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L p Error estimate for minimal norm SBF interpolation

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 the p th 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.

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 [1], Freeden et al. [2], and Muller [3], have moved their interests further to the topics of spherical approximation [17]. Based on the developments, spherical harmonic analysis was established and had some considerable progress. Meanwhile, L p error estimations for the SBF approximation were studied hereafter and stepped forward in the studies of Le Gia et al. [4], Hubbert and Morton [5], and Sloan and Wendland [7] after 2004. In 2007, Chen [8] established error bound for the minimal norm interpolation on a sphere. With this plentiful foundation, Lin and Cao [9] embedded firstly the smooth SBFs in a native space and specified the error bound between the best approximation and the target function via L p metric. As a consecutive study, the paper thus aims to derive a minimal norm interpolation with the L p measure.

For a fixed integer q (q1), S q is a unit sphere in R q + 1 , i.e., S q ={x=( x 1 , x 2 ,, x q + 1 ) R q + 1 , x 1 2 + x 2 2 ++ x q + 1 2 =1}, and represents a sufficient small elemental area on the spherical surface S q . The total surface area of S q can hence be expressed as

ω q = S q dω= 2 π q + 1 2 Γ ( q + 1 2 ) .

In regard to , the L p ( S q ) inner product of functions f and g is given as

( f , g ) L p ( S q ) = S q f(x)g(x)dω(x),

and the L p ( S q ) norm is defined as

f p := f L p ( S q ) := { ( S q | f ( x ) | p d ω ) 1 p , 1 p < + ; max x S q | f ( x ) | , p = + ,

where L p ( S q ) represents a function space constructed by a complex function, f: S q C, fulfilling f p <+. An identical function in L p ( S q ) is characterized as a function identical to the functions which have the same output values everywhere with the same inputs.

For an integer k0, H k q denotes a linear subspace in R q + 1 constructed by all the k th order homogeneous harmonic polynomials p(x) restricted by S q , and Π n q denotes the function constructed by all the k th order, kn, spherical harmonic polynomials. The relationship between Π n q and H k q can primarily be given as

Π n q = k = 0 n H k q .

Obviously, the dimension of H k q is

d k q =dim H k q = { 2 k + q 1 k + q 1 ( 2 k + q 1 k + q 1 ) , k 1 ; 1 , k = 0 .

In H k q , we select a set of functions Y k , j (j=1,2,, d k q ) to form a standard orthogonal basis Π n q ={ Y k , j ,j=1,2,, d k q ,k=0,1,2,,n}. The subspace H k q also confirms L 2 ( S q )= k = 0 H k q ¯ and the famous additive law:

j = 1 d k q Y k , j (ξ) Y k , j (η)= d k q ω q P k (q+1:ξη),

where ξη denotes the inner product of ξ and η. P k (q+1:x) denotes the k th order Legendre polynomial satisfying P k (q+1:1)=1 and complies with

1 1 P k (q+1:x) P l (q+1:x) ( 1 x 2 ) q 2 1 dx= ω q ω q 1 d k q δ k , l .

By giving A={{ A n , j }: A n , j R,n=0,1,2,,j=1,, d n q }, a series { A n }A, A n >0, n=0,1,2, , can also be expressed as { A n , j }A, A n , j = A n , j=1,, d n q . Using the series { A n , j }, a space defined as

H s ( S q ) = { F L 2 ( S q ) , n = 0 j = 1 d n q A n s ( F , Y n , j ) 2 < }

has the inner product ( , ) H s ( S q ) :

( F , G ) H s ( S q ) = n = 0 j = 1 d n q A n s (F, Y n , j )(G, Y n , j ),

and the corresponding norm is consequently defined as

F H s ( S q ) = ( ( F , F ) H s ( S q ) ) 1 2 .

As known from the definition here, H s ( S q ) is a Hilbert space, i.e., when s> q 2 , H s ( S q ) has a reproducing kernel

K(ξ,η)= n = 0 j = 1 d n q A n s Y n , j (ξ) Y n , j (η)

if A n satisfies A n 1+ n 2 (see the reference [7]).

Hereafter, we assume consistently in this study that H s ( S q ) is a reproducing kernel Hilbert space (RKHS) constructed by the reproducing kernel and call the space a native space of K(ξ,η) (see [10, 11]). From the additive law, we have

K(ξ,η)= n = 0 j = 1 d n q A n s Y n , j (ξ) Y n , j (η)= n = 0 d n q ω q A n s P n (q+1,ξη),ξ,η S q ,

where P n (q+1,x) is the n th order Legendre polynomial. Here K(ξ,η) is obviously a spherical radial basis function.

Let us assume that X={ x 1 , x 2 ,, x N } is a set of N points taken from the unit sphere S q and has N output values y 1 , y 2 ,, y N corresponding to x 1 , x 2 ,, x N through a function F:

I N (y)= { F H s ( S q ) : F ( x i ) = y i , i = 1 , 2 , , N } .

If there exists F I N (y) such that F can be minimized, we say that the minimization of F is a problem subject to a minimal norm. By denoting S N (ξ) as an interpolation with the minimal norm, there is a kernel basis expression for the interpolation S N (ξ) (see the references [2, 7, 8]):

S N (ξ)= i = 0 N α i K( x i ,ξ),ξ S q .

Suppose that y i , i=1,2,,N, are produced by the function f(x), i.e., f( x i )= y i , i=1,,N. The interpolation problem becomes intrinsically a function approximation problem to estimate the error between f(x) and S N (x). The approximation order of the interpolation can then be determined by the grid norm, h, and the input x i X. Here, h is defined as

h:= sup x S q min x i X dist(x, x i ),

where dist(ξ,η) is the spherical distance between ξ and η, i.e., dist(ξ,η)=arccos(ξ,η), ξ,η S q .

Because H s ( S q ) is a RKHS constituted by the reproducing kernel K(ξ,η), the error between f(x) and S N (x) in the knot x i will vanish, i.e.,

( S N f , K ( , x i ) ) H s ( S q ) = S N ( x i )f( x i )=0

for an arbitrary f H s ( S q ), if S N is interpolated via the minimal norm.

Suppose V x :=span{K(, x 1 ),,K(, x N )}, we thus have

( S N f , v ) H s ( S q ) =0

for a given arbitrary v V x . It implies that S N is an orthogonal projection of f to the span V x in the space H s ( S q ).

Furthermore, the L p (p=+) approximation of an interpolation via minimal norm is consequently bounded as follows. By assuming there exists a positive integer m such that the grid norm h 1 2 m for X={ x 1 , x 2 ,, x N } is taken from S q , we deduce that the L p (p=+) approximation via the minimal norm must satisfy

sup x S q | f ( x ) S N ( x ) | C ( n = m + 1 d n q A n 2 ) f
(1.1)

for an arbitrary f H s ( S q ) (refer to [8]). Coming with these contributions, this study is sought to extend the significant results to a more general case of L p (1p+).

Theorem 1.1 Assume that X={ x 1 , x 2 ,, x N } is a set of N points taken from the unit sphere S q , h is the corresponding grid norm, and there exists a positive integer m such that h 1 2 m . For an arbitrary function f H s ( S q ), s> q 2 , and its corresponding interpolation via the minimal norm S N , there must exist a positive constant C, which is independent of f and h, such that the following estimations are satisfied:

f S N p C h s + q p q 2 ( n = m + 1 d n q A n 2 ) 1 2 f S N H s ( S q ) C h s + q p q 2 ( n = m + 1 d n q A n 2 ) 1 2 f H s ( S q ) , 2 p <
(1.2)

and

f S N p C h s ( n = m + 1 d n q A n 2 ) 1 2 f S N H s ( S q ) C h s ( n = m + 1 d n q A n 2 ) 1 2 f H s ( S q ) , 1 p < 2 .
(1.3)

Theorem 1.2 Assume that X={ x 1 , x 2 ,, x N } is a set of N points taken from the unit sphere S q , and the grid norm satisfies the condition h 1 2 m . For an arbitrary function f H s ( S q ), s> q 2 , and its corresponding interpolation via the minimal norm S N , we have

f S N p C h 2 s + q p q 2 ( n = m + 1 d n q A n 2 ) 1 2 f H 2 s ( S q ) ,2p<
(1.4)

and

f S N p C h 2 s ( n = m + 1 d n q A n 2 ) 1 2 f H 2 s ( S q ) ,1p<2,
(1.5)

where C denotes a positive constant independent of f and h.

To prove the main results of Theorems 1.1 and 1.2, the concept of spherical cap should be introduced first: With center x 0 and radius γ, the spherical cap G( x 0 ,γ) is defined as

G( x 0 ,γ)= { x S q , x x 0 > cos γ } ,

and the surface area, referred to as G(γ) of G(x,γ), can be given as

G(γ)= 0 γ ω q sin q 1 θdθ.
(1.6)

2 Lemmas

To completely prove Theorems 1.1 and 1.2, five lemmas are given as follows.

Lemma 2.1 (refer to [5, 6, 9])

Assume an integer q1, constants M=2 q and δ q = 1 ( 4 q + 1 ) 3 2 , and h 0 := θ M + M 1 + δ q with an arbitrary positive number M 1 , and θ(0, π 3 ), then there exists a point set Z h S q with an arbitrary h(0, h 0 ) satisfying

S q = z Z h G(z,Mh).
(2.1)

If F A represents the characteristic function of a given set A S q , there exists a positive integer Q independent of h satisfying

z Z h F G ( z , M ¯ h ) Q,
(2.2)

where M ¯ =M+ M 1 . Furthermore, there exists a constant C Q independent of h such that | Z h | C Q h q .

Lemma 2.2 (refer to [5, 6, 9])

By giving constants S0, M 1 , C, M ¯ =M+ M 1 , h 0 = C 3 M ¯ and h(0, h 0 ), there must exist an arbitrary f H s ( S q ) such that

z Z h f H s ( G ( z , M ¯ h ) ) 2 Q f H s ( S q ) 2 .
(2.3)

Lemma 2.3 Assume that η 1 is an evaluation functional corresponding to η 1 in H s ( S q ), η 1 in H s ( S q ) can alternatively be expressed as K( η 1 ,ξ).

Lemma 2.4 Assume that x , x 1 , x 2 ,, x N are evaluation functionals in H s ( S q ) and their expressions in H s ( S q ) are u, u 1 ,, u N . Together with S N , which is interpolated via the minimal norm along x 1 , x 2 ,, x N , and z, which is the optimized approximation of u in the span{ u 1 , u 2 ,, u N }, we have

| f ( x ) S N ( x ) | uz f s N 2 .
(2.4)

It should be noted that Lemmas 2.3 and 2.4 can be directly obtained from [9].

Lemma 2.5 (refer to [4, 5, 7])

There exist two constants C>0 and h 1 >0, dependent only on s and d, such that a function g (g H s ( S q ), g | X =0 for arbitrary X S d , h h 1 ) satisfies

g 2 C h s g H s ( S q ) ,
(2.5)

when s> d 2 .

3 Proof of theorems

Proof of Theorem 1.1 From Lemma 2.1, we have

f S N p p = S q | ( f S N ) ( ξ ) | p d ω ( ξ ) z Z h G ( z , M h ) | ( f S N ) ( ξ ) | p d ω ( ξ )

for arbitrary 1p< and M=2 q . We consider first the local error estimation. Since f S N is continuous on G ( z , M h ) ¯ , and G ( z , M h ) ¯ is a compact support of S q , there must exist ξ z G ( z , M h ) ¯ such that f S N is maximized at ξ z . We consequently have

f S N p p z Z h | ( f S N ) ( ξ z ) | p G ( z , M h ) d ω ( ξ ) C q h q z Z h | ( f S N ) ( ξ z ) | p ,

where C q is a constant dependent only on q and satisfying G(Mh)= C q h q .

(a) When p2: Taking ξ z as an evaluation functional, S ξ z is its best approximation in span{ u 1 , u 2 ,, u N }. From the Jensen inequality (refer to [9, 12]), i = 1 N a i p ( i = 1 N a i 2 ) p 2 , Lemmas 2.3 and 2.4, we obtain

f S N p p C h q ( z Z h | ( f S N ) ( ξ z ) | 2 ) p 2 C h q ( z Z h ξ z S ξ z 2 f S N L 2 ( G ( z , M h ) ) 2 ) p 2 C h q ( z Z h f S N L 2 ( G ( z , M h ) ) 2 K ( ξ z , ) i = 1 N α i K ( x i , ) 2 ) p 2 C h q ( z Z h f S N L 2 ( G ( z , M h ) ) 2 ) p 2 ( z Z h K ( ξ z , ) i = 1 N α i K ( x i , ) 2 ) p 2 C h q ( z Z h f S N L 2 ( G ( z , M h ) ) 2 ) p 2 ( z Z h n = m + 1 d n q A n 2 ) p 2 .

It should be noted that one of the following inequalities (refer to [8]) is used in the last step of the derivations above

K ( x , ) i = 1 N α i K ( x i , ) 2 C n = m + 1 d n q A n 2 .
(3.1)

For sure, by taking h 0 ¯ = 1 2 m (where h 0 ¯ =min{ h 0 , h 1 }), we can go further

f S N p p C h q + s p ( z Z h f S N H s ( G ( z , M h ) ) 2 ) p 2 ( n = m + 1 d n q A n 2 ) p 2 C Q h p q 2 C h q + s p p q 2 ( n = m + 1 d n q A n 2 ) p 2 f S N H s ( S q ) p

with h(0, h 0 ¯ ). By using both M< M ¯ and the consequence | Z h | C Q h q , Lemma 2.5, Lemma 2.2, and Lemma 2.1, we achieve

f S N p C h s + q p q 2 ( n = m + 1 d n q A n 2 ) 1 2 f S N H s ( S q ) .
(3.2)

(b) When 1p<2: From the inequality (refer to [9, 12])

i = 1 N a i p N 1 p 2 ( i = 1 N a i 2 ) p 2 ,

and the fact which Z h is less than C h q , we obtain first

f S N p p C h q ( z Z h | ( f S N ) ( ξ z ) | 2 ) p 2 h ( 1 p 2 ) q C h p q 2 ( z Z h ξ z S ξ z 2 f S N L 2 ( G ( z , M h ) ) 2 ) p 2 C h s p ( n = m + 1 d n q A n 2 ) p 2 f S N H s ( S q ) p .

Following similarly the steps in the case of p2, we also have

f S N p C h s ( n = m + 1 d n q A n 2 ) 1 2 f S N H s ( S q ) .
(3.3)

With the fact that S N is the orthogonal projection of f in H s ( S q ), the proof of Theorem 1.1 is thus completed. □

Proof of Theorem 1.2 From the property of orthogonal projection of S N , the Cauchy-Schwarz inequality, and Lemma 2.5, we can easily obtain

f S N H s ( S q ) p = ( f S N H s ( S q ) 2 ) p 2 = ( f S N , f S N ) H s ( S q ) p 2 = ( f S N , f ) H s ( S q ) p 2 = ( n = 0 j = 1 d n q A n s ( f S N , Y n , j ) ( f , Y n , j ) ) p 2 ( ( n = 0 j = 1 d n q A n 2 s ( f , Y n , j ) 2 ) 1 2 ( n = 0 j = 1 d n q ( f S N , Y n , j ) 2 ) 1 2 ) p 2 = ( f H 2 s ( S q ) f S N L 2 ( S q ) ) p 2 C h s p 2 f H 2 s ( S q ) p 2 f S N H s ( S q ) p 2 .

Therefore

f S N H s ( S q ) p C h s p f H 2 s ( S q ) p .
(3.4)

Together with the results of Theorem 1.1, Theorem 1.2 is thus established. □

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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).

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Correspondence to Jianjun Wang.

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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.

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Wang, J., Yang, CY. & Gu, Z. L p Error estimate for minimal norm SBF interpolation. J Inequal Appl 2013, 510 (2013). https://doi.org/10.1186/1029-242X-2013-510

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Keywords

  • spherical basis function
  • minimal norm
  • L p approximation
  • error estimate