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<strong>Radial basis functions</strong> are means to approximate
multivariable (also called ''multivariate'') functions by linear combinations of 
terms based on a single univariate function (the '''radial basis function'''). This is radialised so that in can be used in more than one dimension. They are usually applied to approximate functions or data ([[#ApproximationTheoryandMethods|Powell 1981]],[[#Cheney|Cheney 1966]],[[#Davis|Davis 1975]]) which
are only known at a finite number of points (or too difficult to
evaluate otherwise), so that then evaluations of the approximating
function can take place often and efficiently. 

Primarily in computational applications, functions of many variables often
need to be approximated by other functions that are better understood or more readily
evaluated. This may be for the purpose of displaying them frequently on a computer screen for instance, so computer graphics are a field of practical use. Radial basis functions are one efficient, frequently used way to do this. Further applications include the important fields of neural networks and learning theory.
Since they are radially symmetric functions which are shifted by points in multidimensional Euclidean space and then linearly combined, they form data-dependent approximation spaces. This data-dependence
makes the spaces so formed suitable for providing approximations to large classes of given functions. It also opens the door to existence and uniqueness results for  
interpolating scattered data by radial basis functions in very
general settings (in particular in many dimensions). 


Indeed, one of the greatest advantages of
this method lies in its applicability in almost any dimension (whence its
versatility) because there are generally little restrictions on the way the data
are prescribed. Examples below include positive definite kernels where there are no restrictions on the data except that they need to be at distinct points.
This should be contrasted to, e.g., multivariable ''polynomial interpolation'' (but see [[#deBoorRon|de Boor and Ron 1990]] for an especially flexible approach) or ''splines''. 
A further advantage is their high accuracy or [[Convergence|fast convergence]] to the approximated
target function in many cases when data become dense.

For applications it is indeed desirable that there are few conditions
on the geometry or the directions in which the data points have to be placed in
space. No triangulations of the data points or the like are
required for radial basis function algorithms, whereas for instance finite element ([[#Brenner|Brenner and Scott 1994]],[[#Ciarlet|Ciarlet 1978]]) or multivariate spline methods ([[#deBoor|de Boor 1993]],[[#LaiSchumaker|Lai and Schumaker 2007]]) normally
need triangulations. In fact, the advance structuring of the
data that some other approximation schemes depend on  can be prohibitively
expensive to compute in applications, especially in more than two dimensions.
Therefore our approximations here are considered as '''meshfree approximations''', also for instance to be
used to facilitate the numerical solution of partial differential equations ([[#Fasshauer|Fasshauer 2007]]). On the other hand, as will be mentioned below, 
advanced numerical methods for computing the radial basis function approximations are
required when the data set is large while more standard software is required for
instance for finite elements.

== Definition of the method ==

=== The given data ===

The method usually works in <math>n</math> dimensional Euclidean
space which we call <math> R ^n</math> fitted with the Euclidean norm <math>\|\cdot\|\ .</math> Other choices of spaces are possible. There are <math>m</math> 
points in this space at which the function to be
approximated is known, call them <math>x_1, x_2, \ldots, x_m\ .</math> These
points are usually assumed to be all different from each other, otherwise the problem will become ''singular'' when interpolation is used. Both
<math>n</math> and <math>m</math> are positive integers.
 
The given values
at the points may be called <math>f(x_1), f(x_2), \ldots, f(x_m),</math>
the idea being that they come from a function <math>f:  R ^n\to
 R </math> 
which is evaluated at the respective points. This
function may be completely unknown except at those <math>m</math> 
points or, for
example, too difficult or time-consuming to evaluate otherwise. Although this
is a useful ansatz, other approaches without an underlying function are 
possible which allow coalescing points with different values, using the idea of
spline smoothing ([[#Wahba|Wahba 1990]]). 

Functions  <math>f:  R^n\to  R^k \ ,</math> where <math>k</math> is a positive integer
as well, are also admitted in the concept of radial basis functions; approximations are then carried out componentwise in the
<math>k</math> components of <math>f\ :</math> <math>f=(f_1,f_2,\ldots,f_k), f_i:R^n\to R, 1\leq i\leq k,</math> each. Generalising the radial basis function approach to matrix-valued kernels, alternative ideas are also available (e.g., [[#Lowitzsch|Lowitzsch 2005]]).

=== The approximation scheme ===

Given this information, we create the sought approximant by a sum

:<math>\label{Interpolant}
s(x)  = \sum_{j=1}^m \lambda_j \phi (\| x-x_j  \|),\qquad x\in R^n,

</math>

where
*the mentioned <math>x_j</math> are the data points, at which we know <math>f\ ,</math> which lie in <math> R ^n\ ,</math>
*the <math>x</math> is a free variable at which we wish to evaluate our approximant later,
*the <math>\phi</math> is a ''univariate'', normally continuous function <math>\phi:R_+\to R\ ,</math> namely the '''radial basis function''' (examples  will be given below),
*the <math>\|\cdot\|</math> denotes a norm <math>\|\cdot\|:  R ^n\to
 R \ ,</math> normally the Euclidean norm but there are more general approaches, and
*the <math>\lambda_j</math> are scalar parameters.

In fact, norms other than Euclidean are possible, but rarely chosen, and at any rate the individual terms would then of course no longer be radially symmetric about the <math>x_j.</math>

The scalar parameters are chosen so that <math>s</math> ''approximates''
<math>f</math> in a useful way at all desired points other than at the
<math>x_j\ ,</math> where it is known anyway. 

Thus, in general, the unknown (or difficult to
evaluate) function <math>f</math> is approximated by a linear expression, which
remains conceptually
simple even if <math>m</math> or  <math>n</math> become very
large. The flexibility of the approach is also based on the radial
symmetry of each term (although not, of course, of the whole
expression \eqref{Interpolant}) since its definition 
essentially depends only on a univariate
function. 

=== Choice of interpolation as approximation method ===


Most often, radial basis function approximations are used in
combination with '''interpolation''', i.e. the scalar parameters
<math>\lambda_j</math> are chosen, if possible, such that <math>s</math> matches
<math>f</math> exactly at the given <math>m</math> points
<math>x_j\ .</math> This is called interpolation and can be defined by
the conditions

:<math>\label{Interpolationconditions}
s(x_j) = f(x_j),\qquad j=1,2,\ldots,m.
</math>


These, in combination with the form \eqref{Interpolant}, result in a square, <math>m\times m</math> linear system of
equations in the <math>\lambda_j</math> which may or may not be
uniquely solvable. Its ''interpolation matrix'' is the square symmetric matrix
:<math>\label{Interpolationmatrix}
A=\Bigl(\phi(\| x_j-x_\ell \|)\Bigr)
</math>

whose non-singularity will guarantee the unique existence of the coefficients
<math>\lambda_j\ .</math> They are then to be computed in standard ways
of solving linear systems (using matrix decompositions) if <math>m</math> is small or in non-standard ways (see below) if it is large. When the kernel function in form of the radial basis function is strictly ''positive definite'', the
interpolation matrix is a positive definite matrix and non-singular (positive definite functions were considered in the classical paper [[#Schoenberg|Schoenberg 1938]] for example). Positive definite functions, and their generalisations conditionally positive definite functions, see below, are closely related to 
reproducing kernel Hilbert spaces (see literature under further reading).

Indeed, for those radial basis functions which are most often used in practice, the linear system is often hard to solve numerically because of high condition numbers but there exists specialised software to compute the sought coefficients even for very large <math>m\ .</math>

== Examples of radial basis functions ==

Clearly, a good choice of the <math>\phi</math> is important for the
quality of the approximation and for the existence of the
interpolants. Many choices guarantee the unique existence of \eqref{Interpolant} satisfying
\eqref{Interpolationconditions} for all
<math>m</math> and <math>n</math> solely under the condition that the
data points are all different ([[#Micchelli|Micchelli 1986]]). This is the case for

*linear radial basis function <math>\phi(r)=r\ ,</math> so long as <math>m>1\ ,</math>
*(Hardy) multiquadrics radial basis function <math>\phi(r)=\sqrt{r^2+c^2}\ ,</math> which contains another scalar parameter <math>c</math> which may be adjusted to improve the approximation, where the choice <math>c=0</math> gives the previous example, 
*Gaussian kernel <math>\phi(r)=\exp(-c^2 r^2) \ ,</math> which also contains another scalar parameter <math>c\neq0</math> which may be adjusted to adapt  the approximation,  or finally
*(Hardy) inverse  multiquadrics radial basis function <math>\phi(r)=1/\sqrt{r^2+c^2}\ ,</math> which contains another scalar parameter <math>c\neq0</math> which provides further flexibility.

== Further example with linear additional terms ==

Sometimes, the unique existence of interpolants can be guaranteed with
a small variation on the concept by adding low order polynomials to
<math>s</math> and some mild extra conditions. For example

:<math>
s(x)  = \sum_{j=1}^m \lambda_j \phi (\| x-x_j  \|)+a+b^T x,\qquad x\in R^n,
</math>

where <math>a</math> is a scalar and <math>b</math> is a vector in
<math> R ^n</math> will give unique existence of interpolating <math>s</math>
using ''thin-plate splines'' <math>\phi(r)=r^2\log r\ ,</math> with its value at the origin declared to be zero, so long as
the <math>x_j</math> are not collinear and the extra conditions

:<math>\label{Sideconditions}
\sum_{j=1}^m\lambda_j=0,\qquad \sum_{j=1}^m\lambda_j x_j=(0,0,\ldots,0)^T, 
</math>


are fulfilled. Here, <math>(0,0,\ldots,0)^T</math> denotes the zero vector in <math>n</math> dimensions. Thus, under these conditions, the scalars <math>\lambda_j, a</math> and the vector  <math>b</math> can be solved for uniquely.

These extra conditions \eqref{Sideconditions} take up the new degrees of
freedom that come in with <math>a</math> and <math>b\ .</math> Many
further examples of radial basis functions <math>\phi</math> exist
which are equally useful and sometimes require that 
polynomials of degree more than one are added and suitable extra conditions similar to
\eqref{Sideconditions} are imposed. But the polynomials are normally not 
of very high degree, constant to cubic being typical. The side conditions \eqref{Sideconditions} have to be adjusted to different order conditions when polynomials of degree other than one are used. Also the geometric conditions (centres not being collinear in the case \eqref{Sideconditions}) will have to be strengthened accordingly  ([[#Duchon|Duchon 1976]]). Such kernels are no longer positive definite as mentioned above, but conditionally positive definite due to the aforementioned side conditions.

== Properties of the method ==

=== Convergence ===

In spite of the simplicity of the idea even in high dimensions, good
convergence properties have been observed when the <math>x_j</math>
become dense, for example in compact sub-sets of the space <math>R^n\ .</math> In particular
Duchon has studied the thin-plate splines and related radial basis
functions when the scattered data points are becoming dense, other
examples of radial basis functions that are included in his theory
being ''pseudo cubics'' <math>\phi(r)=r^3</math> and
<math>\phi(r)=r^4\log r</math> ([[#Duchon|Duchon 1976]]), ([[#UniformConvergence|Powell 1994]]), or with multiquadrics ([[#Mad|Madych and Nelson 1992]]).

For the convergence analysis, one assumes sometimes that the data
points are on equispaced grids in <math> R ^n\ ,</math> so that
infinitely many data are given. The spacing being denoted by
<math>h\ ,</math> one then lets <math>h\to0\ .</math> We find in cases
that include most of the radial basis functions mentioned above that
the uniform difference between <math>s</math> and <math>f</math> (the ''error'')
goes to zero at the same rate as some power of <math>h</math> ([[#Buhmann|Buhmann 2003]], [[#Wend|Wendland 2005]]). One also looks
specifically for the best possible powers there (''saturation orders'') when <math>f</math> satisfies suitable conditions ([[#Johnson|Johnson 2000]]).

An example of a uniform convergence result in <math>n</math> dimensions states that multiquadric
interpolation on an infinite uniform grid of spacing <math>h</math> provides convergence of
<math>O(h^{n+1})</math> to sufficiently smooth <math>f</math> with certain partial derivatives bounded, i.e., the uniform error is bounded above by a fixed multiple of <math>h^{n+1}\ .</math> It should be noted here that the exponent of <math>h</math> increases with the dimension.

More general convergence theory is given for instance in ([[#WuSchaback|Wu and Schaback 1993]],[[#NarcWaWend|Narcowich, Ward and Wendland 2005]]).

=== Computation of the approximations ===

An unwelcome aspect appears when the linear systems are solved for
large <math>m</math> since for most radial basis functions the matrix
is full and ill-conditioned ([[#Ward|Narcowich and Ward 1991]]). An exception is provided by the  [[Compactly supported radial basis functions|radial basis functions of compact support]] described below.

A typical case is the multiquadric function,
where the interpolation matrix has spectral properties which depend both on the
parameter and on the distances of the data-points. They can lead to large
condition numbers, and of course the matrix is not ''sparse''. 

Some preconditioning and iterative
methods are to be applied (for an early approach see [[#Dyn|Dyn and Levin 1983]]). The case when multiquadrics are used is very important since they are most often used in applications, other important choices the aforementioned thin-plate splines and exponential functions. One class of particularly successful methods for computing interpolants with many centres are Krylov space
methods ([[#IterativeMethod|Powell 1994]]), others contain  particle methods and far field
expansions ([[#Beatson|Beatson et al. 1998]], see also the article of Beatson and Greengard in [[#Light|Levesley et al. 1997]]). General preconditioning methods are also useful, especially if <math>m</math> is not too large ([[#Fasshauer|Fasshauer 1999]]).

Other approaches which avoid the difficulty of ill-conditioned interpolation matrices include the idea of quasi-interpolation (e.g. see [[#Buhmann|Buhmann 2003]] for a number of useful examples of quasi-interpolation), or the aforementioned spline smoothing. 

=== Aspects of the parameters in multiquadrics and exponentials ===

In applications, the parameters <math>c</math> in multiquadrics,
inverse multiquadrics and exponentials play an important role. The
aforementioned ill-conditioning problems become very severe if the parameters go to
limits (e.g., <math>c\to\infty</math> in (inverse)
multiquadrics, or to zero in Gaussian kernels, where the entries of the matrix become constant asymptotically). 
Nonetheless, methods have been developed to handle
this situation and it has even been observed that sometimes the whole
approximants <math>s</math> tend to polynomial limits in those cases ([[#Larsson|Larsson and Fornberg 2005]]).

== Compactly supported radial basis functions ==

Compactly supported radial basis functions have been invented for the
purpose of getting finite-element type approximations ([[#Brenner|Brenner and Scott 1994]]). They give rise to sparse interpolation matrices and can be
used to solve numerically partial differential equations ([[#Fasshauer|Fasshauer 1999]]). Some of them
are piecewise-polynomial as a one-dimensional function
<math>\phi</math> (usually only two pieces) ([[#Wendland|Wendland 1995]] where there are useful lists of examples provided together with the theory). Under suitable conditions
on degree and dimension <math>n\ ,</math> they give rise to positive
definite interpolation matrices <math>A</math> that are banded, therefore sparse, and
then of course also regular (for further choices see [[#CompactSupport|Buhmann 2001]]).

A little less flexibility stems from restrictions on <math>n</math> which may not be arbitrarily large anymore,
but there are still no upper bounds on <math>m\ .</math> Also, the positive definiteness of the interpolation matrices (similarly as with Gaussian kernels and inverse multiquadrics) makes the radial
basis functions useful in statistics. 

== Applications ==

Applications are manifold, they include
* finite element or spectral methods for the solution of partial differential equations ([[#Fornberg|Fornberg 1999]]),
* neural networks with radial basis functions,[http://en.wikipedia.org/wiki/Radial_basis_function_network], and machine learning ([[#CuckerSmale|Cucker and Smale 2002]], [[#SchabackWendland|Schaback and Wendland 2006]]),
* approximations on spheres, ([[#Freeden|Freeden et al. 1998]], [[#GoliLight|Golitschek and Light 2000]])
* statistical approximations, where positive definite kernels are very important, see ([[#NEM|Nisbet et al. 2009]]) or, for instance, the paper on Nonstandard Kernels and their Applications by DeMarchi and Schaback on [http://drna.di.univr.it/],
* geophysical research ([[#Freeden|Freeden et al. 1998]]),
* and many engineering applications ([[#Ferreira|Ferreira et al. 2007]]).

== Further reading ==

*{{Bibitem book 1|Radial basis functions: theory and implementations|Cambridge University Press|Cambridge|2003|Buhmann|Martin|isbn=ISBN 978-0-521-10133-2.}}<span id=''Buhmann''></span>
*{{Bibitem book 1|Meshfree Approximation Methods with Matlab|World Scientific|Singapore|2007|Fasshauer|Greg}}<span id=''Meshfree''></span>
*{{Bibitem book 1|Approximation Theory and Methods|Cambridge University Press|Cambridge|1981|Powell|M J D}}<span id=''ApproximationTheoryandMethods''></span>
*{{Bibitem book 1|Scattered data approximation|Cambridge University Press|Cambridge|2005|Wendland|H}}<span id=''Wend''></span>

<!-- Authors, please check this list and remove any references that are irrelevant. This list is generated automatically to reflect the links from your article to other accepted articles in Scholarpedia. -->
<b>Internal references</b>

* Rob Schreiber (2007) [[MATLAB]]. Scholarpedia, 2(7):2929.

* Kendall E. Atkinson (2007) [[Numerical analysis]]. Scholarpedia, 2(8):3163.

* Andrei D. Polyanin, William E. Schiesser, Alexei I. Zhurov (2008) [[Partial differential equation]]. Scholarpedia, 3(10):4605.

* David Gottlieb and Sigal Gottlieb (2009) [[Spectral methods]]. Scholarpedia, 4(9):7504.


== References ==

*{{Bibitem article 3|Fast fitting of radial basis functions: methods based on preconditioned GMRES iteration|Advances in Computational Mathematics|11|1998|253-270|Beatson|R|Cherrie|J|Mouat|C}}<span id=''Beatson''></span>
*{{Bibitem article 1|Multivariate piecewise polynomials|Acta Numerica|2|1993|65-109|de Boor|C| doi=10.1017/s0962492900002348}}<span id=''deBoor''></span>
*{{Bibitem article 2|On multivariate polynomial interpolation|Constructive Approximation|6|1990|287-302|de Boor|C|Ron|A| doi=10.1007/bf01890412}}<span id=''deBoorRon''></span>
*{{Bibitem book 2|The mathematical theory of finite elements|Springer|New York|1994|Brenner|S|Scott|L}}<span id=''Brenner''></span>
*{{Bibitem article 1|A new class of radial basis functions with compact support| Mathematics of Computation|70|2001|307-318|Buhmann|Martin| doi=10.1090/s0025-5718-00-01251-5}}<span id=''CompactSupport''></span>
*{{Bibitem book 1|Introduction to Approximation Theory|McGraw-Hill|New York|1966|Cheney|E W}}<span id=''Cheney''></span>
*{{Bibitem book 1|The finite element method for elliptic problems|North-Holland|Amsterdam|1978|Ciarlet|P}}<span id=''Ciarlet''></span>
*{{Bibitem article 2|On the mathematical foundations of learning|Bulletin AMS|39|2002|1-49|Cucker|F|Smale|S| doi=10.1090/s0273-0979-01-00923-5}}<span id=''CuckerSmale''></span>
*{{Bibitem book 1|Interpolation and  Approximation |Dover|New York|1975|Davis|P J}}<span id=''Davis''></span>
*{{Bibitem article 2|Iterative solution of systems originating from integral equations and surface interpolation|SIAM Journal of Numerical Analysis|20|1983|377-390|Dyn|N|Levin|D| doi=10.1137/0720026}}<span id=''Dyn''></span>
*{{Bibitem article 1|Sur l'erreur d'interpolation des fonctions de plusieurs variables pas les Dm-splines|Rev. Francaise Automat. Informat. Rech. Oper. Anal.  Numer.|10|1976|5-12|Duchon|Jean}}<span id=''Duchon''></span>
*{{Bibitem article 1|Solving differential equations with radial basis functions: multilevel methods and smoothing|Advances in Computational Mathematics|11|1999|139-159|Fasshauer|Greg}}<span id=''Fasshauer''></span>
*{{Bibitem book 4|Second Eccomas Thematic Conference on Meshless Methods 2007|University of Porto|Porto|2007|Ferreira|Antonio|Kansa|Ed|Fasshauer|Greg|Leitao|Vitor|isbn=ISBN 978-972-8826-12-3.}}<span id=''Ferreira''></span>
*{{Bibitem book 1|A practical guide to pseudospectral methods|Cambridge University Press|Cambridge|1999|Fornberg|Bengt}}<span id=''Fornberg''></span>
*{{Bibitem book 3|Constructive approximation on the sphere|Clarendeon Press|Oxford|1998|Freeden|W|Gervens|T|Schreiner|M|isbn=ISBN 0-19-853682-8.}}<span id=''Freeden''></span>
*{{Bibitem article 2|Interpolation by polynomials and radial basis functions on spheres|Constructive Approximation|17|2000|1-18|Golitschek|M|Light|W| doi=10.1007/s003650010028}}<span id=''GoliLight''></span>
*{{Bibitem article 1|The L_2-approximation order of surface spline interpolation|Mathematics of Computation|70|2000|719-737|Johnson|Michael| doi=10.1090/s0025-5718-00-01301-6}}<span id=''Johnson''></span>
*{{Bibitem book 2|Spline functions on triangulations|Cambridge University Press|Cambridge|2007|Lai|M J|Schumaker|L L}}<span id=''LaiSchumaker''></span>
*{{Bibitem article 2|Theoretical and computational aspects of multivariate interpolation with increasingly flat radial basis functions|Comput. Math. Appl.|49|2005|103-130|Larsson|E|Fornberg|B| doi=10.1016/j.camwa.2005.01.010}}<span id=''Larsson''></span>
*{{Bibitem book 3|Wavelets, Multilevel Methods and Elliptic PDEs|Oxford University Press|Oxford|1997|Levesley|J|Light|W|Marletta|M}}<span id=''Light''></span>
*{{Bibitem article 1|Error estimates for matrix-valued radial basis function interpolation|Journal of Approximation Theory|137|2005|234-249|Lowitzsch|S| doi=10.1016/j.jat.2005.09.008}}<span id=''Lowitzsch''></span>
*{{Bibitem article 2|Bounds on multivariate polynomials and exponential error estimates for multiquadric interpolation|Journal of Approximation Theory|70|1992|94-114|Madych|W|Nelson|S| doi=10.1016/0021-9045(92)90058-v}}<span id=''Mad''></span>
*{{Bibitem article 1| Interpolation of scattered data: distance matrices and conditionally positive definite functions|Constructive Approximation|1|1986|11-22|Micchelli|Charles| doi=10.1007/bf01893414}}<span id=''Micchelli''></span>
*{{Bibitem article 2|Norms of inverses and condition numbers of matrices associated with scattered data|Journal of Approximation Theory|64|1991|69-94|Narcowich|F|Ward|J| doi=10.1016/0021-9045(91)90087-q}}<span id=''Ward''></span>
*{{Bibitem article 3|Sobolev bounds on functions with scattered zeros, with applications to radial basis function surface fitting|Mathematics of Computation|74|2005|643-763|Narcowich|F|Ward|J|Wendland|H| doi=10.1090/s0025-5718-04-01708-9}}<span id=''NarcWaWend''></span>
*{{Bibitem book 3|Statistical Analysis and Data Mining|Academic Press|New York|2009|Nisbet|R|Elder|J|Miner|G}}<span id=''NEM''></span>
*{{Bibitem article 1|The uniform convergence of thin-plate spline interpolation in two dimensions|Numerische Mathematik|67|1994|107-128|Powell|M J D| doi=10.1007/s002110050051}}<span id=''UniformConvergence''></span>
*{{Bibitem article 1|A new iterative algorithm for thin plate spline interpolation in two dimensions|Annals of Numerical Mathematics|4|1997|519-527|Powell|M J D}}<span id=''IterativeMethod''></span>
*{{Bibitem article 1|Error estimates and condition numbers for radial basis function   interpolation|Advances in Computational Mathematics|3|1995|251-264|Schaback|Robert| doi=10.1007/bf02432002}}<span id=''Schaback''></span>
*{{Bibitem article 2|Kernel techniques: from machine learning to meshless methods|Acta Numerica|15|2006|543-639|Schaback|R|Wendland|H| doi=10.1017/s0962492906270016}}<span id=''SchabackWendland''></span>
*{{Bibitem article 1|Metric spaces and positive definite functions|Transactions of the AMS|44|1938|522-536|Schoenberg|I J| doi=10.1090/s0002-9947-1938-1501980-0}}<span id=''Schoenberg''></span>
*{{Bibitem article 1| Piecewise polynomial, positive definite and compactly supported radial functions of minimal degree''|Advances in Computational Mathematics|4|1995|389-396|Wendland|Holger| doi=10.1007/bf02123482}}<span id=''Wendland''></span>
*{{Bibitem book 1|Spline Models for Observational Data|SIAM|Philadelphia|1990|Wahba|G}}<span id=''Wahba''></span>
*{{Bibitem article 2|Local error estimates for radial basis function interpolation of scattered data|IMA Journal of Numerical Analysis|13|1993|13-27|Wu|Z M|Schaback|R| doi=10.1093/imanum/13.1.13}}<span id=''WuSchaback''></span>



== See also ==

Finite elements, multivariate splines, multivariate approximation theory, kernel space methods.

[[Category: Approximation theory]]

[[Category:Numerical Analysis]]