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Let $0 < f ∈ L^1(\mathbb{R^2})$ almost everywhere. Prove: $|\hat{f}(x)| < \hat{f}(0) \quad \forall x \in \mathbb{R^d}\backslash\{0\}$.

Could someone please help me with this question? I tried solving this with the step function but I don't seem to get anywhere. Any advice would be much appreciated. Thank you in advance.

user14978538
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1 Answers1

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$$|\hat{f}(\xi)| < \int_{\mathbb{R}^{d}} |e^{-i x \xi}| f(x) dx=\int_{\mathbb{R}^{d}} f(x) dx=\hat{f}(0)$$

mover002
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  • Thank you for your help. Could you elaborate as to why the first inequality holds? I could explain it using the triangle inequality, but then it would be a <=, right? – user14978538 Jan 14 '21 at 20:54
  • the set on which $e^{-i x \xi}f(x)$ is negative has nonzero measure. – mover002 Jan 14 '21 at 21:00