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== Definition ==
A radial function is a function <math display="inline">\varphi:[0,\infty) \to \mathbb{R}</math>. When paired with a norm on a vector space <math display="inline"> \|\cdot\|:V \to [0,\infty)</math>, a function of the form <math display="inline"> \varphi_\mathbf{c} = \varphi(\|\mathbf{x}-\mathbf{c}\|) </math> is said to be a '''radial kernel''' centered at <math display="inline"> \mathbf{c} \in V </math>. A
{{bulleted list
| The kernels <math>\varphi_{\mathbf{x}_1}, \varphi_{\mathbf{x}_2}, \dots, \varphi_{\mathbf{x}_n}</math> are linearly independent (for example <math>\varphi(r)=r^2</math> in <math>V=\mathbb{R}</math> is not a radial basis function)
| The kernels <math>\varphi_{\mathbf{x}_1}, \varphi_{\mathbf{x}_2}, \dots, \varphi_{\mathbf{x}_n}</math> form a basis for a [[Haar space|Haar Space]], meaning that the [[radial basis function interpolation|interpolation matrix]] (given below) is non-singular.<ref>{{cite book |last1=Fasshauer |first1=Gregory E. |title=Meshfree Approximation Methods with MATLAB |date=2007 |publisher=World Scientific Publishing Co. Pte. Ltd. |___location=Singapore |isbn=9789812706331 |pages=17–25}}</ref><ref name="wendland2005">{{cite book |last1=Wendland |first1=Holger |title=Scattered Data Approximation |date=2005 |publisher=Cambridge University Press |___location=Cambridge |isbn=0521843359 |pages=11, 18-23,64-66}}</ref>
}}▼
{{NumBlk||<math display="block">
\begin{bmatrix}
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\vdots & \vdots & \ddots & \vdots \\
\varphi(\|\mathbf{x}_1 - \mathbf{x}_n\|) & \varphi(\|\mathbf{x}_2 - \mathbf{x}_n\|) & \dots & \varphi(\|\mathbf{x}_n - \mathbf{x}_n\|) \\
\end{bmatrix}
</math>|{{EquationRef|1}}}}
▲}}
=== Examples ===
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