# Talk:Riemann–Lebesgue lemma

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Field:  Analysis

## which "special cases" in proof?

The first sentence of the proof is confusing, and it is not clear which "special cases" it refers to, and which if any of them is the "first one". Perhaps it could be replaced with something like the following:

"The proof can be organized into steps, proving increasingly general special cases; the 4th step extends the result to the original formulation."

but personally I think that it might be better to remove that sentence altogether --AmitAronovitch (talk) 18:35, 21 May 2010 (UTC)

The statement of the Theorem seems to be flawed. f is assumed to be a measurable function from R to C. But the proof deals with an interval [a,b]. Should we say f:[a,b] to C? or should the proof omit [a,b]? —Preceding unsigned comment added by 69.142.45.52 (talk) 01:22, 29 April 2011 (UTC)
Also the proof is only for f on R, while the statement of the theorem has f on R^d — Preceding unsigned comment added by Bzhao2017 (talkcontribs) 04:10, 2 December 2019 (UTC)

## Proof seems to be a bit unclear

You have written:

We'll focus on the one-dimensional case, the proof in higher dimensions is similar. Suppose first that f is a compactly supported smooth function. Then integration by parts in each variable yields

${\displaystyle \left|\int f(x)e^{-izx}dx\right|=\left|\int {\frac {1}{iz}}f'(x)e^{-izx}dx\right|\leq {\frac {1}{|z|}}\int |f'(x)|dx\rightarrow 0{\mbox{ as }}z\rightarrow \pm \infty .}$

How do you know that if f(x) is L1 function then its derivative is L1 function too? Consider:

${\displaystyle f(x)={\frac {\sin(e^{x})}{1+x^{2}}}}$.

or

${\displaystyle f(x)={\begin{cases}x^{2}e^{-x^{2}}\sin(1/x^{2})&|x|\in (0,1]\\0&{\text{otherwise}}\end{cases}}}$

Concluding: If you just write that you take into account only the situation where f(x) is differentiable and additionally f'(x) is L1 function too,then proof is more clear and universal.

 — Preceding unsigned comment added by 89.79.154.60 (talk) 19:50, 27 February 2014 (UTC)


Consider :

${\displaystyle f(x)={\frac {1_{\mathbb {Q} }2-1}{1+x^{2}}}}$

It's L1${\displaystyle \scriptstyle \mathrm {(} \mathbb {R} ),}$ integrable function, but ${\displaystyle \lim _{|\omega |\to +\infty }\int \limits _{\mathbb {R} }f(x)e^{-i\omega x}dx\neq 0}$ — Preceding unsigned comment added by 89.76.155.25 (talk) 18:05, 24 December 2014 (UTC)

What is ${\displaystyle {1_{\mathbb {Q} }2}}$? If it's a complex number, I think we're okay. 178.39.163.55 (talk) 11:50, 5 July 2015 (UTC)

## Article in "References" section (Self-promotion?)

The Researchgate article included in the references appears to have been added by User:Anilped. User:Anilped claims to be "Prof Anil Pedgaonkar", which also happens to be the alias of that article's author. I've made a brief skim of the article, and it seems inconsequential. But, then again, I've never had cause to use the lemma (yet). Maybe the article is of value, in a way I can't see. Can someone with a background in this area check that his generalizations are actually of sufficient novelty and power that we ought be directing readers there?

(If not, note that we should probably remove the last comment in "Other Versions" too.)

2601:240:C400:D60:902F:C0CF:8076:BA68 (talk) 03:21, 17 July 2017 (UTC)

## Abstract measure spaces

The proof in the research gate article is wrong, the reference was removed. — Preceding unsigned comment added by 81.243.243.97 (talk) 21:10, 21 November 2018 (UTC)