Use of Dirac delta to define an inverse

In summary, We are looking for properties of functions that satisfy the equation ∫dx f(y-x) g(x-z) = δ(y-z), where δ is the Dirac delta and g is the inverse function of f. This can be seen as the continuous limit of matrix multiplication by its inverse, where the identity matrix is represented by the δ function. We are also interested in finding the function g given f. Taking the Fourier transform of both sides of the equation, we can show that g(y)=∫dk (1/f(k)) eikx, up to a factor of 2π, where f(k) is the Fourier transform of f(x).
  • #1
lukluk
8
0
I was wondering which are the properties of functions defined in such a way that

∫dx f(y-x) g(x-z) = δ(y-z)

where δ is Dirac delta and therefore g is a kind of inverse function of f (I see this integral
as the continuous limit of the product of a matrix by its inverse, in which case the δ becomes the identity matrix). Does anyone know where this type of "inverse functions" are
discussed? or how to obtain the function g given f ?
 
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  • #2
update:

I found that taking the Fourier transform of both sides one can show that
g(y)=∫dk (1/f(k)) eikx
up to some factor of 2π, where f(k) is the Fourier transform of f(x).
 

1. What is the Dirac delta function and how is it used?

The Dirac delta function, also known as the unit impulse function, is a mathematical function that is zero everywhere except at a single point, where it is infinite. It is commonly used in physics and engineering to represent an impulse or spike in a system.

2. How is the Dirac delta function related to inverse functions?

The Dirac delta function can be used to define an inverse function for certain types of systems. This is because the Dirac delta function is the identity element for convolution, which is a mathematical operation used to define inverse functions.

3. Can the Dirac delta function be used to solve differential equations?

Yes, the Dirac delta function can be used in the solution of certain types of differential equations. This is because the Dirac delta function can be used to represent discontinuities or impulses in a system, making it useful for solving problems involving sudden changes or spikes.

4. How is the Dirac delta function used in signal processing?

In signal processing, the Dirac delta function is used to represent impulse signals. This is useful for analyzing and manipulating signals that have sudden changes or spikes, such as audio signals.

5. Are there any limitations to using the Dirac delta function to define an inverse?

Yes, there are some limitations to using the Dirac delta function to define an inverse. This method only works for certain types of systems and may not be applicable to all situations. Additionally, the use of the Dirac delta function can sometimes result in non-physical solutions or inaccuracies in the solution.

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