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 α_{ν} and β_{ν} 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(x_{k}) = f_{k}(k = 0, 1, ..., 2n) and the following system of nonlinear equations is obtained:
where λ accounts for the maximum error in the approximation. Since x_{k} = k/2n, then zν = e^{βν/2n} and Eq. (1) can be written as:
The nonlinear Eq. (2) for the unknowns α_{ν}, z_{ν} (σ = 1, ..., n) and λ (note that β_{ν} = 2n ln z_{ν}) 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
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