Talk:Riemann-Lebesgue lemma

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[edit] extension

I think you can apply the theorem to functions over the whole real line, not just intervals, provided they're still L¹ of course. The proof has an extra technical step, but this is a much more useful result. For example, see Reed and Simon (though they prove it quite differently, and they also prove that not only does the Fourier Transform of the L¹ function exist, but that it is ). Lavaka 18:13, 13 September 2006 (UTC)

[edit] intuitively

I understood everything in this article except for the statement: "Intuitively, the lemma says that if a function oscillates rapidly around zero, then the integral of this function will be small." How is this related to the Riemann-Lebesgue lemma? Why is a rapidly-oscillation necessarily in L¹? And how does the lemma imply a small integral? --Zvika (talk) 14:39, 6 December 2007 (UTC)

"Function" refers not to f, but to the integrand f(x)e^izx. Does that help? -- Jitse Niesen (talk) 14:54, 6 December 2007 (UTC)

I think I understand what you mean now, but I don't think this is what the theorem says. There can be rapidly oscillating functions whose integral does not tend to zero, if they are not absolutely integrable. As I understand the lemma, the point is not that rapidly oscillating functions have small integrals, but that absolutely summable functions have limited oscillations (which also sounds, to me, like a much more powerful result). In any case, even if you do want to keep the current explanation, I think its relation to the theorem should be made clearer. Cheers, --Zvika (talk) 20:08, 6 December 2007 (UTC)