R E S E A R C H
Open Access
L
p
Error estimate for minimal norm SBF
interpolation
Jianjun Wang
1*, Chan-Yun Yang
2and Zhigang Gu
1*Correspondence:
1School of Mathematics and
Statistics, Southwest University, Chongqing, 400715, China Full list of author information is available at the end of the article
Abstract
By the method of spherical splitting, the interpolation capability of the spherical basis function (SBF) is investigated. As the main result, we deduce the error estimate for the minimal norm SBF interpolation in the metric of thepth Lebesgue integral function space on the sphere. The result shows that the interpolation capability of SBF depends not only on the smoothness of the target function, but also on the geometric distributions of the interpolation knots.
MSC: 41A36; 41A25
Keywords: spherical basis function; minimal norm;Lpapproximation; error estimate
1 Introductions and main results
Over the past decades, research in the field of function approximation of scattered data points gradually drifted from the polynomials to the radial basis functions (RBFs), and later to the spherical basis functions (SBFs) in a spherical coordinate. Recently, people, such as Wang and Li [], Freedenet al.[], and Muller [], have moved their interests fur-ther to the topics of spherical approximation [–]. Based on the developments, spherical harmonic analysis was established and had some considerable progress. Meanwhile,Lp error estimations for the SBF approximation were studied hereafter and stepped forward in the studies of Le Giaet al.[], Hubbert and Morton [], and Sloan and Wendland [] after . In , Chen [] established error bound for the minimal norm interpolation on a sphere. With this plentiful foundation, Lin and Cao [] embedded firstly the smooth SBFs in a native space and specified the error bound between the best approximation and the target function viaLpmetric. As a consecutive study, the paper thus aims to derive a minimal norm interpolation with theLpmeasure.
For a fixed integerq(q≥),Sqis a unit sphere inRq+,i.e.,Sq={x= (x
,x, . . . ,xq+)∈ Rq+,x +x+· · ·+xq+= }, anddωrepresents a sufficient small elemental area on the spherical surfaceSq. The total surface area ofSqcan hence be expressed as
ωq=
Sqd
ω= π
q+
(q+ ).
In regard todω, theLp(Sq) inner product of functionsf andgis given as
(f,g)Lp(Sq)=
Sqf(x)g(x)dω(x),
and theLp(Sq) norm is defined as
fp:=fLp(Sq):=
(Sq|f(x)|pdω)
p, ≤p< +∞;
maxx∈Sq|f(x)|, p= +∞,
whereLp(Sq) represents a function space constructed by a complex function,f :Sq→C, fulfillingfp< +∞. An identical function inLp(Sq) is characterized as a function identical to the functions which have the same output values everywhere with the same inputs.
For an integerk≥,Hkqdenotes a linear subspace inRq+constructed by all thekth order
homogeneous harmonic polynomialsp(x) restricted bySq, andqndenotes the function constructed by all thekth order,k≤n, spherical harmonic polynomials. The relationship betweenqnandHkqcan primarily be given as
qn= n
k= Hkq.
Obviously, the dimension ofHkqis
dkq=dimHkq=
k+q–
k+q– k+q–
k+q–
, k≥;
, k= .
InHkq, we select a set of functions Yk,j(j= , , . . . ,dqk) to form a standard orthogonal basisqn={Yk,j,j= , , . . . ,dqk,k= , , , . . . ,n}. The subspaceH
q
k also confirmsL(Sq) =
∞
k=H
q
k and the famous additive law:
dqk
j=
Yk,j(ξ)Yk,j(η) =
dqk
ωq
Pk(q+ :ξ·η),
where ξ ·ηdenotes the inner product ofξ andη. Pk(q+ :x) denotes the kth order Legendre polynomial satisfyingPk(q+ : ) = and complies with
–
Pk(q+ :x)Pl(q+ :x)
–x q
–dx= ωq
ωq–dkq
δk,l.
By givingA={{An,j}:An,j∈R,n= , , , . . . ,j= , . . . ,dqn}, a series{An} ∈A,An> ,n= , , , . . . , can also be expressed as{An,j} ∈A,An,j=An,j= , . . . ,dnq. Using the series{An,j},
a space defined as
HsSq=
F∈LSq,
∞
n=
dqn
j=
Asn(F,Yn,j)<∞
has the inner product (·,·)Hs(Sq):
(F,G)Hs(Sq)= ∞
n=
dqn
j=
and the corresponding norm is consequently defined as
FHs(Sq)=(F,F)Hs(Sq)
.
As known from the definition here,Hs(Sq) is a Hilbert space,i.e., whens>q
,H
s(Sq) has
a reproducing kernel
K(ξ,η) =
∞
n=
dqn
j=
A–nsYn,j(ξ)Yn,j(η)
ifAnsatisfiesAn∼ +n(see the reference []).
Hereafter, we assume consistently in this study that Hs(Sq) is a reproducing kernel Hilbert space (RKHS) constructed by the reproducing kernel and call the space a native space ofK(ξ,η) (see [, ]). From the additive law, we have
K(ξ,η) =
∞
n=
dqn
j=
A–nsYn,j(ξ)Yn,j(η) =
∞
n= dqn
ωq
A–nsPn(q+ ,ξ·η), ξ,η∈Sq,
wherePn(q+ ,x) is thenth order Legendre polynomial. HereK(ξ,η) is obviously a spher-ical radial basis function.
Let us assume thatX={x,x, . . . ,xN} is a set ofNpoints taken from the unit sphere
Sq and hasN output valuesy,y, . . . ,yN corresponding tox,x, . . . ,xN through a func-tionF:
IN(y) =
F∈HsSq:F(xi) =yi,i= , , . . . ,N
.
If there existsF∈IN(y) such thatFcan be minimized, we say that the minimization ofFis a problem subject to a minimal norm. By denotingSN(ξ) as an interpolation with the minimal norm, there is a kernel basis expression for the interpolationSN(ξ) (see the references [, , ]):
SN(ξ) = N
i=
αiK(xi,ξ), ξ∈Sq.
Suppose that yi, i= , , . . . ,N, are produced by the function f(x), i.e., f(xi) =yi, i= , . . . ,N. The interpolation problem becomes intrinsically a function approximation prob-lem to estimate the error betweenf(x) andSN(x). The approximation order of the interpo-lation can then be determined by the grid norm,h, and the inputxi∈X. Here,his defined as
h:=sup
x∈Sq min
xi∈Xdist(x,xi),
where dist(ξ,η) is the spherical distance between ξ andη, i.e., dist(ξ,η) =arccos(ξ,η),
BecauseHs(Sq) is a RKHS constituted by the reproducing kernelK(ξ,η), the error be-tweenf(x) andSN(x) in the knotxiwill vanish,i.e.,
SN–f,K(·,xi)
Hs(Sq)=SN(xi) –f(xi) =
for an arbitraryf ∈Hs(Sq), ifS
Nis interpolated via the minimal norm. SupposeVx:=span{K(·,x), . . . ,K(·,xN)}, we thus have
(SN–f,v)Hs(Sq)=
for a given arbitraryv∈Vx. It implies thatSN is an orthogonal projection off to the span
Vxin the spaceHs(Sq).
Furthermore, theLp (p= +∞) approximation of an interpolation via minimal norm is consequently bounded as follows. By assuming there exists a positive integermsuch that the grid normh≤m forX={x,x, . . . ,xN}is taken fromSq, we deduce that theLp (p= +∞) approximation via the minimal norm must satisfy
sup
x∈Sq
f(x) –SN(x)≤C
∞
n=m+ dqn
A
n
f (.)
for an arbitraryf ∈Hs(Sq) (refer to []). Coming with these contributions, this study is sought to extend the significant results to a more general case ofLp(≤p≤+∞).
Theorem . Assume that X={x,x, . . . ,xN}is a set of N points taken from the unit sphere
Sq,h is the corresponding grid norm,and there exists a positive integer m such that h≤m.
For an arbitrary function f ∈Hs(Sq), s> q
,and its corresponding interpolation via the minimal norm SN,there must exist a positive constant C,which is independent of f and h,
such that the following estimations are satisfied:
f–SNp≤Chs+
q p–q
∞
n=m+ dqn
A
n
f –SNHs(Sq)
≤Chs+ q p–
q
∞
n=m+ dqn
A
n
fHs(Sq), ≤p<∞ (.)
and
f–SNp≤Chs
∞
n=m+ dnq
A
n
f–SNHs(Sq)
≤Chs
∞
n=m+ dnq
A
n
fHs(Sq), ≤p< . (.)
Theorem . Assume that X={x,x, . . . ,xN}is a set of N points taken from the unit sphere
Sq,and the grid norm satisfies the condition h≤
s>q,and its corresponding interpolation via the minimal norm SN,we have
f–SNp≤Chs+
q p–
q
∞
n=m+ dqn
A
n
fHs(Sq), ≤p<∞ (.)
and
f–SNp≤Chs
∞
n=m+ dqn
A
n
fHs(Sq), ≤p< , (.)
where C denotes a positive constant independent of f and h.
To prove the main results of Theorems . and ., the concept of spherical cap should be introduced first: With centerxand radiusγ, the spherical capG(x,γ) is defined as
G(x,γ) =x∈Sq,x·x>cosγ,
and the surface area, referred to asG(γ) ofG(x,γ), can be given as
G(γ) =
γ
ωqsinq–θdθ. (.)
2 Lemmas
To completely prove Theorems . and ., five lemmas are given as follows.
Lemma . (refer to [, , ]) Assume an integer q≥,constants M= √q and δq=
(q+),and h:=
θ
M+M+δq with an arbitrary positive number M,andθ∈(,
π
),then there exists a point set Zh⊂Sqwith an arbitrary h∈(,h)satisfying
Sq= z∈Zh
G(z,Mh). (.)
If FA represents the characteristic function of a given set A∈Sq ,there exists a positive
integer Q independent of h satisfying
z∈Zh
FG(z,Mh¯ )≤Q, (.)
whereM¯ =M+M.Furthermore,there exists a constant CQindependent of h such that |Zh| ≤CQh–q.
Lemma .(refer to [, , ]) By giving constants S≥,M,C,M¯ =M+M,h=CM¯ and h∈(,h),there must exist an arbitrary f ∈Hs(Sq)such that
z∈Zh
fHs(G(z,Mh¯ ))≤QfHs(Sq). (.)
Lemma . Assume thatη∗ is an evaluation functional corresponding toηin Hs(Sq),η∗ in Hs(Sq)can alternatively be expressed as K(η
Lemma . Assume that x∗,x∗,x∗, . . . ,x∗N are evaluation functionals in Hs(Sq)and their
expressions in Hs(Sq)are u,u
, . . . ,uN.Together with SN,which is interpolated via the
min-imal norm along x,x, . . . ,xN,and z,which is the optimized approximation of u in the
span{u,u, . . . ,uN},we have
f(x) –SN(x)≤ u–zf–sN. (.)
It should be noted that Lemmas.and.can be directly obtained from[].
Lemma .(refer to [, , ]) There exist two constants C> and h> ,dependent only on s and d,such that a function g(g∈Hs(Sq),g|X= for arbitrary X∈Sd,h≤h)satisfies
g≤ChsgHs(Sq), (.)
when s>d.
3 Proof of theorems
Proof of Theorem. From Lemma ., we have
f–SNpp=
Sq
(f–SN)(ξ) p
dω(ξ) ≤
z∈Zh
G(z,Mh)
(f–SN)(ξ)pdω(ξ)
for arbitrary ≤p<∞andM= √q. We consider first the local error estimation. Since
f –SN is continuous onG(z,Mh), andG(z,Mh) is a compact support ofSq, there must existξz∈G(z,Mh) such thatf –SN is maximized atξz. We consequently have
f–SNpp≤ z∈Zh
(f –SN)(ξz)p
G(z,Mh) dω(ξ)
≤Cqhq z∈Zh
(f–SN)(ξz) p
,
whereCqis a constant dependent only onqand satisfyingG(Mh) =Cqhq.
(a) When p≥: Takingξz∗ as an evaluation functional,Sξ∗z is its best approximation inspan{u,u, . . . ,uN}. From the Jensen inequality (refer to [, ]),
N i=a
p i ≤(
N
i=ai)
p
, Lemmas . and ., we obtain
f–SNpp≤Chq z∈Zh
(f–SN)(ξz)
p
≤Chq
z∈Zh
ξz∗–S∗ξz
f–SNL(G(z,Mh))
p
≤Chq
z∈Zh
f–SNL(G(z,Mh))
K(ξz,·) – N
i=
αiK(xi,·)
p
≤Chq
z∈Zh
f–SNL(G(z,Mh))
p
z∈Zh
K(ξz,·) – N
i=
αiK(xi,·)
p
≤Chq
z∈Zh
f–SNL(G(z,Mh))
p
z∈Zh ∞
n=m+ dqn
A
n
p
.
It should be noted that one of the following inequalities (refer to []) is used in the last step of the derivations above
K(x,·) –
N
i=
αiK(xi,·)
≤C ∞
n=m+ dnq
A
n
. (.)
For sure, by takingh¯=m(whereh¯=min{h,h}), we can go further
f–SNpp≤Chq+sp z∈Zh
f–SNHs(G(z,Mh)) p
∞
n=m+ dnq
A
n
p
CQh–
pq
≤Chq+sp–pq
∞
n=m+ dqn
A
n
p
f –SNpHs(Sq)
with h∈(,h¯). By using bothM<M¯ and the consequence|Zh| ≤CQh–q, Lemma ., Lemma ., and Lemma ., we achieve
f–SNp≤Chs+
q p–
q
∞
n=m+ dqn
A
n
f –SNHs(Sq). (.)
(b) When ≤p< : From the inequality (refer to [, ])
N
i=
api ≤N–p
N
i= ai
p
,
and the fact whichZhis less thanCh–q, we obtain first
f–SNpp≤Chq z∈Zh
(f–SN)(ξz)
p
h–(–p)q
≤Chpq z∈Zh
ξz∗–Sξ∗z
f –SNL(G(z,Mh))
p
≤Chsp
∞
n=m+ dqn
A
n
p
f –SNpHs(Sq).
Following similarly the steps in the case ofp≥, we also have
f–SNp≤Chs
∞
n=m+ dnq
A
n
With the fact thatSNis the orthogonal projection off inHs(Sq), the proof of Theorem .
is thus completed.
Proof of Theorem . From the property of orthogonal projection of SN, the Cauchy-Schwarz inequality, and Lemma ., we can easily obtain
f–SNpHs(Sq)=
f–SNHs(Sq) p
= (f –SN,f –SN)
p
Hs(Sq)
= (f –SN,f)
p
Hs(Sq)
=
∞
n=
dqn
j=
Asn(f –SN,Yn,j)(f,Yn,j)
p
≤
∞
n=
dnq
j= As
n(f,Yn,j)
∞
n=
dqn
j=
(f –SN,Yn,j)
p
=fHs(Sq)f–SNL(Sq) p
≤Chspf
p
Hs(Sq)f–SN
p
Hs(Sq).
Therefore
f–SNpHs(Sq)≤Chspf
p
Hs(Sq). (.)
Together with the results of Theorem ., Theorem . is thus established.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
In this paper, JW had the main role in providing derivations. CY and ZG also contributed significantly with their corresponding effort to finish the work.
Author details
1School of Mathematics and Statistics, Southwest University, Chongqing, 400715, China.2Department of Electrical
Engineering, National Taipei University, No.151, University Rd., San Shia District, New Taipei City, 23741, Taiwan.
Acknowledgements
The first author and the third author would like to thank the Natural Science Foundation of China (Nos. 61273020, 11001227), the Fundamental Research Funds for the Central Universities (No. XDJK2010B005).
Received: 25 February 2013 Accepted: 27 August 2013 Published:08 Nov 2013
References
1. Wang, KY, Li, LQ: Harmonic Analysis and Approximation on the Unit Sphere. Science Press, Beijing (2000) 2. Freeden, W, Gervens, T, Schreiner, M: Constructive Approximation on the Sphere. Oxford University Press, New York
(1998)
3. Müller, C: Spherical Harmonics. Lecture Notes in Mathematics, vol. 17. Springer, New York (1966)
4. Le Gia, QL, Narcowich, F, Ward, J, Wendland, H: Continuous and discrete least-squares approximation by basis functions on spheres. J. Approx. Theory143, 124-133 (2006)
5. Hubbert, S, Morton, TM:Lp-Error estimates for radial basis function interpolation on the sphere. J. Approx. Theory129,
58-77 (2004)
6. Hubbert, S, Morton, TM: A Duchon framework for the sphere. J. Approx. Theory129, 28-57 (2004)
8. Chen, ZX: Error estimates of minimal norm interpolant approximation by the spherical radial basis functions. Acta Math. Appl. Sin.30(1), 159-167 (2007) (in Chinese)
9. Lin, SB, Cao, FL:LpApproximation by shifts of smooth kernels on spheres. Chin. Ann. Math., Ser. A31(5), 615-624
(2010) (in Chinese)
10. Wendland, H: Scattered Data Approximation. Cambridge University Press, Cambridge (2005)
11. Narcowich, F, Schaback, R, Ward, J: Approximation in Sobolev spaces by kernel expansions. J. Approx. Theory114(1), 70-83 (2002)
12. Brenner, SC, Scott, RL: The Mathematical Theory of Finite Element Methods. Springer, New York (1994)
10.1186/1029-242X-2013-510
Cite this article as:Wang et al.:LpError estimate for minimal norm SBF interpolation.Journal of Inequalities and