This is a continuation of our earlier study of the shape parameter c contained in the famous multiquadrics (−1) ⌈ β 2 ⌉ (c 2 + x 2 ) β 2 , β > 0, and the inverse multiquadrics (c 2 + x 2 ) β , β < 0. In the previous two papers we presented criteria for the optimal choice of c, based on the exponentialtype error bound. In this paper a new set of criteria is developed, based on the improved exponentialtype error bound. This results in much sharper error estimates when c is chosen appropriately, with the same size of fill distance. What is important is that the optimal value of c can be successfully predicted without any search when fill distance is of reasonable size, making it practically useful. The drawback is that the distribution of the data points is not purely scattered. However it seems to be harmless.
It's well-known that there is a very powerful error bound for Gaussians put forward by Madych and Nelson in 1992. It's of the form| f (x) − s(x)| ≤ (Cd) c d f h where C, c are constants, h is the Gaussian function, s is the interpolating function, and d is called fill distance which, roughly speaking, measures the spacing of the points at which interpolation occurs. This error bound gets small very fast as d → 0. The constants C and c are very sensitive. A slight change of them will result in a huge change of the error bound. The number c can be calculated as shown in [9]. However, C cannot be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer this question.
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