Does the Klein Gordon Operator Preserve Compactness of Function Supports?

[naima]

Hi PF

I have an operator in the set of indefinitely derivable functions on R^4.
How can we caracterize the operators which send f with supp f compact to a function with compact support?
My example uses the Klein Gordon operator K an its green's function adv and ret.
let us take f such that $g_1 = \int ret. f$ and $g_2 = \int adv. f$
K sends g1 to f and ret sends K(g1) to g2
does g1 -> g2 conserves compactness of supports?
thanks

 

[jvicens]

Hello,

Thank you for your post. Characterizing operators that send functions with compact support to functions with compact support can be done by examining the properties of the operator and its action on the functions. In your example, the Klein Gordon operator K and its Green's functions adv and ret can be used to illustrate this concept.

First, let's define the Klein Gordon operator K as an operator that maps a function f to another function K(f). This operator is defined on the set of indefinitely derivable functions on R^4. In other words, K takes a function f and produces a new function K(f) that is also indefinitely derivable.

Next, let's consider the Green's functions adv and ret. These are functions that are used to solve differential equations, such as the Klein Gordon equation. In your example, you have defined g1 as the integral of ret multiplied by f, and g2 as the integral of adv multiplied by f. This means that g1 and g2 are functions that are obtained by solving the Klein Gordon equation with ret and adv as the Green's functions, respectively.

Now, let's examine the action of K on g1. Since K is an operator on the set of indefinitely derivable functions, it can act on g1 to produce a new function K(g1). This new function will also have compact support, as it is obtained by applying K to a function with compact support.

Similarly, the action of ret on K(g1) will produce a new function g2. Since ret is a Green's function, it is used to solve the Klein Gordon equation and will also have compact support. Therefore, g2 will also have compact support, and the operator g1->g2 conserves the compactness of supports.

In conclusion, operators that send functions with compact support to functions with compact support can be characterized by examining their properties and actions on functions. In this case, the Klein Gordon operator and its Green's functions illustrate this concept. I hope this helps to clarify your question. Please let me know if you have any further questions.