Inverse and Saturation Theorems for Radial
Basis Function Interpolation
Robert Schaback and Holger Wendland
Abstract. While direct theorems for interpolation with radial basis func-
tions are intensively investigated, little is known about inverse theorems so
far. This paper deals with both inverse and saturation theorems. As an
inverse theorem we especially show that a function that can be approximated suciently fast must belong to the native space of the basis function
in use. In case of thin plate spline interpolation we also give certain saturation theorems.
1. Introduction
Direct and inverse theorems play an important role in classical approximation
theory. Examples can be found in [2, 8]. The main idea can be described as follows. Suppose the elements of a linear space (V; kk) should be approximated by
elements of nite dimension subspaces Vh V , where h serves as a discretisation index. Denote the approximation process by Sh : V ! Vh . Then the direct
theorems conclude error estimates from additional information on the elements
to be approximated: If f is an element of a subspace G V then the error can
be bounded by
(1.1)
kf Sh f k cf h :
On the other hand the inverse theorems try to conclude information on f from
the way f can be approximated: If f 2 V satises (1.1) then f must belong to
a certain subspace G V . The situation is optimal if the subspaces and the
approximation orders coincide in both the direct and the inverse theorems.
Finally, saturation theorems give upper bounds on the possible approximation
order: If f 2 G can be approximated by
kf Sh f k cf h ;
where is a certain number larger than , then f must belong to a trivial
subspace N V .
It is the aim of this paper to give both inverse and saturation theorems in
the context of radial basis function interpolation. In case of direct and inverse
theorems we shall take the native space G
; , which we introduce in the third
AMS classication : 41A05, 41A17, 41A27, 41A30, 41A40
Key words and phrases : positive denite functions, approximation orders
1
2
R. Schaback and H. Wendland
section, as the space G of smoother functions. We will look for the approximation
order we can achieve from this fact and for the order we need, to show that a
function belongs to the native space.
In case of saturation theorems we restrict ourselves to thin plate spline interpolation and show that functions that can be approximated with a high order
are necessarily polyharmonic functions.
2. Radial basis function approximation
The theory of interpolation by radial basis functions has become popular in the
last years to reconstruct multivariate functions from scattered data. The starting point of the reconstruction process is the choice of a conditionally positive
denite function : IRd ! IR.
Denition 2.1. A continuous and even function : IRd ! IR is said to be
conditionally positive denite of order m 2 IN0 , i for all N 2 IN, all sets
of pairwise distinct centers X = fx1 ; : : : ; xN g IRd ; and all 2 IRN n f0g
satisfying
N
X
the quadratic form
j =1
j p(xj ) = 0
N
X
j;k=1
for all p 2 Pmd
j k (xj xk )
is positive. Here, PMd denotes the set of all d-variate polynomials with a total
degree less then m. We will denote the set of all conditionally positive denite
functions of order m by cpd(m).
Having a 2 cpd(m) the interpolant sf;X to a function f in X = fx1 ; : : : ; xN g
is given by
(2.2)
sf;X (x) =
N
X
j =1
j (x xj ) +
Q
X
j =1
j pj (x)
where p1 ; : : : ; pQ form a basis of Pmd . To cope with the additional degrees of
freedom, the interpolation conditions
(2.3)
sf;X (xj ) = f (xj ); 1 j N;
are completed by the further conditions
(2.4)
N
X
j =1
j pk (xj ) = 0;
1 k Q:
We summarize some standard statements on the interpolation process in
Inverse Theorems for RBF
3
Lemma2.2. The interpolating function sf;X is well dened and unique, if X
contains a Pmd -unisolvent subset. In this case the operator that maps f to sf;X
is linear and reproduces polynomials up to degree m.
In this paper we only want to deal with conditionally positive denite functions
of order m that possess a generalized Fourier transform that coincides with a
continuous function b on IRd nf0g. We are mainly interested in cases, where this
Fourier transform decays only algebraically, i. e. there exist constants 0 < c1 c2
with
c1 k!k2 d s1 b(!) c2 k!k2 d s1
(2.5)
for k!k ! 1. The upper bound is important for the direct theorems, while
the lower bound is necessary for the inverse theorems. This decay condition
is, for instance, covered by the thin plate splines and the compactly supported
radial basis functions of minimal degree (cf. [13]). But we shall state the inverse
theorems also in case of exponentially decaying Fourier transforms which covers
Gaussian and (inverse) multiquadrics.
3. Direct theorems
There are several papers dealing with direct theorems, but only few have tried
to establish inverse theorems. We will briey repeat direct theorems as far as we
need them for our further analysis.
To state error estimates two preparing steps have to be done. On the one hand
the function space has to be introduced for which the error bounds shall apply.
On the other hand a measure of the data density has to be given. We start with
the function space by introducing the native space.
Let IRd be given. Let us denote by
(Pmd )?
= f;X =
M
X
j =1
j xj : M 2 IN; j 2 IR; xj 2 ; ;X jPmd 0g
the set of all point evaluation functionals of nite support in vanishing on Pmd .
Every conditionally positive denite function of order m allows us to equip
(Pmd )?
with an inner product
(; ) = x y (x y)
where x means the action of with respect to the variable x.
Then we can follow [6, 7] to introduce the function space
G
; = ff 2 C (
) : j(f )j cf kk for all 2 (Pmd )?
g:
We denote the smallest constant cf in the denition of G
; by kf k, i. e.
kf k := max j(f )j :
2(Pmd )?
nf0g
kk
4
R. Schaback and H. Wendland
Then k k is a semi-norm on G
; with null space Pmd . Thus
F
; := G
; =Pmd
is a normed linear space which turns out to be complete. There are several
other possiblities to introduce the native space (cf. [10, 11, 14]) but the chosen
approach serves our purposes best.
Not only the space Pmd is a subspace of G
; , but also all interpolating functions (2.2) are contained.
Lemma 3.1. The map
F : (Pmd )?
! F ((Pmd )?
) G
;
;X 7! y;X ( y)
is well dened and bijective. Furthermore, we have the relations
k;X k = kF (;X )k
and
;X (F (;Y )) = (;X ; ;Y ) = ;Y (F (;X )):
The proof is straightforward and will be omitted.
The rst step in bounding the interpolation error is to dene the power function as the norm of the pointwise error functional
P (x) = sup jf (x) sf;X (x)j :
X;
f 2G
; nPmd
kf k
which leads immediately to
jf (x) sf;X (x)j PX; (x)kf k:
Then the power function has to be bounded in terms of the ll distance, dened
by
hX;
hX := sup xmin
kx xj k2
2X
x2
j
which was done in [14], for instance.
Theorem 3.2. Let 2 cpd(m) satisfy (2.5). Let be a bounded and open
domain satisfying an interior cone condition. Then there exist constants h0 , C ,
such that for all sets of centers X with hX h0 and all x 2 the power function
can be bounded by
(3.6)
PX; (x) ChsX1 =2
yielding the error bound
(3.7)
kf sf;X kL1(
) Chs1 =2 kf k
for f 2 G
; .
Inverse Theorems for RBF
5
Actually, in [14] the theorem is stated in a more localized version, but the proof
holds true in this situation. There are several other papers giving error bounds
of this form, some of them are [1, 3, 5, 12].
Next, we need a stability result on the interpolation process. Therefore, we
dene the separation distance
kx xk k2
qX := 12 min
j 6=k j
and cite from [9]
Theorem 3.3. Let 2 cpd(m) satisfy the decay condition (2.5). For X =
fx1 ; : : : ; xN g denote by AX; the matrix
AX; = ((xj xk ))1j;kN
and by X the smallest non vanishing eigenvalue of AX; . Then the following
holds true:
1) (Stability) For all 2 IRN satisfying (2.4) we have
and therefore
T AX; X kk2 cqXs1
k(AX; jVX ) k ; cqXs1
with VX := f : ;X 2 (Pmd )?
g and a constant c depending only on .
2) (Uncertainty Relation) For all x 2 n X we have
(3.8)
PX; (x) X [fxg :
1
22
2
4. Inverse theorems concerning Principally, there are two possibilities to state inverse theorems for interpolation
by radial basis functions. The rst one is based only on the basis function and
draws conclusions on the basis function from the fact that the power function
can be bounded like (3.6). This will be done in this section. The second is to
draw conclusions on f from estimates like (3.7) which will be subject of the sixth
section.
Theorem 4.1. Let 2 cpd(m) satisfy (2.5). Let be bounded and open, satisfying an interior cone condition. If there exist constants ; c > 0 such that the
power function PX; can bounded by
2
(4.9)
kPX; kL1(
) ch=
X
for all sets X with suciently small hX , then
s1 must be satised.
6
R. Schaback and H. Wendland
Proof. On account of the conditions on there exists a > 0 and quasiuniform sets X = fx ; : : : ; xN g with respect to this > 0. Here, we call
a set of pairwise distinct centers X = fx ; : : : ; xN g quasi-uniform with
1
1
respect to > 0, i
1) X n fxj g is Pmd -regular for 1 j N ,
2) qX hX .
Then we have (cf. [9]) hX nfxj g 2hX for hX suciently small. Therefore we
can use (2.5), (4.9) and the Uncertainty Relation (3.8) to derive
c2 hX chX nfxj g PX2 nfxj g (xj ) X c qXs1 c s1 hsX1 :
Choosing a sequence of such X with hX ! 0, this leads to s1 .
Theorem 4.1 shows that the decay (4.9) of the power function determines the
decay of the generalized Fourier transform of the basis function and therefore the
smoothness of the basis function itself. It also shows that there is no possibility
to improve error estimates of the form (3.7) based on upper bounds of the power
function.
5. Characterisation of the native space
Our next result characterises the functions f from the native space G
; by
uniform boundedness of their interpolating functions with respect to the seminorm of the native space.
Theorem 5.1. Denote by sf;X the interpolant (2.2) to a function f 2 C (
) on
X using a basis function 2 cpd(m). Then f belongs to the native space G
;
if and only if there exists a constant cf such that ksf;X k cf for all X .
Proof. Assume f 2 G
;. Then sf;X is the best approximation to f from
spanf( x) : x 2 X g + Pmd . with respect to the k k-semi-norm. Thus we
have
kf sf;X k + ksf;X k = kf k;
which gives the bound ksf;X k kf k at once.
Now, let us assume ksf;X k cf for all X . For an arbitrary
2
;X :=
N
X
j =1
2
2
j xj 2 (Pmd )?
we choose sf;X to be the interpolant on X to f satisfying the interpolation
conditions (2.3) and (2.4). Then sf;X belongs to G
; and we have
;X (f sf;X ) = 0:
Inverse Theorems for RBF
7
Thus we can estimate
j;X (f )j j;X (f sf;X )j + j;X (sf;X )j
k;X k ksf;X k
cf k;X k
As this holds for all ;X we have f 2 G
; .
6. Inverse theorems concerning f
Now we draw conclusions about a function from L1-convergence orders of its
interpolants. To be more precise, we show that a function f 2 C (
) which
can be approximated sucently fast by radial basis function interpolants in the
L1 -norm must belong to the native space of the basis function.
Theorem 6.1. Let IRd be a bounded and open domain satisfying an interior cone condition. The basis function 2 cpd(m) should satisfy the decay
condition (2.5). Suppose further that for some f 2 C (
) there exist constants
> 0 and cf > 0 such that kf sf;X kL1 cf hX for all X with hX
suciently small. If 2 > s1 + d, then f must belong to the native space G
; .
Proof. All sets of centers X that may appear in this proof shall be quasiuniform with respect to a xed > 0.
Every interpolant sf;X denes a linear functional from (Pmd )?
which we shall
(
)
denote by ;X . From Theorem 3.3 and Lemma 3.1 we have
ksf;X k = k;X k
2
(6.10)
2
= T AX; = T AX; (AX; jVX ) 1 (AX; jVX )
k(AX; jVX ) 1 k2;2 kAX k2L (X )
= k(AX; jVX ) 1 k2;2 ksf;X pX k2L (X )
2
2
Here, VX denotes again the space f 2 IRjX j : ;X 2 (Pmd )?
g, and pX denotes
the polynomial in the denition of sf;X .
If we have two sets of centers X Y and compare the two interpolating
functions sf;Y and sf;X , we can interpret sf;X as the interpolant to sf;Y and use
the polynomial reproduction property of Lemma 2.2 to get
sf;X sf;Y =
jX j
X
j =1
Xj (
xj )
jY j
X
j =1
Yj ( yj ):
On the other hand the dierence can be interpreted as the interpolating function
on Y to sf;Y sf;X . This leads us to
8
R. Schaback and H. Wendland
ksf;Y sf;X k k(AY;jVY ) k ; ksf;Y sf;X kL
X
jf (y) sf;X (y)j
1
2
1
2
22
Y y 2Y
c 1 qY s1 jY jc2f h2X :
Y)
2(
2
In what follows c will denote a generic constant. Now we consider a special family
of quasi-uniform sets of centers. We assume Xn to satisfy jXn j c2nd and
c1 2 n qXn hXn c2 2 n :
Such a choice is always possible because of the assumption made on . If we
take X = Xk Y = Xn with n k we get
ksf;Xn sf;Xk k c2s1n
2
dn 2k
= c2(d+s1)n 2k
= c22(n k) 2n ;
+
where > 0 is dened by d + s1 + 2 = 2. Thus we can estimate the -norm
of two succeeding interpolants by
ksf;Xk
sf;Xk k c2 k :
+1
A telescoping sum argument leads to
ksf;XK k K
X
k=0
c
ksf;Xk
1
X
j =0
+1
sf;Xk k + ksf;X k
0
2 k + ksf;X k :
0
1 c2 + ksf;X k :
0
Thus, the sequence ksf;Xk k is bounded. But for n k the interpolant sf;Xk
is also the interpolant to sf;Xn and therefore a best approximant to sf;Xn from
S (Xk ) := spanf( x) : x 2 Xk g + Pmd . This leads to
ksf;Xn sf;Xk k + ksf;Xk k = ksf;Xn k
which shows that the sequence ksf;Xk k is also increasing and therefore convergent. Furthermore, (6.11) implies that sf;Xn is a Cauchy sequence in G
; with
(6.11)
2
2
2
a limit s~, which is uniquely determined up to a polynomial of degree less than
m.
Inverse Theorems for RBF
9
Finally, we have to show that f coincides on with s~ modulo Pmd . Let us
choose a xed Pmd -unisolvent set = f1 ; : : : ; Q g and denote with uj (x) 2
Pmd , 1 j Q a Lagrange basis with respect to . Then the functional
Q
X
(x) := x
j =1
uj (x)j
lies in (Pmd )?
for every x 2 . Thus we have
j x (f s~)j j x (f sf;Xn )j + j x (sf;Xn s~)j
( )
( )
( )
k x kks~ sf;Xn k + j(f sf;Xn )(x) +
( )
k x k ks~ sf;Xn k + chXn :
Q
X
j =1
uj (x)(f sf;Xn )(j )j
( )
Thus we can derive
(x) (f ) = (x) (~s)
for all x 2 or in other words
f (x) = s~(x) +
N
X
j =1
uj (x)(~s f )(j ):
This implies
;X (f ) = ;X (~s) cs~ k;X k
for general ;X 2 (Pmd )?
, which completes the proof.
Note that there is a gap of d=2 between the necessary and sucient approximation order for functions in the native space G
; . A closer look shows that
the direct theorem 3.2 implies for f 2 G
;:
(6.12)
kf sf;X kL1 C hs1 = kf sf;X k:
(
2
)
The hs1 =2 term comes from the estimate on the power function and is optimal
in the sense of Theorem 4.1. On the other hand
kf sf;X kL1 Cf hs1 = hd= "
is so far necessary for showing f 2 G
; via Theorem 6.1. Thus the gap could
(
)
2
be closed either by showing
kf sf;X k Cf hd= " ;
2+
or by improving our inverse theorem.
2+
10
R. Schaback and H. Wendland
Before we come to inverse theorems for Gaussian and multiquadrics, let us
remark that in case of an unconditionally positive denite function and a
quasi-uniform set X equation (6.10) can be rewritten as
ksf;X k chX s1
(
d=
+ ) 2
ksf;X kL1 ;
(
)
which can be seen as a kind of Bernstein inequality.
Now, let us assume for the rest of the section that the Fourier transform
satises
b(!) ce c~ k!k :
(6.13)
1
2
This leads to estimates of the form
k(AX; jVX ) k ; cec hX
(6.14)
1
1
22
where we used the subspace VX again and where c always denotes a generic
constant. In case of multiquadrics and Gaussians the constants , c~1 and c1 can
be found in [9].
Theorem 6.2. Let IRd be a bounded and open domain satisfying an interior cone condition. The basis function 2 cpd(m) should satisfy the decay
condition (6.13). Suppose further that for some f 2 C (
) there exist constants
c > c and cf > 0 such that kf sf;X kL1 cf e c hX for all X with
hX suciently small. Then f must belong to the native space G
; .
2
1
(
2
)
Proof. The proof of Theorem 6.1 applies completely if we show that the native
space norm of the dierence of two interpolants can be bounded in such a way
that the telescoping sum argument still works. But this is the case: for X Y
we can derive
ksf;X sf;Y k cec hX
1
ce
X
y2Y
c3 hx jf (y) sf;X (y)j
2
jY j
with c3 := c2 c1 > 0. Taking the same sequence of sets of centers Xn as in
Theorem 6.1 we see that the cardinality of Y is only polynomial in hY , which
means that we can bound two succeeding interpolants by
ksf;Xk
+1
sf;Xk k ce c~ 2 n = :
3
(
This ensures the convergence of the telescoping sum.
) 2
Inverse Theorems for RBF
11
7. Saturation for thin plate spline interpolation
In this section we concentrate on interpolation by thin plate or polyharmonic
splines. To be more precise we consider the functions d;` = d;`(k k2 ) with
(7.15)
2` d
odd d
d;` (r) = ced;` rr2` d log r for
for even d
d;`
with d 2` where the constants
( 1)` d2 `
cd;` = 22` d=2 (`) ;
(d 2)=2
:
ed;` = 2` 1 d=( 2 1)
2 (` 1)! ` d2 !
are determined by the fact that these functions should be the fundamental solutions of the iterated Laplacian (see Lemma 7.2).
The functions d;` are conditionally positive denite of order m with m =
` bd=2c + 1 on IRd and possess a generalized Fourier transform bd;` which is
k k2 2` up to a constant factor. Thus interpolants come from the space
S (X ) = spanfd;`( x) : x 2 X g + Pmd
and Theorem 3.2 leads to the error bound
(7.16)
d
kf sf;X kL1 chX` kf kd;` :
(
2
)
For a restricted set of functions f , an improvement in [10] yields
(7.17)
kf sf;X kL1 cf hX` d:
(
2
)
In [1] the following improved error estimate is given:
Theorem 7.1. Suppose
is a cube in IRd and the set of centers Xh are given
d
by the grid points hZZ \ . If f 2 Lip(2` + 1; ), then the error can be bounded
by
(7.18)
kf sf;Xh kL1 K cf h `
(
)
2
for every compact subset K of the interior of as h ! 0.
See [1] for the exact denition of the space Lip(2`+1; ). Note that this estimate
is based on three additional assumptions:
{ The function f is supposed to be smoother than f 2 G;
. This is a natural
assumption.
{ The domain has to be a cube and the centers have to form a grid. This is a
consequence of the proof given in [1]. A generalization to arbitrary centers
would be useful.
12
R. Schaback and H. Wendland
{ The estimates are restricted to compact subsets of the interior of the cube.
This means that boundary eects are neglected.
Nonetheless, the result gives a hint on the possible local convergence order
and we shall show that this order is also the saturation order. But before we can
do that, we need two auxiliary results:
Lemma 7.2. For every test function 2 C 1(IRd) and every y 2 IRd we have
0
Z
d
IR
d;`(x y)` (x)dx = (y):
A proof can be found in [4].
Lemma 7.3. Suppose X = fx ; : : : ; xN g is given. Suppose further that
2 C 1 (
) satises X \ supp = ;. Then for every s 2 S (X ):
1
0
(` s; )L (
) = 0:
2
Proof. Choose an arbitrary s(x) = PNj j d;`(x xj ) + p(x) 2 S (X ). As
` Pmd 0 we can use Lemma 7.2 to obtain
=1
(` s; )L (
) =
2
=
N
X
j =1
N
X
j =1
j
Z
` (x)d;` (x xj )dx
d
IR
j (xj )
= 0:
Now we can give our saturation result.
Theorem 7.4. Let d;` be any of the thin plate splines dened in (7.15). Suppose IRd to be open and bounded, satisfying an interior cone condition.
Suppose that for f 2 C ` (
) the interpolating functions sf;X on X satisfy
2
kf sf;X kL1 K = o(hX` )
(
as hX ! 0
2
)
for every compact subset K of . Then f satises
` f = 0
on :
Proof. Fix x 2 . Choose X to be quasi-uniform with respect to a xed
> 0, such that minx2X kx x k = c hX with c independent of hX . Choose
aR test function 2 C 1 (IRd ) with supp = B (0) = fx 2 IRd : kxk 1g and
(x)dx = 1. Set h~ = c hX =2 and h := h~ d (( x )=h~ ). Then the support
0
0 2
0
0
0
0
0
0
0
1
0
2
0
Inverse Theorems for RBF
13
of h is given by Bh~ (x0 ) and satises Bh~ (x0 ) \ X = ;. Thus we can use Lemma
7.3 to get
(` f; h )L (
) = (` (f s); h )L (
)
= (f s; ` h )L (
)
2
2
2
kf skL Bh x k`h kL Bh x
c h~ d kf skL1 Bh x k` h kL Bh x
2(
~ ( 0 ))
2
(
with s = sf;X . And because of
k`h k2L2 (Bh~ (x0 )) = h~ 2d
2(
~ ( 0 ))
Z
d
IR
Z
~ ( 0 ))
2(
~ ( 0 ))
j` h (x)j dx
2
= h~ d 4` d j0 (x)j2 dx
IR
=: h~ d 4` c20
we can conclude
(` f; h )L (
) ch~ 2`kf skL1(Bh (x )) :
On account of the assumptions this leads to
2
~
0
lim (` f; h )L (
) = 0:
h!0+
On the other hand we have
lim (` f; h )L2 (
) = hlim
h!0+
!0+
which proves ` f (x0 ) = 0.
2
Z
d
IR
(` f )(x0 + hx)0 (x)dx = ` f (x0 )
Note that our proof also applies to the situation of classical splines. As in
the latter case, functions in the saturation class are already determined by their
values on the boundary of the domain:
Corollary 7.5. Suppose in addition to the assumptions of the last theorem that
has a C ` boundary @
. Then f is already determined by the values of @ j f=@ j ,
0 j ` 1, on the boundary @
. Here denotes the outer unit normal vector.
This sheds some light on the inuence of boundary conditions on the possibilities to improve the approximation order ` d=2 of (7.16) towards 2`.
Finally, we want to draw the reader's attention to the d=2-gap arising not only
in the discussion around (6.12), but also in (7.17) when compared to (7.18). If
(7.16) could be improved by hd=2, then (7.17) would by [10] improve to h2` and
coincide with (7.18). We consider closing the d=2-gap to be a challenging research
task.
14
R. Schaback and H. Wendland
References
1. Bejancu, A., Local accuracy for radial basis function interpolation on nite uniform grids,
Preprint Cambridge 1997.
2. DeVore, R. A. and Lorentz, G. G., Constructive Approximation, Springer, Berlin, 1993.
3. Duchon, J., Sur l'erreur d'interpolation des fonctions de plusiers variables par les Dm splines, R.A.I.R.O. Analyse numerique 12 (1978), 325-334.
4. John, F., Plane Waves and Spherical Means Applied to Partial Dierential Equations,
Interscience Publishers, Inc., New York, 1955.
5. Light, W. and Wayne, H., On power functions and error estimates for radial basis function
interpolation, J. of Approx. Theory 92 (1998), 245-266.
6. Madych, W. R. and Nelson, S. A., Multivariate interpolation and conditionally positive
denite functions, Approx. Theory and its Appl. 4 (1988), 77-89
7. Madych, W. R. and Nelson, S. A., Multivariate interpolation and conditionally positive
denite functions II, Math. Comp. 54 (1990), 211-230.
8. Schumaker, L. L., Spline Functions: Basic Theory, John Wiley & Sons, New York,1981.
9. Schaback, R., Error estimates and condition numbers for radial basis function interpolation, Advances in Comp. Math. 3 (1995), 251-264.
10. Schaback, R., Improved error bounds for radial basis function interpolation, Preprint
Gottingen, 1997, to appear in Math. Comp.
11. Schaback, R., Native spaces for radial basis functions I, Preprint Gottingen 1998, to
appear in the Proceedings of IDoMat 98.
12. Wendland, H., Sobolev-type error estimates for interpolation by radial basis functions, in:
A. LeMehaute, C. Rabut, and L.L Schumaker (eds.), Surface Fitting and Multiresolution
Methods , pp 337-344, Vanderbilt University Press, Nashville, TN, 1997.
13. Wendland, H., Error estimates for interpolation by compactly supported radial basis
functions of minimal degree, J. Approx. Theory 93 (1998), 258-272.
14. Wu,Z. and Schaback, R. Local error estimates for radial basis function interpolation of
scattered data, IMA J. of Numerical Analysis 13 (1993), 13-27.
R. Schaback, H. Wendland
Institut fur Numerische und
Angewandte Mathematik
Universitat Gottingen
Lotzestr. 16-18
D-37083 Gottingen, Germany