Two-parameter regularization of ill-posed spherical pseudo-differential equations in the space of continuous functions

0.54 MB·English

by  Hui Cao

manuscript No.
(will be inserted by the editor)
Two-parameter regularization of ill-posed spherical
pseudo-differential equations in the space of
continuous functions
Hui Cao · Sergei V. Pereverzyev ·
Ian H. Sloan · Pavlo Tkachenko
5 Received:date/Accepted:date
1
0
2
Abstract In this paper, a two-step regularization method is used to solve an

n
ill-posed spherical pseudo-differential equation in the presence of noisy data.
a
For the first step of regularization we approximate the data by means of a
J
spherical polynomial that minimizes a functional with a penalty term consist-
2
ing of the squared norm in a Sobolev space. The second step is a regularized

] collocation method. An error bound is obtained in the uniform norm, which
A
ispotentiallysmallerthanthatforeitherthenoisereductionaloneorthereg-
N ularized collocation alone. We discuss an a posteriori parameter choice, and
. presentsomenumericalexperiments,whichsupporttheclaimedsuperiorityof
h
the two-step method.
t
a
m
Keywords Two-parameter regularization · spherical pseudo-differential
[
equations · quasi-optimality criterion

1
v
2
H.Cao
6
Guangdong Province Key Laboratory of Computational Science, Sun Yat-sen University,
3
Guangzhou,China
0 E-mail:[e-mail]
0
. S.V.Pereverzyev
1 Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy
0 ofSciences,Altenbergerstrasse69,4040Linz,Austria
5 E-mail: [e-mail]
1
I.H.Sloan
:
v SchoolofMathematicsandStatistics,UniversityofNewSouthWales,SydneyNSW2052,
i Australia
X E-mail:[e-mail]
r P.Tkachenko
a
Johann Radon Institute for Computational and Applied Mathematics, Austrian Academy
ofSciences,Altenbergerstrasse69,4040Linz,Austria
E-mail:[e-mail]
2 HuiCaoetal.
1 Introduction
Mathematical models appearing in geoscience commonly have the form of an
ill-posed spherical pseudo-differential equation
Ax=y, (1)
where A is a pseudo-differential operator that relates continuous functions
x∈C(Ω ) and y ∈C(Ω ) defined on concentric spheres Ω ,Ω ∈R3 of radii
R ρ R ρ
R ≤ ρ. For example, in satellite geodesy, this approach has been introduced
in [5,25] where the spheres Ω ,Ω are models of respectively the surface of
R ρ
the Earth and the surface traversed by the satellite.
Recall that a spherical pseudo-differential operator A:C(Ω )→C(Ω ) is
R ρ
a linear operator that assigns to any x∈C(Ω ) a function
R
∞ 2k+1 (cid:18) (cid:19)
X X 1 ·
Ax:= a x Y ∈C(Ω ), (2)
k bk,jρ k,j ρ ρ
k=0 j=1
where
(cid:28)1 (cid:16) · (cid:17) (cid:29) 1 Z (cid:16)τ (cid:17)
x = Y ,x(·) := x(τ)Y dΩ (τ)
bk,j R k,j R R k,j R R
L2(ΩR) ΩR
are the spherical Fourier coefficients, and Y (·),j = 1,2,...,2k+1, are the
k,j
sphericalharmonics[16]ofdegreek whichareL -orthonormalwithrespectto
2
the unit sphere Ω ∈R3, as a result of which
1
(cid:28)1 (cid:16) · (cid:17) 1 (cid:16) · (cid:17)(cid:29)
Y , Y =hY ,Y i =δ δ . (3)
R k,j R R k0,j0 R k,j k0,j0 L2(Ω1) kk0 jj0
L2(ΩR)
Thesequenceofrealnumbers(a )∞ isreferredtoasthesphericalsymbolof
k k=0
A. We shall assume that a is positive, and converges monotonically to zero.
k
In the case when the symbol sequence a tends to zero fast enough the
k
operator A is compact. Therefore, its inverse A−1 is unbounded, and keeping
inmindHadamard’sdefinitionofawell-posedproblem(existence,uniqueness,
andcontinuityofinverse),weconcludethatforthiscasethefirstandthethird
conditions are violated, and the problem (1), (2) with y ∈C(Ω ) becomes ill-
R
posed. Therefore, a regularization technique should be employed for solving
it [4].
As examples of the ill-posed problem (1), (2) we can mention the satellite-
(cid:16) (cid:17)k
to-satellitetrackingproblem(SST-problem)witha = k+1 R ,thesatellite
k ρ ρ
(cid:16) (cid:17)k
gravity gradiometry problem (SGG-problem) with a = (k+1)(k+2) R , etc.
k ρ2 ρ
(formoredetailsontheseandotherexampleswecanreferaninterestedreader
to [5]). These problems are severely ill-posed because of the occurrence of the
geometric factor (R/ρ)k.
Two-parameterregularization 3
Itisworthmentioningthatmanypracticalapplicationsuseafinitedimen-
sional approximation of the solution of (1), (2). For example, Earth Gravity
Models, such as EGM96 or EGM2008 [20] are parametrized by the spherical
Fourier coefficients up to some prescribed degree M.
Note that in applications the function y is assumed to be continuous.
However, in practice one is provided just with a finite number of points
{t }N ⊂Ω atwhichinformationaboutthevaluesofy iscollected.Itshould
i i=1 ρ
benotedalsothatthepointwisedatay(cid:15)(t )containmeasurementerrors,which
i
can be modeled, for example, in the following way:
|y(cid:15)(t )−y(t )|≤(cid:15) , i=1,...,N.
i i i
We assume for convenience that there exists a function y(cid:15) ∈ C(Ω ) standing
ρ
for the noisy version of the original function y, such that
ky(cid:15)−yk ≤(cid:15):= max {(cid:15) },
C(Ωρ) 1≤i≤N i
where (cid:15) are measurement errors.
i
In such a setup the problem (1), (2) is reduced to the following spherical
pseudo-differential operator equation
M 2k+1 (cid:18) (cid:19)
X X 1 ·
A x:= a x Y =y(cid:15). (4)
M k bk,jρ k,j ρ
k=0 j=1
Severalregularizationtechniquescanbeusedfortreating(4).Forexample,
the method where the discretization level M plays the role of regularization
parameter was discussed in [2,3,9]. However, in our case we assume that the
value of M is prescribed, thus this approach cannot be used.
The problem (4), but without noise in the right-hand side, was studied
in [13]. In this noise-free case the solution x can be approximated by solving
the equation
AMx=VbMy,
where VbM : C(Ωρ) → PM(Ωρ) is the so-called quasi-interpolatory operator.
Here by P (Ω ) we denote the set of all spherical polynomials of degree less
M ρ
thanorequaltoM,orinotherwordstherestrictiontoΩ ofthepolynomials
ρ
in R3 of degree less than or equal to M. Thus, in [13] the authors suggest
constructingapolynomialapproximationofy fromtheoriginalpointwisedata
y(t ),i=1,2,...,N, and then formally inverting the operator A .
i M
In principle, this idea can be used also for the ill-posed case (4). However,
in contrast to the approximation of a noise-free continuous function y on the
sphere by means of polynomials, which has been discussed by many authors
(see,forexample,[7,24,26,28,30]),theapproximationofnoisyfunctionsy(cid:15) has
been studied only recently in [1,21], to the best of our knowledge. Applying
the method from [1,21] we first perform so-called data-smoothing (or noise
reduction). After this data preprocessing the formal inversion of A should
M
4 HuiCaoetal.
be safer. However, for performing the noise reduction step one needs a priori
information about the smoothness of the function y, which is usually not
available,afactthatmakesthedirectapplicationofthescheme[21]notalways
appropriate.
In a different direction, for estimating the Fourier coefficients x directly
bk,j
from noisy measurements y(cid:15)(t ) a regularized collocation method has been re-
i
centlypresentedin[18].Thismethodisbasedonthestandardandwidelyused
Tikhonov-Phillipsregularization.However,itiswell-knownthattheTikhonov-
Phillips method suffers from saturation [15,19], meaning that the accuracy of
reconstruction cannot be improved regardless of the smoothness of the solu-
tion x. Another point is that while the Tikhonov-Phillips method has been
well studied in the Hilbert space L , to the best of our knowledge, no analysis
2
has been done in the space of continuous functions, a natural choice for our
noise model.
In the present study we combine these two approaches. Moreover, in con-
trasttothepreviousresults,wewillanalyzetheapproximationinthespaceof
continuous functions. A combination of two regularization methods into one
can be seen as a two-step regularization, in which we use the composition
R ◦T of data smoothing operator T , and a regularized collocation
α,M λ,M λ,M
operator R . In the literature there are not so many studies on two-step
α,M
regularization,andwecanreferonlyto[11,12].However,theanalysisinthose
papers does not correspond to the setting of our problem (1)–(4).
The paper is organized as follows. In the next section we present the regu-
larization method for noise reduction and define the data smoothing operator
T . In Section 3 we will give a short overview of the regularized collocation
λ,M
method,anddefinetheoperatorR .Section4isdevotedtotheoreticalerror
α,M
bounds for the constructed two-parameter regularization. We will show that
the approximation has the potential to perform at least as well as the better
of the one-parameter regularizations which are involved in the composition.
Finally, in the last section we discuss an a posteriori parameter choice rule,
and present some numerical experiments supporting the claimed superiority
of our method.
2 Data noise reduction
At the first step of our scheme we approximate the noisy continuous function
y(cid:15) ∈C(Ω )bymeansofasphericalpolynomialp ∈P (Ω ).Asdiscussedin
ρ M M ρ
the Introduction, instead of y we are provided only with pointwise measure-
ments y(cid:15)(t ),i = 1,2,...,N. Therefore we introduce the sampling operator
i
S :C(Ω )→RN for which
N ρ ω
S y(cid:15) :=(y(cid:15)(t ),y(cid:15)(t ),...,y(cid:15)(t )).
N 1 2 N
By RN we denote the vector space RN equipped with the inner product
ω
Two-parameterregularization 5
N
X
hη,γi := ω η γ , η,γ ∈RN;
ω i i i
i=1
the corresponding norm is kηk = hη,ηi1/2. Here ω ,ω ,...,ω are positive
ω ω 1 2 N
weights in a cubature formula, and t ,t ,...,t ∈ Ω are the corresponding
1 2 N ρ
cubaturepoints,withthecubaturerulehavingthepropertyofbeingexactfor
all polynomials of degree up to 2M,
N Z
X
∀p∈P (Ω ), ω p(t )= p(ζ)dΩ (ζ). (5)
2M ρ i i ρ
i=1 Ωρ
A method for approximating y from S y(cid:15) ∈ RN based on such a cubature
N ω
rule (but with all weights equal) was proposed recently in [1], while its gener-
alization to other positive weights was presented in [21].
We briefly outline the method from [21] and its extension to pseudo-
differential operator equations. We start with the observation that the space
P (Ω ) of spherical polynomials p of degree at most M can be considered as
M ρ
the reproducing kernel Hilbert space (RKHS) H generated by the kernel
K
M 2k+1 (cid:18) (cid:19) (cid:18) (cid:19)
X X 1 t 1 τ
K(t,τ)= β−2 Y Y , t,τ ∈Ω , (6)
k ρ k,j ρ ρ k,j ρ ρ
k=0 j=1
where β = (β ,β ,...,β ,...) is a non-decreasing sequence of positive pa-
1 2 M
rameters. Note that the inner product of H associated with this kernel can
K
be written as
XM β2 2Xk+1(cid:28) (cid:18)·(cid:19) (cid:29) (cid:28) (cid:18)·(cid:19) (cid:29)
hf,gi = k Y ,f Y ,g . (7)
HK ρ2 k,j ρ k,j ρ
k=0 j=1 L2(Ωρ) L2(Ωρ)
The reproducing property hf,K(·,τ)i = f(τ) for τ ∈ Ω can be verified
HK R
easily. In this paper the numbers β ,β ,...,β are assumed to be given a
1 2 N
priori. We shall see their role later.
The approximation p studied in [1,21] appears as the minimizer of the
min
following functional
n o
p =argmin kS p−S y(cid:15)k2 +λkpk2 , p∈P (Ω ) , (8)
min N N ω HK M ρ
where λ ≥ 0 is a regularization parameter. Note that (8) can be seen as the
definition of the data smoothing operator T :C(Ω )→P (Ω ) such that
λ,M ρ M ρ
p =T y(cid:15). Note that the regularization term involves the H -norm of p,
min λ,M K
and hence depends on the rate of growth of the parameters β . The solution
k
of (8) is given by the following theorem.
6 HuiCaoetal.
Theorem 1 Assume that the points {t } and weights {ω } are such that (5)
i i
holds. Then the minimizer T y(cid:15) =p in (8) has the form
λ,M min
M 2k+1 (cid:18) (cid:19) N (cid:18) (cid:19)
(T y(cid:15))(·)=X 1 X 1Y · Xω 1Y ti y(cid:15)(t ). (9)
λ,M 1+λβ2 ρ k,j ρ iρ k,j ρ i
k=0 k j=1 i=1
Proof It is known that the minimizer of the functional in (8) can be written
as
T y(cid:15) =(λI+S∗S )−1S∗S y(cid:15), (10)
λ,M N N N N
where S∗ : RN → H is the adjoint of S , and I is the identity operator in
N ω K N
H . By definition, for (η )N ∈RN we have
K i i=1 ω
N N
X X
hη,S fi = η f(t )ω = η hK(t ,·),f(·)i ω
N ω i i i i i HK i
i=1 i=1
* N +
X
= ω K(t ,·)η ,f(·) ,
i i i
i=1 HK
thus there holds
N
X
(S∗η)(·)= ω K(t ,·)η , ∀η ∈RN,
N i i i ω
i=1
and thereby
N
X
(S∗S f)(·)= ω K(t ,·)f(t ).
N N i i i
i=1
Inserting this expression into (10), we observe that p solves
min
N N
X X
λp (·)+ ω K(t ,·)p (t )= ω K(t ,·)y(cid:15)(t ) (11)
min i i min i i i i
i=1 i=1
The spherical harmonic expansion of the polynomial p =T y(cid:15) is
min λ,M
M 2k+1(cid:28) (cid:18) (cid:19)(cid:29) (cid:18) (cid:19)
X X 1 · 1 t
p (t)= p , Y Y , t∈Ω . (12)
min min ρ k,j ρ ρ k,j ρ ρ
k=0 j=1 L2(Ωρ)
To find the coefficients in this expansion, we write the second term on the
left-hand side of (11), on using (5) and then (6), as
N
X
ω K(t ,t)p (t )=hp ,K(·,t)i
i i min i min L2(Ωρ)
i=1
N 2k+1(cid:28) (cid:18) (cid:19)(cid:29) (cid:18) (cid:19)
X X 1 · 1 t
= β−2 p , Y Y .
k min ρ k,j ρ ρ k,j ρ
k=0 j=1 L2(Ωρ)
Two-parameterregularization 7
After expanding the right-hand side of (11) using (6), and then equating the
(cid:16) (cid:17)
coefficients of Y t , we obtain
k,j ρ
(cid:28) (cid:18) (cid:19)(cid:29) N (cid:18) (cid:19)
t X t
(λ+β−2) p ,Y =β−2 ω Y y(cid:15)(t ), t∈Ω ,
k min k,j ρ k i k,j ρ i ρ
L2(Ωρ) i=1
which on substituting into (12) yields the desired result.
tu
At this point the solution of the first step depends on the regularization
parameter λ, and the penalization weights β . As mentioned in the Introduc-
k
tion, the choice of the regularization parameter λ will be addressed in the last
section. A data-driven choice of the penalization weights β has been recently
k
discussed in [21].
The assumption that the function x on Ω is continuous implies that
R
x∈L2(Ω ),andhencethatitsFouriercoefficients(cid:10)1Y (cid:0) · (cid:1),x(cid:11) with
R R k,j R L2(ΩR)
respect to the basis of spherical harmonics are square-summable, i.e,
X∞ 2Xk+1(cid:12)(cid:12)(cid:28)1 (cid:16) · (cid:17) (cid:29) (cid:12)(cid:12)2
(cid:12) Y ,x (cid:12) <∞.
(cid:12) R k,j R (cid:12)
k=0 j=1 (cid:12) L2(ΩR)(cid:12)
Any additional smoothness of x can be measured in terms of the summability
ofFouriercoefficientswithsomeincreasingweightsdependingonthesequence
(β ) or, as it is usual for the regularization theory (see, e. g., [14]), on the
k
symbol(a ).Therefore,itisconvenienttointroducetwoSobolevspacesWφ,β,
k
and Wψ,a, the first depending on the sequence (β ) and the second on the
k
symbol (a ), and defined by
k
Wφ,β :=g∈L2(ΩR):Xk∞=02Xjk=+11(cid:12)(cid:12)(cid:12)(cid:10)R1Yk,j(cid:0)φR2·((cid:1)βk,−g2(cid:11))L2(ΩR)(cid:12)(cid:12)(cid:12)2 =:kgk2Wφ,β <∞, (13)
Wψ,a:=g∈L2(ΩR):X∞ 2Xk+1(cid:12)(cid:12)(cid:12)(cid:10)R1Yk,j(cid:0)ψR·2(cid:1)(a,2g)(cid:11)L2(ΩR)(cid:12)(cid:12)(cid:12)2 =:kgk2Wψ,a <∞,(14)
 k=0 j=1 k 
where φ,ψ are non-decreasing functions such that φ(0) = 0 and ψ(0) =
0. In the literature, see, e.g., [14], the functions φ,ψ go under the name of
“smoothness index functions”, and are usually unknown.
3 Regularized collocation method after noise reduction
After the first step of our method the reduced original equation (4) is trans-
formed into the equation
8 HuiCaoetal.
A x=T y(cid:15). (15)
M λ,M
The regularized collocation method (see [18]) is now applied to this equation,
yielding an approximate solution x(cid:15) of the equation (15), defined as the
α,λ
minimizer of the functional
kB x−S T y(cid:15)k2 +αkxk2 , (16)
N,M N λ,M ω L2(ΩR)
where B := S A : L (Ω ) → RN, and α ≥ 0 is the regularization
N,M N M 2 R ω
parameter for the second step. The minimizer of (16) can be written in the
form
x(cid:15) =(αI+B∗ B )−1B∗ S T y(cid:15), (17)
α,λ N,M N,M N,M N λ,M
where B∗ :RN →L (Ω ) is the adjoint of B , given by
N,M ω 2 R N,M
(B∗ η)(·)=XM a 2Xk+1 1Y (cid:16) · (cid:17)XN ω 1Y (cid:18)ti(cid:19)η ,
N,M k R k,j R iρ k,j ρ i
k=0 j=1 i=1
from which it follows, using (4), that
M 2k+1
X X 1 (cid:16) · (cid:17)
B∗ B x= a2 Y x .
N,M N,M k R k,j R bk,j
k=0 j=1
Using (17) and (9) we then obtain explicitly
x(cid:15) :=XM ak 1 2Xk+1 1Y (cid:16) · (cid:17)XN ω 1Y (cid:18)ti(cid:19)y(cid:15)(t ). (18)
α,λ α+a2 1+λβ2 R k,j R iρ k,j ρ i
k=0 k k j=1 i=1
If we now define R :P (Ω )→P (Ω ) as
α,M M ρ M R
(R p)(·):=(cid:16)(cid:0)αI+B∗ B (cid:1)−1B∗ S p(cid:17)(·)
α,M N,M N,M N,M N
= XM ak 2Xk+1 1Y (cid:16) · (cid:17)XN ω 1Y (cid:18)ti(cid:19)p(t ) (19)
α+a2 R k,j R iρ k,j ρ i
k=0 k j=1 i=1
= XM ak 2Xk+1 1Y (cid:16) · (cid:17)(cid:28)1Y ,p(cid:29) . (20)
α+a2 R k,j R ρ k,j
k=0 k j=1 L2(Ωρ)
then we can write x(cid:15) = R T y(cid:15). Thus R reflects the regularized
α,λ α,M λ,M α,M
collocation second step of the method, whereas T corresponds to the noise
λ,M
reduction first step.
Asonecanseefrom(18),intheendweapproximatetheunknownsolution
x on Ω by means of a spherical polynomial of degree M. In this context a
R
question arises about the best approximation of a continuous function x by
means of a polynomial of degree M. In turn the quality of best polynomial
Two-parameterregularization 9
approximation is determined by the smoothness of x. We shall assume that x
liesintheintersectionofWφ,β andWψ,a,see(13)or(14),thusthesmoothness
ofxisencodedinφandβ ,orψ anda .Forexample,ifthesmoothnessindex
k k
function φ(t) and the sequence β ={β } increase polynomially with t and k,
k
then Jackson’s theorem on the sphere (see [22], Theorem 3.3) tells us that for
x∈Wφ,β there is ν >0 such that
inf kx−pk =O(cid:0)M−ν(cid:1). (21)
p∈PM C(ΩR)
At the same time, if the sequence β = {β } increases exponentially then
k
for polynomially increasing φ and x∈Wφ,β we have
inf kx−pk =O(cid:0)e−qM(cid:1),
p∈PM C(ΩR)
where q is some positive number that does not depend on M.
Intheerroranalysisofthenextsectionweshallmakeuseofaconstructive
polynomial approximation introduced in [26], in which x∈C(Ω ) is approxi-
R
mated by V x∈P (Ω ), given by
M M R
XM (cid:18) k (cid:19)2Xk+1 1 (cid:18) t (cid:19)(cid:28)1 (cid:16) · (cid:17) (cid:29)
(V x)(t)= h Y Y ,x , t∈Ω ,
M M R k,j R R k,j R R
k=0 j=1 L2(ΩR)
(22)
where h (a “filter function”) is a continuously differentiable function on R+
satisfying
(cid:26)
1, t∈[0,1/2],
h(t)= 0≤h(t)≤1fort∈R+
0, t∈[1,∞),
Explicit examples of suitable filter functions h can be found in [27]. It is im-
portantforourlateranalysisthat,asshownin[26],V xis(uptoaconstant)
M
an optimal approximation in the uniform norm, in the sense that,
kx−V xk ≤c inf kx−pk , (23)
M C(ΩR) p∈P[M/2] C(ΩR)
where c is a generic constant, which may take different values at different
occurrences, and [·] denotes the floor function.
In view of (21), for polynomially increasing φ,β and for x∈Wφ,β we have
kx−V xk ≤c[M/2]−ν ≤cM−ν.
M C(ΩR)
Ontheotherhand,forexponentiallyincreasingβ andpolynomiallyincreasing
φthetheory[23]suggeststakingh(t)≡1,t∈[0,1].Inthiscasetheright-hand
√
sideof (23)hastobemodifiedbymultiplyingby M,andbyreplacing[M/2]
by M, thus for x∈Wφ,β there holds
√ √
kx−V xk ≤c M inf kx−pk ≤c Me−qM.
M C(ΩR) p∈PM C(ΩR)
10 HuiCaoetal.
4 Error estimation in uniform norm
Inthissectionwewillestimatetheuniformerrorofapproximationofxbythe
polynomial x(cid:15) given by (18). It is clear that
α,λ
(cid:13) (cid:13) (cid:13) (cid:13)
(cid:13)x−x(cid:15)α,λ(cid:13)C(ΩR) =(cid:13)x−Rα,MTλ,My(cid:15)(cid:13)C(ΩR) (24)
(cid:13) (cid:13)
≤kx−VMxkC(ΩR)+(cid:13)VMx−Rα,MUMy(cid:13)C(ΩR)
(cid:13) (cid:13)
+(cid:13)(Rα,M −Rα,MTλ,M)UMy(cid:13)
C(ΩR)
(cid:13) (cid:13) (cid:0) (cid:1)
+(cid:13)Rα,MTλ,M(cid:13)C(Ωρ)→C(ΩR) kUMy−ykC(Ωρ)+ky−y(cid:15)kC(Ωρ) ,
where U y is the same as V x in (22) except that it is an approximation on
M M
the larger sphere Ω instead of on Ω ,
ρ R
M (cid:18) (cid:19)2k+1 (cid:18) (cid:19)(cid:28) (cid:18) (cid:19) (cid:29)
X k X 1 t ·
(U y)(t)= h Y Y ,y(·) . (25)
M M ρ2 k,j ρ k,j ρ
k=0 j=1 L2(Ωρ)
It is natural to assume that kU y−yk < (cid:15), since otherwise data noise
M C(Ωρ)
is dominated by the approximation error and no regularization is required.
We also restrict ourselves to the case when kx−V xk < c(cid:15), otherwise
M C(ΩR)
the term kx−V xk , representing the error of almost best polynomial
M C(ΩR)
approximation, will dominate the error bound.
Then the bound (24) can be reduced to the following one
(cid:13)(cid:13)x−x(cid:15)α,λ(cid:13)(cid:13)C(ΩR) ≤kVMx−Rα,MUMykC(ΩR) (26)
+k(R −R T )U yk +c(1+kR T k )(cid:15).
α,M α,M λ,M M C(ΩR) α,M λ,M C(Ωρ)→C(ΩR)
An estimate of the coefficient in the last term is given by the following
theorem.
Theorem 2 Under the conditions of Theorem 1
(cid:12) (cid:12)
kRα,MTλ,MkC(Ωρ)→C(ΩR)≤ R1ρtm∈ΩaxR(cid:12)(cid:12)(cid:12)(cid:12)XN ωiXM 4π(α(+2ka+2k)(11)a+kλβk2)Pk(cid:16)tR·ρti(cid:17)(cid:12)(cid:12)(cid:12)(cid:12),
i=1 k=0
where P are the Legendre polynomials of degree k.
k
Proof Inviewofthedefinition(9)and(19),togetherwith(3)withRreplaced
by ρ and the property (5) of the cubature rule, we can write
(R T y)(t)
α,M λ,M
M 2k+1 (cid:18) (cid:19) N (cid:18) (cid:19)
= X ak 1 X 1Y t Xω 1Y ti y(t )
α+a2 1+λβ2 R k,j R iρ k,j ρ i
k=0 k k j=1 i=1
M N (cid:18) (cid:19)
= 1 X2k+1 ak 1 Xω P t·ti y(t ), ∀t∈Ω ,
Rρ 4π α+a2 1+λβ2 i k Rρ i R
k=0 k k i=1