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EXPONENTIAL SUMS

DOI: 10.1615/AtoZ.e.exponential_sums

In the analysis of decay processes, an empirical function f(x) is approximated on a real interval [a, b] by a finite sum of the from

where αν and βν are real numbers. This is a problem in nonlinear approximation theory. For a fixed integer n > 1 , the unit interval [0, 1] is considered with an equidistant partition of length l/2n;

If, at these 2n + 1 points, the values of the function to be approximated are known, then f(xk) = fk(k = 0, 1, ..., 2n) and the following system of nonlinear equations is obtained:

(1)

where λ accounts for the maximum error in the approximation. Since xk = k/2n, then zν = eβν/2n and Eq. (1) can be written as:

(2)

The nonlinear Eq. (2) for the unknowns αν, zν (σ = 1, ..., n) and λ (note that βν = 2n ln zν) can then be solved. The Newton iteration method is applied to improve on this first approximation.

REFERENCES

Braess, D. (1986) Nonlinear Approximation Theory, VI. Springer-Verlag, Berlin.

Meinardus, G. (1967) Approximation of Functions: Theory and Numerical Methods, 10. Springer-Verlag, Berlin. DOI: 10.1016/S0016-0032(68)90588-7

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