Approximation of solutions to the Helmholtz equation from scattered data

0.48 MB·English

by  Gilles Chardon

Approximation of solutions to the Helmholtz equation
from scattered data ∗
Gilles Chardon, Albert Cohen and Laurent Daudet
December 11, 2013
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2
Abstract
n
a WeconsidertheproblemofreconstructinggeneralsolutionstotheHelmholtzequation∆u+λ2u=0,
J for some fixed λ > 0, on some domain Ω ⊂ R2 from the data of these functions at scattered points

2
x ,...,x ⊂Ω. This problem typically arises when sampling acoustic fields with n microphones for the
1 n

purposeofreconstructingthisfieldoveraregionofinterestΩthatiscontainedinalargerdomainDover
]
A whichtheacousticfieldisgenerated. Inmanyappliedsettings,theboundaryconditionssatisfiedbythe
N acoustic field on ∂D are unknown as well as the exact shape of D. Our reconstruction method is based
. ontheapproximationofageneralsolutionubylinearcombinationsofFourier-Besselfunctionsorplane
h
waves e (x) := eik·x with |k| = λ. We study two different ways of discretizing the infinite dimensional
t k
a space V of solutions to the Helmholtz equation, leading to two different types of finite dimensional
m λ
approximation subspaces, and we analyze the convergence of the least squares estimates to u in these
[
subspacesbasedonthesamples(u(x )) . Ouranalysisdescribestheamountofregularizationthat
i i=1,...,n

1 is needed to guarantee the convergence of the least squares estimate towards u, in terms of a condition
v thatdependsonthedimensionoftheapproximationsubspaceandthesamplesizen. Thisconditionalso
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involvesthedistributionofthesamplesandrevealstheadvantageofusingnon-uniformdistributionsthat
3
2 havemorepointsnearorontheboundaryofΩ. Numericalillustrationsshowthatourapproachcompares
0 favorably with reconstruction methods using other basis functions, and other types of regularization.
.
1
0
1 Introduction
3
1
:
v Acommonproblem in acoustics isto obtainapreciseapproximation ofthe soundfield over aspatial domain
i Ω of interest, using the smallest possible number of pointwise measurements, e.g. as provided by micro-
X
phones. Forinstance, onemaywishtomeasurethecomplexradiationpatternofanextendedsource(source
r
a identification problem), to localize a number of point sources within a spatial domain (source localization
problem), or to optimize the output of a sound reproduction system over a large control area, to name only
a few applications. In practice, the main difficulty that one is usually faced with is how to handle reverber-
ation : the reverberant field might well be of a magnitude comparable to the direct sound, and it depends
in a non-trivial way on both the geometry of the domain D where the acoustic field is defined and the type
of boundary conditions on ∂D (with Dirichlet or Neumann as ideal cases, but more likely in engineering
problems with a frequency-dependent mixed behaviour).
Thegoalofthispaperistostudytheaccuracythatcanbeachievedwhenapproximatingtheacousticfield
over the domain Ω ⊂ D, based on a set of point measurements, without precise knowledge on the geometry
∗ThisresearchissupportedbytheANRprojectD´efi08ECHANGE.
1
of D and boundary conditions on ∂D. A general setting is the following: the soundfield p(x,t) is measured
at microphones located at positions x ,...,x ∈ Ω, and over a (discretized) time interval [0,T]. After
1 n
application of the (discrete) Fourier transform F in the time variable, and considering a given frequency ω,
the function
u(x):=Fp(x,ω)
is a solution on D to the Helmholtz equation
∆u+λ2u=0, (1.1)
where λ=ω/c, with c denoting the wave velocity, and where the boundary conditions are unknown to us.
Depending on the applications, the geometry of the domain Ω may either be 2-D (membranes) or 3-D
(rooms). Here, we restrict our attention to the 2-D setting, although certain features of our approach are
readily extendable to the 3-D setting. Our problem therefore amounts to reconstructing, on some domain
Ω ⊂ R2, a general solution to the Helmholtz equation (1.1) from its sampling at points x ,...,x ∈ Ω.
1 n
These samples may be measured exactly or up to some additive noise. We denote by
y =u(x )+η , l=1,...,n, (1.2)
l l l
these samples, where η represent the additive noise.
l
Reconstruction from scattered points is a widely studied topic, and a variety of methods have been
proposedandanalyzed. Manyexistingmethodscanbeviewedasreconstructingsomeformofapproximation
totheunknownfunctionubysimplerfunctionssuchassplines,partialFouriersumsorradialbasisfunctions.
The success of these methods therefore relies in good part on the quality of the approximation of u by such
simpler functions, which is typically governed by the smoothness of u.
In our present setting, the fact that u obeys the Helmholtz equation, may be used in addition to its
smoothness in order to guarantee the accuracy of certain approximation schemes, which are well adapted to
such solutions. We introduce the Fourier-Bessel functions
b (x):=eijθJ (λr) (1.3)
λ,j j
where (r,θ) are the polar coordinates of x and J is the j-th Bessel function of the first kind. b is solution
j λ,j
to the Helmholtz equation (1.1) over R2 if and only if its parameter λ is the same as in (1.1). Denoting V
λ
the set of the solutions of (1.1), it is known [4] that
L2(Ω)
V =span{b } , (1.4)
λ λ,j
and that the solutions of (1.1) can be approximated by elements of the subspaces Vb = span{b ,−m ≤
m λ,j
j ≤m} as m grows.
An alternative approximation scheme uses planes waves defined by
e (x):=eik·x (1.5)
k
which are solutions of (1.1) if and only if |k|=λ. The spaces Ve, spanned by the particular plane waves
m
(cid:16) (cid:16) 2jπ (cid:17) (cid:16) 2jπ (cid:17)(cid:17)
e :=e , k :=λ cos ,sin , j =−m,...,m, (1.6)
j kj j 2m+1 2m+1
can also be used to approximate solutions of (1.1) as m grows [4].
2
The most widely used approach to approximate u in a finite dimensional space V , from its data at
m
points x ,...,x , is the least squares method, namely with m≤n solving the minimization problem
1 n
n
1 (cid:88)
π =argmin |y −v(x )|2. (1.7)
v∈Vmn i i
i=1
The effectiveness of the least squares approximation is governed by a certain trade-off in the choice of the
dimension m of the approximation:
• A small value of m leads to a highly regularized reconstruction of u, which is usually robust but has
poor accuracy.
• A large value of m may lead to unstable and therefore inaccurate reconstructions although the space
V contains finer approximants to u.
m
Let us observe that regularization is relevant even in a noiseless context where the function is measured
exactly: for example choosing m=n corresponds to searching for an exact interpolation of the data which
may be very unstable and inaccurate, similar to the Runge phenomenon in polynomial approximation.
In this paper, we discuss the amount of regularization which is needed when applying the least squares
methodusingthefinitedimensionalsubspacesVb andVe extractedfromV . Withsuchdiscretizations,the
m m λ
distribution of the sampling points x ,...,x has an influence on the above described trade-off. Our main
1 n
theoreticalresult, establishedinthecaseofadisc, showsthathighervaluesofm, leadingthereforetobetter
accuracy, can be used if the x are not uniformly distributed on Ω in the sense that a fixed fraction of these
i
points are located on the boundary ∂Ω. This result is confirmed by numerical experiments.
Therestofthispaperisorganizedasfollow: wegiveabriefaccountin§2onapproximationofsolutionsto
(1.1)byFourier-BesselfunctionsandplanewaveswhichreliesonVekua’stheory,andin§3ongeneralresults
onthestabilityandaccuracyofleast-squaresapproximationsrecentlyestablishedin[2]. Wethenstudyin§4
the spaces Ve and Vb in more detail, in the particular case where Ω is a disc, and use the above mentioned
m m
results to compare least-squares approximations on these spaces based on different sampling strategies. In
§5,wepresentnumericalteststhatillustratethevalidityofthiscomparison. Wealsoshowthatourapproach
comparesfavorablywithreconstructionsbasedonotherapproximationschemessuchaspartialFouriersums
(that do not exploit the fact that u is a solution to (1.1)) and to other form of regularizations such as
Orthogonal Matching Pursuit.
2 Approximation by Fourier-Bessel functions and plane waves
Results on the approximation of solutions to (1.1) by Fourier-Bessel functions and plane waves given in
[4] are based on the theory developed in the 1950’s by Vekua [6]. This theory generalizes approximation
results for holomorphic functions, viewed as solutions of ∆u=0, to solutions of more general elliptic partial
differential equations, by means of appropriate operators that link the two types of solutions.
In the case of the Helmholtz equation on a domain Ω, that is star-shaped with respect to a point which
is fixed as the origin 0, these operators (in their version mapping harmonic functions to solutions of (1.1))
have the explicit expression
1
λ|x|(cid:90) 1 √
V φ(x)=φ(x)− √ J (λ|x| 1−t)dt, (2.1)
1 2 1−t 1
0
3
and
1
λ|x|(cid:90) 1 √
V φ(x)=φ(x)+ I (λ|x| 1−t)dt, (2.2)
2 2 (cid:112)t(1−t) 1
0
where |x| stands for the euclidean norm of x, J is the Bessel function of the first kind of order 1 and I the
1 1
modifiedBesselfunctionofthefirstkindoforder1,see[3]formoredetails. Theseoperatorshaveimportant
properties:
• Their are linear.
• V maps harmonic functions to solutions of the Helmholtz equation, and V does the converse.
1 2
• They are continuous in the Sobolev Hk norms for all k ≥0.
• They are inverse to each other on these spaces.
As a consequence, any approximation method for harmonic functions can be translated as an approxi-
mation method for solutions of the Helmholtz equation. In particular, approximation of harmonic functions
by harmonic polynomials of degree m translates as approximation of solution of the Helmholtz equation by
the so-called generalized harmonic polynomials m which are their image by V . The generalized harmonic
1
polynomialsofdegreemcanbeexpressedaslinearcombinationsoftheFourier-Besselfunctions(1.3),leading
therefore to results for the approximation of solutions to (1.1) by elements of Vb in Sobolev norms. More
m
precisely, the following result can be obtained when the domain Ω is convex, see theorem 3.2 of [4]
(cid:18)logm(cid:19)p−k
min (cid:107)u−v(cid:107) ≤C (cid:107)u(cid:107) , (2.3)
v∈Vb Hk m Hp
m
where the constant C depends on p, k, λ and the geometry of Ω. This results still holds for more general,
star-shaped convex domains, with a slower convergence.
Planes waves and Bessel functions are related by the Jacobi-Anger identity
(cid:88)
e = imJ (λr)eim(θ−φ). (2.4)
kφ m
m∈Z
where k :=λ(cos(φ),sin(φ)), and its converse, the Bessel integral
φ
π
1 (cid:90)
J (λr)einθ = e einφdφ, (2.5)
n 2πin kφ
−π
Approximatingtheintegralin(2.5)byadiscretesum,withuniformlysamplingofthewavevectorsk onthe
φ
circleofdiameterλ,leadstoapproximationsofsolutionsto(1.1)bylinearcombinationsofthe2m+1planes
waves e , that is, by elements of Ve. It is also known (see theorem 5.2 of [4]) that such approximations
kj m
have the same convergence properties, e.g., for a convex domain,
(cid:18)logm(cid:19)p−k
min (cid:107)u−v(cid:107) ≤C (cid:107)u(cid:107) . (2.6)
v∈Vb Hk m Hp
e
4
3 Least-square approximations
The results of the previous section quantify how a general solution u of the Helmholtz equation can be
approximated by functions from spaces Ve or Vb. We are now interested in understanding the quality of
m m
approximations from these spaces built by the least squares methods based on scattered data (x ,y ) .
l l l=1,...,n
In particular, we want to understand the trade-off between the dimension m and the number of samples
n. Ideally we would like to choose m large in order to benefit of the approximation properties (2.3) and
(2.6), however not too large so that stability of the least-square method is ensured. We are also interested
in understanding how the spatial distribution of the sample x influences this trade-off.
l
This problem was recently studied in [2], in a general setting where the x are independently drawn ac-
l
cordingtoagivenprobabilitymeasureν definedonΩ. Thismeasurethereforereflectsthespatialdistribution
of the samples. For example, the uniform measure
dν :=|Ω|−1dx, (3.1)
tends to generate uniformly spaced samples. In order to present the general result of [2], we assume that
(V ) is an arbitrary sequence of finite dimensional spaces of functions defined on Ω with dim(V )=m.
m m≥1 m
We introduce the L2 norm with respect to the measure ν
(cid:16)(cid:90) (cid:17)1/2
(cid:107)v(cid:107):= |v|2dν , (3.2)
Ω
and we define the best approximation error for a function u in this norm as
σ (u):= min (cid:107)u−v(cid:107) (3.3)
m
v∈Vm
Note that, in the noiseless case, the least squares method amounts to computing the best approximation of
u onto V with respect to the norm
m
(cid:16)1 (cid:88) (cid:17)1/2
(cid:107)v(cid:107) := |v(x )|2 . (3.4)
n n l
Thisnormcanbeviewedasanapproximationofthenorm(cid:107)v(cid:107)basedonthedraw,anditisthereforenatural
to compare the error (cid:107)u−π(cid:107) where π is computed by (1.7) with σ (u). We give below a criterion that
m
describes under which condition on m these two quantities are of comparable size.
Here, we assume that (L ,...,L ) is a basis of V which is orthonormal in L2(Ω,ν). We define the
1 m m
quantity
m
(cid:88)
K(m)=K(V ,ν):=max |L (x)|2, (3.5)
m j
x∈Ω
j=1
which depends both on V and on the chosen measure ν, but not on the choice of the orthonormal basis
m
since it is invariant by rotation.
We also assume that an a-priori bound (cid:107)u(cid:107) ≤ M is known on the function u. We can therefore only
L∞
improve the least squares estimate by defining
u˜:=T (π), (3.6)
M
where T (t) := sign(t)max{|t|,M} and π is given by (1.7). The following result was established in [2], in
M
the case of noiseless data, i.e. η =0 in (1.2).
l
5
Theorem 3.1 Let r >0 be arbitrary but fixed and let κ:= 1−log2. If m is such that
2+2r
n
K(m)≤κ , (3.7)
logn
then, one has
E((cid:107)u−u˜(cid:107)2)≤(1+ε(n))σ (u)2+8M2n−r, (3.8)
m
where ε(n):= 4κ →0 as n→+∞.
logn
It is also established in [2] that the condition (3.7) ensures the numerical stability of the least-square
method,withprobabilitylargerthan1−2n−r. Theseresultssuggesttosettheregularizationlevelbypicking
thelargestvalueofm∗ =m∗(n)suchthat(3.7)holds. Thedependanceofm∗(n)withnisobviouslyrelated
to that of K(m) with m. In particular, slower growth of K(m) with m implies faster growth of m∗(n) and
therefore faster convergence of the least squares approximation. Notice that we always have
(cid:90) m
(cid:88)
K(m)≥ |L |2dν =m. (3.9)
j
j=1
Ω
In the next section, we evaluate K(Ve,ν) and K(Ve,ν) in the case where Ω is a disk, for various choices of
m b
the measure ν.
4 Fourier-Bessel functions and plane waves on the disc
As mentioned in the previous section, the quantity K(m) depends both on the space V and the measure
m
ν that reflects the sampling strategy. Here we study the case where V is either one of the spaces of plane
m
waves Ve or of Fourier-Bessel functions Vb defined in the introduction. Note that the dimension of these
m m
spaces is not m but 2m+1, but for notational simplicity we maintain the shorthand K(m) for K(Ve,ν)
m
and K(Ve,ν).
b
We consider two sampling strategies. The first one uses the uniform probability distribution
dx
ν := , (4.1)
0 |Ω|
and the second one combines uniform sampling on the domain and on its boundary, with proportion 0 <
α≤1, according to the probability distribution
dx dσ
ν :=(1−α) +α . (4.2)
α |Ω| |∂Ω|
In order to obtain explicit results, we focus on the simple case where Ω is a disk. Without loss of generality,
we fix
Ω:={x∈R2 : |x|≤1}. (4.3)
4.1 Fourier-Bessel approximation
Fourier-Bessel functions are orthogonal on the disk with respect to any rotation-invariant measure, because
of their angular dependence in einθ. This allows a simple computation of the quantity K(m) for the space
Vb, leading to the following result.
m
6
Theorem 4.1 For the space Vb and the measure ν on the unit disk Ω, one has for sufficiently large m
m α
K(m)≥c +c m2, (4.4)
0 1
when α=0 (that is, for the uniform measure), for any c <1/4, and where c depends on c and λ, and
1 0 1
2m
K(m)≤C+ , (4.5)
α
when α>0, where C depends on λ and α.
Proof: Since the Fourier-Bessel functions are orthogonal in L2(Ω,ν ), we have
α
(cid:13) (cid:13)
K(m)=(cid:13)(cid:13)(cid:13) (cid:88)m |bj|2 (cid:13)(cid:13)(cid:13) . (4.6)
(cid:13) (cid:107)b (cid:107)2(cid:13)
(cid:13)j=−m j (cid:13)
L∞(Ω)
In the case of the uniform measure, we bound K(m) from below. We first write
(cid:13) (cid:13)
K(m)≥(cid:13)(cid:13)(cid:13) (cid:88)m |bj|2 (cid:13)(cid:13)(cid:13) . (4.7)
(cid:13) (cid:107)b (cid:107)2(cid:13)
(cid:13)j=−m j (cid:13)
L∞(∂Ω)
We next bound (cid:107)b (cid:107)2, for j >(cid:100)λ(cid:101) (the case j <−(cid:100)λ(cid:101) is identical, as |b |=|b |), according to
j j −j
1 (cid:90)
(cid:107)b (cid:107)2 = J (λ|x|)2dx
j π j
Ω
1
(cid:90)
= 2 rJ (λr)2dr
j
0
∞
4 (cid:88)
= (j+1+2p)J2 (λ)
λ2 j+1+2p
p=0
∞
4 (cid:88)
≤ (j+1+2p)(λ/j)2+4pJ2(λ)
λ2 j
p=0
(cid:32) (cid:33)
j+1 1 2 (λ/j)4
= 4 + J2(λ)
j2 1−(λ/j)4 j+1(1−(λ/j)4)2 j
where the third equality is identity (11.3.32) of [1] for the third equality, and the first inequality comes from
(A.6) of [5]. As |b (1,θ)|=|J (λ)|, we have
j j
(cid:32) (cid:33)−1
|b (1,θ)2| j2 1 2 (λ/j)4 j
j ≥ + ∼ (4.8)
(cid:107)bj(cid:107)2 4(j+1) 1−(λ/j)4 j+1(1−(λ/j)4)2 4
For any c<1, there exists a j such that when j >j ,
0 0
|b (1,θ)2| j
j ≥c (4.9)
(cid:107)b (cid:107)2 4
j
and
K(m) ≥ (cid:88)j0 |Jj(λ)|2 +2c (cid:88) j (4.10)
(cid:107)b (cid:107)2 4
j
j=−j0 j0<j≤m
≥ (cid:88)j0 |Jj(λ)|2 −cj0(j0+1) + cm(m+1) (4.11)
(cid:107)b (cid:107)2 4 4
j
j=−j0
7
which proves the bound (4.4) when m>j .
0
In the case of mixed sampling, we bound K(m) from above by
m (cid:107)b (cid:107)2
(cid:88) j L∞(Ω)
K(m)≤
(cid:107)b (cid:107)2
j
j=−m
When j >λ, the function r (cid:55)→J (λr) is monotone increasing on [0,1], so that (cid:107)b (cid:107) =J (λ). Thus,
l j L∞(Ω) j
(cid:107)b (cid:107) J (λ)2
j L∞(Ω) = j
(cid:107)b (cid:107)2 2π
n 1−α(cid:82) J (λ|x|)2dx+ α (cid:82) J (λ)2dθ
π j 2π j
D 0
1
<
α
We then have
2(m−(cid:98)λ(cid:99)) (cid:88)(cid:98)λ(cid:99) (cid:107)bj(x)(cid:107)2L∞(Ω)
K(m)≤ +
α (cid:107)b (cid:107)2
j
j=−(cid:98)λ(cid:99)
which proves (4.5).
4.2 Plane wave approximation
Similar results can be obtained for the plane wave approximation:
Theorem 4.2 For the space Ve and the measure ν on the unit disk Ω, one has for sufficiently large m
m α
K(m)≥c +c m2, (4.12)
0 1
when α=0 (that is, for the uniform measure), for any c <1/4, where c depends on c and λ, and
1 0 1
2m
K(m)≤C +C , (4.13)
1 2 α
when α>0, for any C >1, where C depends on C , λ and α.
2 1 2
Proof: Since plane waves are not orthogonal in L2(Ω,ν), the first step is the computation of an orthogonal
basis of the space spanned by these plane waves. Let us consider 2m+1 plane waves, with wave vectors
uniformly distributed on the circle of radius λ. For the measures considered here, the Gram matrix of this
family is a circulant matrix, which is diagonalized in the Fourier basis. An orthogonal family spanning the
same space is therefore given by functions that are linear combinations of plane waves with the coefficients
of the discrete Fourier transform:
m
1 (cid:88)
bm := e2πij/(2m+1)e (4.14)
j (2m+1) kj
j=−m
This formula may be thought as a quadrature for the Bessel integral (2.5): the bm are thus approximations
j
of the Fourier-Bessel functions b . In order to bound the quantity K(m)=K(Ve,ν), we compare it to the
j m
quantity K(Vb,ν) which behaviour is described by Theorem 4.1. Using (8) and (9) from [5] we have
m
(cid:88)
bm = ip(2m+1)b , (4.15)
j j+p(2m+1)
p∈Z
8
and
(cid:12)(cid:12)(cid:12)bmj −1(cid:12)(cid:12)(cid:12)≤ (cid:88) |bj+p(2m+1)|. (4.16)
(cid:12)b (cid:12) |b |
j p∈Z−{0} j
With m≥j ≥λ, we thus have for all 0≤r ≤1 and 0≤θ ≤2π,
 
(cid:12)(cid:12)(cid:12)(cid:12)bbmj((rr,,θθ)) −1(cid:12)(cid:12)(cid:12)(cid:12) ≤ |b (r1,θ)|(cid:88)|bj+(p+1)(2m+1)(r,θ)|+(cid:88)|bj+(p+1)(2m+1)−2j(r,θ)|
j j
p≥0 p≥0
(cid:88)(cid:32) (cid:18)λ(cid:19)−2j(cid:33)(cid:18)λ(cid:19)(p+1)(2m+1)
≤ 1+
j j
p∈N
(cid:16) (cid:17)2m+1 (cid:16) (cid:17)2m+1−2j
λ + λ
j j
=
(cid:16) (cid:17)2m+1
1− λ
j
2λ
≤ j
1− λ
j
where we have used equation (A.6) of [5] to obtain the second inequality.
Using the orthogonality of the b , we have
j
(cid:88)
(cid:107)bm(cid:107)2 = (cid:107)b (cid:107)2, (4.17)
j j+p(2m+1)
p∈Z
and
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:107)bmj (cid:107)2 −1(cid:12)(cid:12)= (cid:88) (cid:107)bj+p(2m+1)(cid:107)2. (4.18)
(cid:12)(cid:107)b (cid:107)2 (cid:12) (cid:107)b (cid:107)2
(cid:12) j (cid:12) p∈Z−{0} j
When j ≥λ and l≥0 we bound (cid:107)b (cid:107)2 according to
j+l
1
(cid:90)
(cid:107)b (cid:107)2 = 2π rJ (λr)2dr
j+l j+l
0
1
(cid:90) (cid:18)λr(cid:19)2l
≤ 2π r J (λr)2dr
j j
0
1
(cid:18)λ(cid:19)2l (cid:90)
≤ 2π J (λr)2dr
j j
0
(cid:18)λ(cid:19)2l
= (cid:107)b (cid:107)2
j j
where we again have used equation (A.6) of [5] to obtain the first inequality. We thus have
(cid:12) (cid:12)
(cid:12)(cid:12)(cid:107)bmj (cid:107)2 −1(cid:12)(cid:12) = (cid:88) (cid:107)bj+p(2m+1)(cid:107)2
(cid:12)(cid:107)b (cid:107)2 (cid:12) (cid:107)b (cid:107)2
(cid:12) j (cid:12) p∈Z−{0} j
 
1 (cid:88) (cid:88)
= (cid:107)b (cid:107)2  (cid:107)bj+(p+1)(2m+1)(cid:107)2+ (cid:107)bj+(p+1)(2m+1)−2j(cid:107)2
j
p≥0 p≥0
9
 
(cid:88)(cid:18)λ(cid:19)2(p+1)(2m+1) (cid:88)(cid:18)λ(cid:19)2((p+1)(2m+1)−2j)
≤  + 
j j
p≥0 p≥0
(cid:16) (cid:17)2(2m+1) (cid:16) (cid:17)2(2m+1−2j)
λ + λ
j j
=
(cid:16) (cid:17)2(2m+1)
1− λ
j
(cid:16) (cid:17)2
2 λ
j
≤
(cid:16) (cid:17)2
1− λ
j
We now consider the case α = 0. Using the above comparison results between b and bm, we can find,
j j
for any c<1 a positive integer j such that for m≥j ≥j ,
c c
|bm(1,θ)|2 J (λ)2
j ≥c j . (4.19)
(cid:107)bm(cid:107)2 (cid:107)b (cid:107)2
j j
We may thus write
(cid:13) (cid:13)
K(Ve,ν) = (cid:13)(cid:13)(cid:13) (cid:88)m |bmj |2 (cid:13)(cid:13)(cid:13)
m (cid:13) (cid:107)bm(cid:107)2(cid:13)
(cid:13)j=−m j (cid:13)
L∞(Ω)
≥ (cid:88)m |bmj (1,0)|2
(cid:107)bm(cid:107)2
j=−m j
≥ (cid:88) |bmj (1,0)|2 +2c(cid:88)m Jj(λ)2,
(cid:107)bm(cid:107)2 (cid:107)b (cid:107)2
j<jc j j=jc j
≥ 2c(cid:88)m Jj(λ)2.
(cid:107)b (cid:107)2
j
j=jc
The last sum can be bounded from below in a similar way as in the proof of Theorem 4.1, proving (4.12).
We next consider the case α>0. We then write
m (cid:107)bm(cid:107)2
(cid:88) j L∞(Ω)
K(m)≤ . (4.20)
(cid:107)bm(cid:107)2
j=−m j
For any C > 1, there is a j > λ such that when j > j , we have 1|b (r,θ)| ≤ |bm(r,θ)| ≤ C|b (r,θ)|, so
C C C j j j
that (cid:107)bm(cid:107) <C(cid:107)b (cid:107) =CJ (λ), and |bm(1,θ)|≥|J (λ)|/C. We then have, for m≥j ≥j ,
j L∞(Ω) j L∞(Ω) j j j C
(cid:107)bm(cid:107)2 (cid:107)bm(cid:107)2
j L∞(Ω) j L∞(Ω)
=
(cid:107)bm(cid:107)2 2π
j 1−α(cid:82) |bm|2dx+ α (cid:82) |bm|2dθ
π j 2π j
D 0
1 C2J (λ)2
≤ j
αJ (λ)2/C2
j
C
≤ 2
α
10