# Property of the convolution product

• Jhenrique
In summary, it is known that the property of convolution holds for both direct and inverse Fourier transforms. This means that: \mathcal{F}\{f\ast g\}=\mathcal{F}\{f\}\mathcal{F}\{g\} and \mathcal{F}\{f g\}=\mathcal{F}\{f\}\ast\mathcal{F}\{g\}. Therefore, the same property should hold for the inverse transform, \mathcal{F}^{-1}, as well. This can be easily shown by applying \mathcal{F}^{-1} to each line of the property.
Jhenrique
It is known that:$$\mathcal{F}\{f\ast g\}=\mathcal{F}\{f\}\mathcal{F}\{g\}$$$$\mathcal{F}\{f g\}=\mathcal{F}\{f\}\mathcal\ast{F}\{g\}$$

But this property is valid for inverse tranform too?$$\mathcal{F}^{-1}\{F\ast G\}=\mathcal{F}^{-1}\{F\} \mathcal{F}^{-1}\{G\}$$$$\mathcal{F}^{-1}\{F G\}=\mathcal{F}^{-1}\{F\}\ast \mathcal{F}^{-1}\{G\}$$

Ben Niehoff said:

The only difference between ##\mathcal{F}## and ##\mathcal{F}^{-1}## is the sign in the exponential in the kernel, thus, appears there isn't reason for the property of the convolution not be valid for inverse transform, I think. Although I never saw this affirmation/property for inverse transform in anywhere.

You don't need to worry about the integral definitions. Just use this:

Jhenrique said:
It is known that:$$\mathcal{F}\{f\ast g\}=\mathcal{F}\{f\}\mathcal{F}\{g\}$$$$\mathcal{F}\{f g\}=\mathcal{F}\{f\}\ast\mathcal{F}\{g\}$$

and apply ##\mathcal F^{-1}## to each line.

1 person
Oh yeah! Thank you!

## 1. What is the property of the convolution product?

The property of the convolution product is a mathematical operation that combines two functions to produce a third function. It is often used in signal processing and statistics to analyze the relationship between two variables.

## 2. How is the convolution product calculated?

The convolution product is calculated by integrating the product of two functions over all possible values of one variable, while keeping the other variable fixed. This is typically represented by an asterisk (*), such as f*g.

## 3. What is the significance of the convolution product in science?

The convolution product is important in science because it allows us to analyze the relationship between two variables and understand how one variable affects the other. It is particularly useful in signal processing, where it can be used to filter out noise and extract relevant information from signals.

## 4. Can the convolution product be applied to non-numeric data?

Yes, the convolution product can be applied to any type of data, including non-numeric data. It can be used to combine any two functions or datasets, as long as they are defined over the same domain.

## 5. Are there any limitations to the convolution product?

One limitation of the convolution product is that it assumes the two functions being convolved are independent of each other. This may not always be the case in real-world scenarios, and can lead to inaccurate results. Additionally, the convolution product can become computationally intensive when dealing with large datasets.

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