# Convolution: Is There an Exception to Gaussian?

• indigojoker
In summary, the conversation discusses the concept of convolution and its relationship to the gaussian function. The participants question whether there are any exceptions to the rule that convolving a function with itself multiple times will ultimately result in a gaussian. They also discuss the idea of a function approaching a gaussian and how this can be measured. It is concluded that almost any function will not end up as a gaussian when convolved multiple times.
indigojoker
I just realized that the convolution of any function with itself many times will ultimately give a gaussian. I was just wondering if there was a function that was an exception to this?

I'm not sure I can make sense out of your question. Over what interval are you taking the convolution? What happens if f(x) is a constant or f(x)= x?

It is true that the convolution, over $-\infty$ to $\infty$, of two Gaussians is a Gaussian so this may be a "fixed point" theorem.

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Well, because the Fourier transform of a convolution is a product and because the Fourier transform of a Gaussian is a Gaussian, the convolution theorem means that rasing any well defined function to a high integer power gives something increasingly similar to a Gaussian. I was wondering if there was an actual counterexample to this

indigojoker said:
rasing any well defined function to a high integer power gives something increasingly similar to a Gaussian.

What makes you think this? In raising a function to a very high power, the parts with |f(x)|>1 get increasingly larger while those with |f(x)|<1 vanish. So, for example, if the function has several narrow peaks whose height is greater than one, its very high powers will have spikes where each of those peaks were and vanish everywhere else.

I don't think indigojoker meant "high power". I think he meant applying the convolution a large number of times.

yep, that's what i mean

Right, but then you noted that a convolution in, say, the time domain corresponds to a product in the frequency domain, so convolving a function with itself n times corresponds to raising its Fourier transform to the nth power. This is correct, but it doesn't follow that you'll always end up with gaussian like functions.

"This is correct, but it doesn't follow that you'll always end up with gaussian like functions."

so my original question was what types of functions will not end up with a gaussian?

I don't see why anything that isn't a guassian should "approach" a gaussian. And this is impossible to tell anyway unless you say exactly what you mean by "approach", ie, how do you measure how similar two functions are?

indigojoker said:
so my original question was what types of functions will not end up with a gaussian?

Think about what convolution means in the frequency domain, and it's clear that almost any function will not end up as a gaussian.

## 1. What is convolution and how does it relate to Gaussian functions?

Convolution is a mathematical operation that combines two functions to produce a third function. It is commonly used in signal processing and image processing. Gaussian functions are a type of continuous probability distribution that is often used in convolution because of their useful properties, such as being smooth and symmetric.

## 2. How is convolution used in real-world applications?

Convolution is used in a variety of fields, including engineering, physics, and statistics. It is commonly used in signal processing to filter noise and enhance signals, in image processing to blur or sharpen images, and in probability theory to model random processes.

## 3. Is there an exception to using Gaussian functions in convolution?

While Gaussian functions are commonly used in convolution, they are not always the best choice. In some cases, other functions may be more appropriate, such as the Laplace distribution for modeling heavy-tailed data or the Poisson distribution for modeling count data.

## 4. Can convolution be applied to discrete data?

Yes, convolution can be applied to both continuous and discrete data. In the case of discrete data, the convolution operation is simply a sum instead of an integral as with continuous data. This allows for the application of convolution to discrete signals and images.

## 5. Are there any limitations to using convolution in data analysis?

While convolution is a powerful tool in data analysis, it does have some limitations. One limitation is that it assumes that the two functions being convolved are independent. Additionally, convolution can be computationally expensive for large datasets, so alternative methods may be used in certain situations.

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