On Density in Radial Basis Approximation

Mon, 04/06/2012 - 14:00

We characterize the radial basis functions whose
scattered shifts form a fundamental system in
 the space $L_{p}(\rrd)$. In particular, we show that for any even function $h$ from the space
 $L_{2,{\rm loc}}(\rrd)$
 the space formed by all possible linear combinations of
 shifted radial basis functions $h(\|x+a\|)$, $a\in \rrd$, is dense in the
 space $L_p(\rrd)$, $1\le p\le 2$, if
 and only if the function $h$ is not a polynomial.