Convolution of a function with itself

In summary, the conversation is discussing whether the operation of convolution of a function, denoted by '*', can be applied to an arbitrary number of times, denoted by 'n'. The speaker also mentions knowing the expression for n=2, which involves integrating the function f(t) with itself and a time shift, but is curious if this can be extended to any value of n.
  • #1
mhill
189
1
given a function f(t) could we define the operation

[tex] f*f*f*f*f*f*f*f**f*f*f*f*...*f [/tex] n times ?

here the operation '*' means convolution of a function if n=2 i know the expression

[tex] (f*f)= \int_{0}^{x}dt f(t)f(t-x) [/tex]

but i would like to see if this can be applied to arbitrary order , thanks.
 
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  • #2
mhill said:
given a function f(t) could we define the operation

[tex] f*f*f*f*f*f*f*f**f*f*f*f*...*f [/tex] n times ?

here the operation '*' means convolution of a function if n=2 i know the expression

[tex] (f*f)= \int_{0}^{x}dt f(t)f(t-x) [/tex]

but i would like to see if this can be applied to arbitrary order , thanks.

There is nothing in mathematics to prevent you from doing it.
 

Related to Convolution of a function with itself

1. What is the definition of convolution of a function with itself?

The convolution of a function with itself is a mathematical operation that combines two functions to create a third function that represents the amount of overlap between the original functions at each point.

2. How is convolution of a function with itself used in signal processing?

Convolution is commonly used in signal processing to filter signals, extract features, and perform other operations such as time shifting and frequency shifting.

3. What is the difference between convolution and cross-correlation?

Convolution and cross-correlation are similar operations, but they have different applications. Convolution is used to measure the overlap between two functions, while cross-correlation is used to measure the similarity between two signals by shifting one of them.

4. How is convolution of a function with itself related to the Fourier transform?

The convolution theorem states that convolution in the time domain is equivalent to multiplication in the frequency domain. This means that convolution of a function with itself can be calculated more efficiently using the Fourier transform.

5. Can convolution of a function with itself be applied to non-mathematical problems?

Yes, convolution can be applied to many real-world problems, such as image processing, pattern recognition, and natural language processing. It is a powerful tool for analyzing and manipulating data in various fields of science and technology.

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