Help with Riemann-Lebesgue theorem pls

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SUMMARY

The Riemann-Lebesgue theorem states that if a function f is integrable over the entire real line, specifically on the interval E = (-∞, ∞), then the limit of the integral of f multiplied by cos(nx) approaches zero as n approaches infinity. This conclusion holds true particularly when f is a step function, as it can be analyzed by breaking it into corresponding intervals. The integral over each interval converges to zero, confirming the theorem's validity.

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  • Understanding of the Riemann-Lebesgue theorem
  • Knowledge of integrable functions
  • Familiarity with step functions
  • Basic concepts of Fourier analysis
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  • Study the proof of the Riemann-Lebesgue theorem in detail
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Mathematicians, students studying real analysis, and anyone interested in Fourier analysis and the properties of integrable functions will benefit from this discussion.

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If f is integrable on [tex](-\infty, \infty) = E[/tex] then [tex]\lim_{n\rightarrow\infty} \int_E f(x)cos(nx) = 0[/tex]

it's supposed to be easy if f is a step function & then i can use another problem that was part of the homework, but i can't seem to figure out how that would work...





(ps- maybe i shouldn't call myself Fourier jr if I'm stuck on a problem like this :rolleyes: )
 
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If f is a step function, break it up into the corresponding intervals. Can you show the integral restricted to anyone of these intervals goes to zero as n goes to infinity?
 
thx for the tip... :smile: i must have been thinking too hard about it... :redface:
 

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