Gottfried Wrote:bo198214 Wrote:Gottfried Wrote:... the goal to extend the operator hierarchy is not always driven by a functional representation ... there is a strong effort to extend the (binary) operator hierarchy solely based on the properties of the usual operators.

What means "driven by functional representation"?

Of course there is this area of the operator hierarchy and then there is this area of non-integer iteration. But these are not different approaches but the non-integer iteration is a way to get the operations demanded in the operator hierarchy.

From the operator hierarchy we have the law:

So we need an operation [4] (with real right operand) which satisfies these conditions. That means for each we need a function

(with certain smoothness) which satisfies:

.

We can obtain such a function by non-integer iteration of , (perhaps there are also other possibilities):

.

And we can perform non-integer iteration via those several approaches we discussed here (natural, diagonalization, regular). And those approaches then may have variants for which we need a series expansion or for which we dont need a series expansion.

So the really concurrent approaches are not operator-hierarchy and series-expansion. But our 3 different approaches to function iteration.

Operator hierarchy and series expansion are just parts of the whole area, where operator hierarchy is a framework into which you can plugin different approaches for non-integer iteration, including those performed by series expansion.