Convolution kernels based on thin-plate splines (Q1904152)

The so-called quasi-interpolation is studied which uses radial basis functions for constructing approximations to continuous functions in many space dimensions. A procedure for generating kernels for quasi-interpolation is discussed, using functions which have series expansions involving terms like \(r^\alpha \log r\). It is shown that such functions are suitable if and only if \(\alpha\) is a positive even integer and the spatial dimension is also even.