# Is this Fourier transformation an eigenproblem?

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• LagrangeEuler
In summary, the conversation discusses questions regarding Fourier transformation, particularly whether it can be called an operator and whether it can be considered symmetric. The answer is yes, it can be called an operator and it is an eigenproblem, but it is not clear what definition of symmetric is being used. The concept of a unitary operator is also brought up and the conversation suggests looking at its definition. The conversation then delves into a specific example where the argument of the function changes, causing confusion. The person asking the questions is looking for a reference or discussion on this topic.
LagrangeEuler
I have two questions regarding Fourier transformation. First of all is it ok to call Fourier transformation operator, or it should be distinct more? For instance, if I wrote
$$F[f(x)]=\lambda f(y)$$
is that eigenproblem, regardless of the different argument of function ##f##? Could I call ##F## unitary operator if the transformation is symmetric?

Office_Shredder said:
The answer is yes, it's an eigenproblem, and it's not clear what the definition of symmetric you are using is.

But it's probably worth looking at the definition of a unitary operator is
https://en.m.wikipedia.org/wiki/Unitary_operator
I asked it because the definition is
$$Af(x)=\lambda f(x)$$
same argument ##x##. And in my question is
$$Af(x)=\lambda f(y)$$. Is there some reference where this is discussed?

Sorry, I missed the change in argument. Can you give some more context to your example? Like, how is the difference between x and y significant here? Is the map f(y) goes to f(x) a trivial isomorphism between functions of y and functions of x?

## 1. What is a Fourier transformation?

A Fourier transformation is a mathematical operation that decomposes a function into its constituent frequencies. It is commonly used in signal processing and image analysis to analyze and manipulate data in the frequency domain.

## 2. What is an eigenproblem?

An eigenproblem is a mathematical problem that involves finding the eigenvalues and eigenvectors of a given matrix. Eigenvalues represent the scaling factor for the corresponding eigenvector, and are often used in applications such as principal component analysis and image compression.

## 3. How is a Fourier transformation related to an eigenproblem?

A Fourier transformation can be seen as an eigenproblem, where the function being transformed is the eigenvector and the transformed function is the eigenvalue. This is because the Fourier transformation is a special case of the more general eigenvalue problem, where the matrix is a continuous and infinite dimensional operator.

## 4. Why is it important to understand if a Fourier transformation is an eigenproblem?

Understanding the relationship between a Fourier transformation and an eigenproblem can provide insights into the properties of the Fourier transformation and its applications. It also allows for the use of techniques and algorithms from eigenvalue problems to solve problems involving Fourier transformations.

## 5. Are there any practical implications of the Fourier transformation being an eigenproblem?

Yes, there are practical implications of the Fourier transformation being an eigenproblem. For example, the eigenvalues and eigenvectors of the Fourier transformation can be used to efficiently compute the transformation or to approximate it using a smaller number of terms. This can be useful in applications where speed or memory usage is a concern.

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