Looking for a source: Fourier inversion of $f \in L^1$

Question

Is there a book where I can find a thorough proof of the following assertion?

Let $f \in L^1(\mathbb{R}^d)$ be continuous at zero and $\hat{f}\ge0$. Then $\hat{f} \in L^1(\mathbb{R}^d)$ and $$f(t) = \int_{\mathbb{R}^d} \hat{f}(\xi)e^{2\pi i\, t\xi} \,d\xi$$ almost everywhere.

I'm looking for the context in which this Lemma is stated, more than the actual proof.

I've finally found the source: E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. $\S$1. Corollary 1.26 (p.15)

@MhenniBenghorbal On this lecture notes, p3. Satz 10. But they are in German. — Lex Luengas, Aug 22 '13 at 18:08

score 2 · Answer 1 · answered Aug 21 '15 at 22:53

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By OP:

I've finally found the source: E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971. §1. Corollary 1.26 (p.15)

answered Aug 21 '15 at 22:53

Silvia Ghinassi

4,707

score 0 · Answer 2 · edited Apr 13 '17 at 12:21

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In terms of the proof please see here: Proof of Fourier Inverse formula for $L^1$ case

Further to another German reference that rather pays attention to context than proof is here http://www.math.ethz.ch/education/bachelor/seminars/hs2007/harm-analysis/FT2.pdf

edited Apr 13 '17 at 12:21

Community

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answered Aug 22 '13 at 19:07

al-Hwarizmi

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The linked answer doesn't treat the interesting part, that if $f$ is continuous at $0$ and $\widehat{f} \geqslant 0$, then $\widehat{f} \in L^1$. – Daniel Fischer Aug 22 '13 at 19:59

Looking for a source: Fourier inversion of $f \in L^1$

2 Answers2