On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants - Texas A&M University (TAMU) Scholar

On cardinal interpolation by Gaussian radial-basis functions: Properties of fundamental functions and estimates for Lebesgue constants
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Suppose is a positive number. Basic theory of cardinal interpolation ensures the existence of the Gaussian cardinal function L (cursive Greek chi) = jZcj exp(-(cursive Greek chi - j)2), cursive Greek chi R, satisfying the interpolatory conditions L (k) = 0k k Z. One objective of this paper is to derive several additional properties of L. For example, it is shown that L possesses the sign-regularity property sgn[L(cursive Greek chi)] = sgn[sin(cursive Greek chi)/(cursive Greek chi)], cursive Greek chi R, and that |L(cursive Greek chi)| 2e8 min{(|cursive Greek chi| + 1)-1, exp(-|cursive Greek chi|)}, cursive Greek chi R. The analysis is based on a simple representation formula for L and employs some methods from classical function theory. A second consideration in the paper is the Gaussian cardinal-interpolation operator , defined by the equation (y)(cursive Greek chi) := kZykL (cursive Greek chi - k), cursive Greek chi R, y = (yk)kZ. On account of the exponential decay of the cardinal function L, is a well-defined linear map from (Z) into L (R). Its associated operator-norm |||| is called the Lebesgue constant of . The latter half of the paper establishes the following estimates for the Lebesgue constant: |||| (Equivalent to) 1, , and |||| (Equivalent to) log(1/), 0+. Suitable multidimensional analogues of these results are also given.