Function case

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:

operatorIf this becomesg ・ f = y

operatorwe can see an aspect of this operatorg ・ f' = λ f',

*g*. This

*f'*is an eigenfunction.

*λ*is still called an eigenvalue.

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