Eine Erweiterung des Riemann-Lebesgue-Lemmas

Abstract

It is proved that iff∈L ₁(ℝ),f'∈L ₁(ℝ) 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:

$$\tilde f_A (\zeta ) = O(\left\| \zeta \right\|)^{ - b_k } with b_k : = (\sum\limits_{j = 1}^k {j!)^{ - 1} } for k \in \mathbb{N}$$

.