(05/25/2021, 07:32 PM)MphLee Wrote: Question 1: for which those spaces are in bijection?

Question 2: Do this bijection preserve some stucture? Idk... are those functions paces closed under piecewise sum, scalar multiplication, piecewise multiplication, do have a metric or topological structure (a system of open sets), a norm?

Question 3: take we have . What is the relationship between and o between and ?

Asap I'll go on the other sections.

Regards

Hey, Mphlee, I'll answer these questions to the best of my ability.

The people at U of T called it a hyper-operator chain; that's not my terminology. I know it can be a tad confusing for this forum; but that's what they call it

; so I stuck with the terminology.

You don't need to include zero; but go right ahead and include it. As these functions are presumed to be entire; the integral at zero is always defined. We are only worried about the behaviour as

with

to ensure the integral converges. As to what kind of arc; they can self intersect; they can loop; they can do what ever; so long as the initial point is

and the end point

and they are contained in

. Since these functions are holomorphic, and

is simply connected; the integral only depends on the initial point and the end point.

Yes, by correspondence I meant

is virtually the same as

; one takes derivatives, the other shifts the variable.

(a) and (b) are exactly as I intend to say it. So yes, your understanding of these seems correct.

I guess your questions after that are about how I order the theorems. I guess it's just personal preference. You can always feed Ramanujan into Euler; that can be done even more generally than how I do it. I'm restricting the cases where you can do this. Because it garners an isomorphic relationship.

1.)

I'm a little confused by your first question;

bijectively. And additionally,

for

; just as well with

. They are in bijection only for the same

; other wise its a different kind of map.

2.)

This is a good question, that has a pretty deep answer. First of all

implies that

and

then

; so this is a linear isomorphism. It's actually a linear isomorphism between hilbert spaces; but it's a little difficult to do this exactly. This would mean there is a norm; and there even is an inner product; but it's spurious to this paper. Id have to dust off my copy of Linear Operators on Hilbert Spaces to remind myself what exactly these are; can't recall off the top of my head.

Now,

; which happens for all

; and therefore if

then

. As to what happens when you apply the mapping to the product; you get a binomial convolution.

I didn't prove this in this paper; and this result is not mine. It's commonly known as the binomial theorem (I think?); you can find it in any text book on fractional calculus; it's usually one of the first things you prove. It's a little difficult; but in the best scenarios I can prove it pretty quickly because;

So if you can show

; they're equivalent by The Identity Theorem you get using Ramanujan's master theorem. This depends on how well

or

behave however. This convolution won't work generally for all

because

may not exist.

You can then, write this as a convolution,

Where sometimes this has the above representation; not always though. What you always get though; which again, isn't in the paper; is the indefinite sum representation.

This representation was more carefully studied in the indefinite sum paper on my ariv that's referenced in this paper. Though I use a slightly less direct isomorphism (forgive me, I wrote that paper a long time ago; but it still gets the job done).

Going in the other direction is more difficult. Recall that

implies that

as

. So this means, if

and

then

; which may or may not belong to an

depending on what

and

are. If they do belong to one then when you put it in the space

; then,

3.

As to the relationship between varying

and

; the best I have is that, the maximal sector in which

converges

, is the maximal set

it belongs to. And additionally; the maximal set

belongs in is

. And the maximal value

in which

is the maximal set

. I'm not sure what else you could be asking here..? Am I missing something?

Regards, James