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Assume that I have N multiquadric radius basis functions with different $ε_i$ and $c_i$, How to prove $\sqrt{1+(ε_i(x−c_i))^2}$ are linear independent, $1\leq i \leq N$.

yang
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  • What have you done? Did you try to use ideas from your previous question (looking at the aymptotics; e.g., the limit of a linear combination around a convenient point, for instance one of the $c_i$'s)?

    (also, as a typo correction: you may want to use the argument $r$ somewhere in the function)

     – Clement C. Dec 26 '14 at 14:31

1 Answers1

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Both Gaussian and multiquadric functions are (strictly) positive definite (Wendland's "Scattered data approximation", pp.74,76). If there were a non-trivial linear combination identically zero for all $x$ then it would be zero for any $x=x_j$ too and you could write a quadratic form $\sum\sum\alpha_j\bar{\alpha_k}\Phi(x_j-x_k)=0$ which would contradict the strict positive definiteness of $\Phi$.

rych
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