Eine Erweiterung des Riemann-Lebesgue-Lemmas |
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Authors: | Dr Wolfgang Stadje |
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Institution: | 1. Institut für Mathematische Statistik und Wirtschaftsmathematik, Universit?t G?ttingen, Lotzestra?e 13, D-3400, G?ttingen, Bundesrepublik Deutschland
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Abstract: | It is proved that iff∈L 1(?),f'∈L 1(?) and ∫∣x∣ i ∣f(x)∣dx<∞ fori=1, ...,k?1 and ifA=(a ij ) is a (k×k)-matrix with non-vanishing determinant, for $$\tilde f_A (\zeta ): = \smallint \exp (i\zeta _1 \sum\limits_{j = 1}^k {a_{1j} x^j } + ... + i\zeta _k \sum\limits_{j = 1}^k {a_{kj} x^j } )f(x)dx$$ the following relation holds: $$\tilde f_A (\zeta ) = O(\left\| \zeta \right\|)^{ - b_k } with b_k : = (\sum\limits_{j = 1}^k {j!)^{ - 1} } for k \in \mathbb{N}$$ . |
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