Is this Fourier transformation an eigenproblem?

[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]

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

 

[LagrangeEuler]

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?

 

[Office_Shredder]

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?