International Journal of Numerical Methods and Applications
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Abstract: For multiquadric interpolation, it is well known that
the most powerful error bound is the so-calledexponential-type error bound. It is of the form where is a constant, C
is a constant, d is the fill distance which roughly speaking measures the spacing
of the data points, is the
interpolating function of and denotes the
norm of f induced by the multiquadric
or inverse multiquadric. This error bound for was put
forward by Madych and Nelson in 1992 and converges to zero very fast as The
drawback is that both C and l get worse as In
particular, very fast as In this
paper, we raise an error bound of the form where can be
independent of the dimension n and Moreover, is only
slightly different from C. What is
noteworthy is that both and can be
computed without slight difficulty. The drawback of our approach should also be
mentioned. In our approximation scheme, the data points are not purely
scattered. They are somewhat evenly spaced in a simplex. The details will be
seen in the text. Although this is a restriction, an invigorating fact is that
our result is proven in a way totally different from that of Madych and Nelson.
No algebraic geometry or the theory of nondegenerate points is involved and the
correctness can easily be checked. As pointed out by Wendland in page 188 of
[11], the only known proof for spectral convergence of multiquadrics is based on
some deep results from algebraic geometry and the theory of nondegenerate
points, which can be found in [10]. This is something of frustrating. We avoid
this trouble completely.
