(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
| \le \theta)
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,
g(y)| \,dy < \infty)
; 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.
g(w) = \sum_{k=0}^\infty \binom{z}{k} f^{(k)}(w) \frac{d^{z-k}}{dw^{z-k}} g(w) = H(z)\\<br />
)
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;
 = \sum_{k=0}^n\binom{n}{k} f^{(k)}(w) g^{(n-k)}(w) = \frac{d^{n}}{dw^{n}} f(w)g(w) \\<br />
)
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,
g(w) = F * G\\<br />
)
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.
G(z-j)\\<br />
)
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
| \le C e^{(\pi/2 - \theta)|\Im(z)|})
as
 \to \pm \infty)
. So this means, if
and
then
G(z)| \le Me^{(\pi - \theta-\kappa)|\Im(z)|})
; 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,
 = \sum_{n=0}^\infty F(n)G(n) \frac{w^n}{n!}\\<br />
)
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
F(z)| \le Ce^{-\theta |Im(z)|})
is the maximal set
. I'm not sure what else you could be asking here..? Am I missing something?
Regards, James
).
go figure.
