Interestingly, the same story is repeated again in function. (Well, ``interesting'' is just my personal feeling. So, many might not agree with this. I found this --- the same story repeated again, but in the different level --- interesting in mathematics. Like Hitchhiker's Guide to the galaxy's jokes have some mathematical structure.) So far, we apply an ``operation'' to a scalar or a vector. Then, we again apply an operation to a function. We want to know what is the substance of the ``operation'' instead of the each result of operation. We could not know the substance at once, but we could know the response of the function with an operation.
Usually, a function is an operation to a scalar or a vector, therefore, it is a bit confusing to think about an operation on an operation. Let's see an example. Let's assume a function f and a scalar or a vector x, this function f can be an operation on x, we write this as y = f(x). Also assume a function g that can operate on f, y' = g(f(x)). This is the meaning of an operation on an operation. Here, interesting matter is what is g of f. We are not so interested in that which x is applied. OK, interesting is subjective matter, so I would like to explain a bit more. For example, assume a noise reduction function f, we want to improve a bit more with a function g. Then, what is the combined noise reduction filter g ・ f? This is usually an interesting question and usually a specific input signal x is not so interesting for a filter design. Because, we are usually interested in a filter that always works, instead of the response of a special input.
We could write down like this:
operator g ・ f = yIf this becomes
operator g ・ f' = λ f',we can see an aspect of this operator g. This f' is an eigenfunction. λ is still called an eigenvalue.