论文信息 - Approximation of 1/x by exponential sums in [1, ∞)

Approximation of 1/x by exponential sums in [1, ∞)

Approximations of 1/x by sums of exponentials are well studied for finite intervals. Here the error decreases like O(exp(-ck)) with the order k of the exponential sum. In this paper we investigate approximations of 1/x in the interval [1, ∞). We prove estimates of the error by O(exp(-c√k)) and confirm this asymptotic estimate by numerical results. Numerical results lead to the conjecture that the constant in the exponent equals c =π √2.

Dietrich Braess | Wolfgang Hackbusch | W. Hackbusch | D. Braess

[1] Lars Grasedyck,et al. Existence and Computation of a Low Kronecker-Rank Approximant to the Solution of a Tensor System with Tensor Right-Hand Side , 2003 .

[2] A A Gončar. ON THE SPEED OF RATIONAL APPROXIMATION OF SOME ANALYTIC FUNCTIONS , 1978 .

[3] Werner Kutzelnigg,et al. Theory of the expansion of wave functions in a gaussian basis , 1994 .

[4] E. Saff,et al. Logarithmic Potentials with External Fields , 1997 .

[5] ON THE LEAST DEVIATIONS OF THE FUNCTION $ \operatorname{sign} x$ AND ITS PRIMITIVES FROM THE RATIONAL FUNCTIONS IN THE $ L_p$ METRICS, $ 0 < p \leqslant \infty$ , 1977 .

[6] Herbert Stahl,et al. Best uniform rational approximation ofxα on [0, 1] , 2003 .

[7] Dietrich Braess,et al. Asymptotics for the approximation of wave functions by exponential sums , 1995 .

[8] D. Braess. Nonlinear Approximation Theory , 1986 .

[9] Boris N. Khoromskij,et al. Hierarchical Kronecker tensor-product approximation to a class of nonlocal operators in high dimensions , 2004 .