Real analysis: Integrable function

In summary, in order to perform a Fourier transform on a function f(x), it must be integrable, meaning that the integral of |f(x)| from negative infinity to positive infinity must be finite. However, this does not necessarily mean that f(x) approaches zero as x approaches positive or negative infinity. A counterexample is the function x^-1, which goes to zero but its integral from negative infinity to positive infinity diverges.
  • #1
Niles
1,866
0

Homework Statement


HI all.

In order to perform a Fourier transform on a function f(x), f(x) must be integrable, i.e.

[tex]
\int_{-\infty}^{\infty}|f(x)|dx < \infty.
[/tex]

Can you confirm that this also implies that f(x) -> 0 for x -> (+/-) infinity?
 
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  • #2
Can you find a function that has a finite area beneath it while not being zero in infinity?
 
  • #3
Cyosis said:
Can you find a function that has a finite area beneath it while not being zero in infinity?

No. But I am wondering if it is a sufficient conditions for the integral to be finite? E.g. f(x) = x-1?
 
  • #4
To avoid any confusion. Do you mean if f(x)->0 when x->+-infinity then the integral of f(x) over +- infinity converges? This is not true and you already gave a counter example. The function x^-1 goes to zero, but the integral of x^-1 from -infinity to infinity diverges.
 
  • #5
Great, thanks!
 

1. What is an integrable function in real analysis?

An integrable function in real analysis is a function that can be integrated over a certain interval to give a finite result. This means that the area under the curve of the function is finite and can be calculated using integration techniques. In other words, an integrable function is a function that satisfies the Riemann integrability condition.

2. How is the integrability of a function determined?

The integrability of a function is determined by checking if it satisfies the Riemann integrability condition. This condition states that for a function to be integrable, the upper and lower Riemann sums must converge to the same value as the partition of the interval approaches zero. This condition can be checked using various tests such as the Cauchy criterion or the Lebesgue's criterion.

3. What is the difference between a Riemann integrable function and a Lebesgue integrable function?

The main difference between a Riemann integrable function and a Lebesgue integrable function is the way the integration is defined. Riemann integration is based on the partitioning of the interval into subintervals and calculating the area under the curve using rectangles. On the other hand, Lebesgue integration is based on the measure of the set over which the function is defined. In general, Lebesgue integration is considered to be a more powerful and flexible method than Riemann integration.

4. Can a function be integrable but not continuous?

Yes, a function can be integrable but not continuous. The continuity of a function is not a necessary condition for its integrability. As long as the function satisfies the Riemann integrability condition, it can be integrated even if it has discontinuities. However, it is important to note that a continuous function is always integrable.

5. What are some applications of integrable functions in real analysis?

Integrable functions have various applications in real analysis, including calculating areas and volumes of irregular shapes, calculating work and energy in physics, solving differential equations, and approximating solutions to complex mathematical problems. They are also used in fields such as economics, engineering, and statistics to model and analyze real-world phenomena.

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