Can different weights help determine the difference between two functions?

[member 428835]

Hi PF!

Can any of you help me determine a good measure for how "different" two functions are from each other?

I've thought of using something like $\int_\Omega (f-g)^2 \, dx$. Can anyone recommend a good technique and direct me to the theory so I can understand it well?

Thanks so much!

Josh

 

[Orodruin]

Your proposal is an option and is a special case of using an inner product for the purpose. You can view the function space as a linear vector space and generally define an inner product by integrating the product of the functions, possibly together with a positive weight function. Your proposal would be equivalent to taking the inner product of the difference function with itself, much like you could determine the distance between two points in any vector space with an inner product.

 

[member 428835]

So does this difference function (with weight 1) seem like a good representation for distance, or is there a better method in your opinion?

 

[Orodruin]

This would depend on exactly what you are looking to do. For some problems, it is natural to use different weights.