  This paper studies the influence of scaling on the behavior of Radial Basis
Function interpolation. It focuses on certain central aspects, but does not try
to be exhaustive. The most important questions are: How does the error of a
kernel-based interpolant vary with the scale of the kernel chosen? How does the
standard error bound vary? And since fixed functions may be in spaces that
allow scalings, like global Sobolev spaces, is there a scale of the space that
matches the function best? The last question is answered in the affirmative for
Sobolev spaces, but the required scale may be hard to estimate. Scalability of
functions turns out to be restricted for spaces generated by analytic kernels,
unless the functions are band-limited. In contrast to other papers, polynomials
and polyharmonics are included as flat limits when checking scales
experimentally, with an independent computation. The numerical results show
that the hunt for near-flat scales is questionable, if users include the flat
limit cases right from the start. When there are not enough data to evaluate
errors directly, the scale of the standard error bound can be varied, up to
replacing the norm of the unknown function by the norm of the interpolant. This
follows the behavior of the actual error qualitatively well, but is only of
limited value for estimating error-optimal scales. For kernels and functions
with unlimited smoothness, the given interpolation data are proven to be
insufficient for determining useful scales.
