Convolutions of Lebesgue integrable functions

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In summary, the conversation discusses the topic of convolutions and how they are defined for two integrable functions. The question arises about the integrability of the convolution and the text suggests that it can be shown by considering the Lebesgue integral. However, there is confusion about the switch from Lebesgue integration to Riemann integration and the text uses the notation $\int_R f(x) dx$ to represent the Lebesgue integral. The person is seeking clarification on this point.
  • #1
redbowlover
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Hello,

Looking through Torchinksy's Real Variables text and I'm infinitely confused about convolutions.

For two integrable functions [tex] f, g \in \ L^1(R) [/tex] we define the convolution [tex] f*g=\int_R f(x-y)g(y) dy , \forall x\in R [/tex].

Then, apparently, [tex]f*g[/tex] is also integrable. But I'm not sure how to prove this. The text says, for now suppose f, g and the integrand are nonnegative. Then [tex]||f*g||_1=\int_R\int_R f(x-y)g(y) dydx[/tex].

But why is this true? Clearly [tex]||f*g||_1=\int_R\int_R f(x-y)g(y) dydm[/tex], where m is the Lebesgue measure. But what allows you to switch to dx? Maybe this speaks to my greater misunderstanding of integrals. Sigh...If anyone could explain this to me I'd be very grateful.

I thought in general you could only switch from Lebesgue integration to Riemann integration when you knew the thing was Riemann integrable.
 
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  • #2
He writes $\int_R f(x) dx$ to mean the Lebesgue integral.
 

Related to Convolutions of Lebesgue integrable functions

1. What is a convolution of Lebesgue integrable functions?

A convolution of Lebesgue integrable functions is a mathematical operation that combines two functions to create a third function. It is defined as the integral of the product of the two functions, where one function is reflected and shifted over the other. This operation is commonly used in signal processing and probability theory.

2. How is a convolution calculated?

To calculate a convolution, you must first multiply the two functions together. Then, you reflect one of the functions and shift it over the other. Next, you integrate the product of the two functions over the entire domain. The result is a new function that represents the combined effect of the original functions.

3. What is the significance of Lebesgue integrable functions in convolutions?

Lebesgue integrable functions are used in convolutions because they provide a more general and flexible framework for integration compared to Riemann integrable functions. This allows for a wider range of functions to be convolved, making it a more powerful tool in mathematics.

4. Can convolutions of Lebesgue integrable functions be applied in real-world scenarios?

Yes, convolutions of Lebesgue integrable functions have numerous applications in real-world scenarios. For example, they are commonly used in signal processing to filter out noise and extract useful information from signals. They also have applications in probability theory, such as calculating the probability distribution of the sum of two random variables.

5. Are there any limitations to convolutions of Lebesgue integrable functions?

One limitation of convolutions of Lebesgue integrable functions is that they can be computationally expensive. This is because they involve integrating over the entire domain, which can be a time-consuming process. Additionally, the functions being convolved must both be Lebesgue integrable, which may restrict the types of functions that can be used in certain applications.

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