tools to discuss the uncertainty products for the two scaling
functions.
It is of particular interest whether these products converge to the minimal value 1 as
the scale J tends to infinity, since a value of 1 indicates that the functions are well
localized in space and frequency.
Theorem 6.3 (Uncertainty Products of the Bernstein and of the Shannon Scaling Function). The uncertainty product of the L2 (Ω)-normalized Bernstein scaling
function Φ̃B
J (·, η) of scale J is given by
∗
∆Φ̃B (·,η) ∆L
Φ̃B
J
J (·,η)
=
2J+1 − 1
2J+1 − 2
12
,
and for J −→ ∞ this product tends to 1.
The uncertainty product of the L2 (Ω)-normalized Shannon scaling function Φ̃SJ (·, η)
of scale J reads as follows
L∗
∆Φ̃S (·,η) ∆Φ̃
S (·,η)
J
J
and it diverges for J −→ ∞.
=
(2J+1 − 1)(2J + 1)
2J+1 − 2
12
THE SPHERICAL BERNSTEIN WAVELET
17
Proof. The product for the Bernstein scaling function of scale J can easily be
computed from Theorem 5.1 and Theorem 6.1. In the Shannon case the result follows from Theorem 5.3 and Theorem 6.2. In both cases the behavior for J −→ ∞ is
obvious.
7. Concluding Remarks. In this article we have demonstrated that there exist
bandlimited scaling functions without oscillations. Furthermore, the Bernstein scaling
function represents an example of a family of kernels that minimizes its uncertainty
product in the limit, but unlike the Gauß-Weierstraß kernels which also possess this
property (cf. [5, 9]) this minimizer is bandlimited. In detail, the uncertainty product
of the Bernstein scaling function tends to 1. To our knowledge it is the first bandlimited kernel presented in literature for which this property has been proven. An
uncertainty product of 1 indicates that the space as well as frequency localization is
well balanced. Note that the dyadic arrangement of the scales is not associated with
the properties presented in this work. All of them hold true also for other types of
scales. Finally, it should be emphasized that the Bernstein scaling function possesses
a closed representation by elementary expressions. Especially the latter allows a fast
and stable evaluation. The numerical properties of this new spherical scaling function
and wavelet will be subject to further studies.
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[1] M. Abramowitz, I.A. Stegun: Handbook of Mathematical Functions. Dover Publications,
Inc., New York, 1972.
[2] M. J. Fengler: Vector Spherical Harmonic and Vector Wavelet Based Non-Linear Galerkin
Schemes for Solving the Incompressible Navier-Stokes Equation on the Sphere. PhD-thesis,
University of Kaiserslautern, Department of Mathematics, Geomathematics Group (submitted), 2005.
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Spherical Harmonics for Solving the Incompressible Navier-Stokes Equation on the Sphere.
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[11] N. Laı́n Fernández: Polynomial Bases on the Sphere. PhD-Thesis, University of Lübeck,
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[12] N. N. Lebedev: Spezielle Funktionen und Ihre Anwendung. B.I.-Wissenschaftsverlag,
Mannheim, Wien, Zürich, 1973.
[13] W. Magnus, F. Oberhettinger, R. P. Soni: Formulas and Theorems for the Special Functions of Mathematical Physics. Die Grundlagen der mathematischen Wissenschaften in
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[14] C. Müller: Spherical Harmonics. Lecture Notes in Mathematics, Vol. 17, Springer, 1966.
18
M. J. FENGLER, W. FREEDEN, M. GUTTING
[15] C. Müller: Foundation of the Mathematical Theory of Electromagnetic Waves. Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellung, Bd. 155, Springer, 1969.
[16] F. J. Narcowich, J. D. Ward: Nonstationary Wavelets on the m-sphere for Scattered Data.
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[17] H. R. Schwarz: Numerische Mathematik. B. G. Teubner, Stuttgart, 1997.
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Informationen:
Prof. Dr. W. Freeden
Prof. Dr. E. Schock
Fachbereich Mathematik
Technische Universität Kaiserslautern
Postfach 3049
D-67653 Kaiserslautern
E-Mail: [email protected]
[email protected]