Tschebyscheff — Approximation mit Exponentialsummen mit Polynomialen Exponenten

Zusammenfassung

In this paper the existence of a best Tschebyscheff-approximation of the class

$${E_{n,m}}: = \{ y \in c(I)|y(x) = \sum\limits_{i = 1}^k {P(x)} {e^{{Q_i}^{(x)}}},\;{P_i},\;{Q_i}\;Polynome\;mit\;\delta {Q_i} \leqslant m\;and\;\sum\limits_{i = 1}^k {(\delta {P_i} + m) \leqslant n \cdot m} \} $$

for any f ∈ C(I) is schown. For that purpose one has to proof some properties of the family E_n,m, where mainly the connection between certain linear differential equations and E_n,m leads to the existence of a best approximation. As E_n,m doesn’t fullfill the Haar-condition, the uniqueness could only be verified for sum of exponentials with nonnegative coefficients. Nevertheless this part of E_n,m, the family

$$E_{n,m}^ + = \left\{ {y \in c(I)\;|\;y(x)\;\sum\limits_{i = 1}^k {{d_i}{e^{{Q_i}^{(x)}}},{d_i} \in {\mathbb{R}^ + },k \leqslant n} } \right\}$$

is of some relevance in practice, which is documented by the closing numerical example.