(05/25/2021, 07:32 PM)MphLee Wrote: Question 1: for which \( \theta \) 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 \( \theta<\kappa \) we have \( S_\theta\subseteq S_\kappa \). What is the relationship between \( {\mathbb S}_\theta \) and \( {\mathbb S}_\kappa \) o between \( {\mathbb E}_\theta \) and \( {\mathbb E}_\kappa \)?
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 \( w \to \infty \) with \( |\arg(w)| \le \theta \) 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 \( 0 \) and the end point \( \infty \) and they are contained in \( S_\theta \). Since these functions are holomorphic, and \( S_\theta \) is simply connected; the integral only depends on the initial point and the end point.
Yes, by correspondence I meant \( \mathbb{E}_\theta \) is virtually the same as \( \mathbb{S}_\theta \); 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; \( \mathbb{E}_\theta \leftrightarrow \mathbb{S}_\theta \) bijectively. And additionally, \( \mathbb{S}_\theta \subset \mathbb{S}_\kappa \) for \( \theta < \kappa \); just as well with \( \mathbb{E}_\theta \subset \mathbb{E}_\kappa \). They are in bijection only for the same \( \theta \); other wise its a different kind of map.
2.)
This is a good question, that has a pretty deep answer. First of all \( f,g \in \mathbb{S}_\theta \) implies that \( f+g \in \mathbb{S}_\theta \) and \( F , G \in \mathbb{E}_\theta \) then \( F+G \in \mathbb{E}_\theta \); 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, \( \int_\gamma |f(y)g(y)| \,dy < \infty \); which happens for all \( f,g \in \mathbb{S}_\theta \); and therefore if \( f,g \in \mathbb{S}_\theta \) then \( f \cdot g \in \mathbb{S}_\theta \). As to what happens when you apply the mapping to the product; you get a binomial convolution.
\(
\frac{d^{z}}{dw^{z}} f(w)g(w) = \sum_{k=0}^\infty \binom{z}{k} f^{(k)}(w) \frac{d^{z-k}}{dw^{z-k}} g(w) = H(z)\\
\)
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;
\(
H(n) = \sum_{k=0}^n\binom{n}{k} f^{(k)}(w) g^{(n-k)}(w) = \frac{d^{n}}{dw^{n}} f(w)g(w) \\
\)
So if you can show \( H \in \mathbb{E}_\theta \); they're equivalent by The Identity Theorem you get using Ramanujan's master theorem. This depends on how well \( g \) or \( f \) behave however. This convolution won't work generally for all \( f,g \) because \( \frac{d^{z-k}}{dw^{z-k}} g(w) \) may not exist.
You can then, write this as a convolution,
\(
\frac{d^{z}}{dw^{z}}|_{w=0} f(w)g(w) = F * G\\
\)
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.
\(
F * G = \sum_{j=0}^z \binom{z}{j}F(j)G(z-j)\\
\)
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 \( F \in \mathbb{E}_\theta \) implies that \( |F(z)| \le C e^{(\pi/2 - \theta)|\Im(z)|} \) as \( \Im(z) \to \pm \infty \). So this means, if \( F \in \mathbb{E}_\theta \) and \( G \in \mathb{E}_\kappa \) then \( |F(z)G(z)| \le Me^{(\pi - \theta-\kappa)|\Im(z)|} \); which may or may not belong to an \( \mathbb{E}_\tau \) depending on what \( \theta \) and \( \kappa \) are. If they do belong to one then when you put it in the space \( \mathbb{S}_\tau \); then,
\(
h(w) = \sum_{n=0}^\infty F(n)G(n) \frac{w^n}{n!}\\
\)
3.
As to the relationship between varying \( \theta \) and \( \kappa \); the best I have is that, the maximal sector in which \( f \) converges \( S_\theta \), is the maximal set \( \mathbb{S}_\theta \) it belongs to. And additionally; the maximal set \( F \) belongs in is \( \mathbb{E}_\theta \). And the maximal value \( \theta \) in which \( |\Gamma(-z)F(z)| \le Ce^{-\theta |Im(z)|} \) is the maximal set \( F \in \mathbb{E}_\theta \). I'm not sure what else you could be asking here..? Am I missing something?
Regards, James
