Functional analysis proof of Ramanujan's Master Theorem According to mathworld, Ramanujan's master theorem is the statement that if $$f(z) = \sum_{k=0}^{\infty} \frac{\phi(k) (-z)^k}{k!}$$ for some function (analytic or integrable) $\phi$, then $$\int_0^{\infty} x^{n-1} f(x) \, \mathrm{d}x = \Gamma(n) \phi(-n).$$
As written it is obviously false as the values of an (analytic or integrable) function $\phi$ at natural numbers do not determine its values anywhere else. However it turns out that $$\int_0^{\infty} x^{s-1} f(x) \, \mathrm{d}x = \Gamma(s) \phi(-s)$$ for arbitrary $s$ under growth conditions on $\phi$.
Recently I came across an elementary "proof": if $T$ denotes the shift operator $T\phi(s) := \phi(s+1),$ then we can write $$f(z) = \sum_{k=0}^{\infty} \frac{(-z)^kT^k \phi(0)}{k!} = e^{-zT}\phi (0)$$ such that $$\int_0^{\infty} x^{n-1} f(x) \, \mathrm{d}x = \int_0^{\infty} x^{n-1} e^{-xT} \phi(0) \, \mathrm{d}x = \Gamma(n) T^{-n}\phi(0) = \Gamma(n) \phi(-n),$$ by plugging $T$ into the Gamma integral $$\int_0^{\infty} x^{n-1} e^{-xs} \, \mathrm{d}x = \Gamma(n) s^{-n}.$$ I am curious whether this argument can be made rigorous with functional analysis on an appropriate function space (which necessarily would have to have some growth conditions).
 A: Write
$$
(M\Psi)(s)=\int^{\infty}_{0}\Psi(t)t^{s-1}dt,
$$
where $s\in A_{\Psi}$ with
$$
A_{\Psi}:=\left\{s\in\textbf{C}:\int^{\infty}_{0}\Psi(t)t^{s-1}dt<\infty\right\}.
$$
Then we have the next Mellin inversion formula
$$
\Psi(z)=\frac{1}{2\pi i}\int^{\sigma+i\infty}_{\sigma-i\infty}(M\Psi)(s)z^{-s}ds,
$$
where $\sigma\in Re\left(A_{\Psi}\right)$.
Theorem 1. If $\Psi(z)$ have power series arround $0$ with radious of convergence $r>0$ and if $x\in\textbf{R}$ such that
$$
\int^{\infty}_{0}\left|\Psi(t)\right|t^{x-1}dt<+\infty.
$$
Then the Mellin transform of $\Psi$ can be analyticaly continued to a meromorphic function in the halphplane $Re(z)<x$, with poles at the points $z=-m$, $m$ is non-negative integers such $m>-x$.
Theorem 2. Let $x>0$ and $f,\Psi$ analytic in $\textbf{C}$ and satisfining the condition
$$
|f(z)(M\Psi)(x+iz)|\leq C (1+|z|)^{\lambda}e^{-\delta |Re(z)|},\tag 1
$$
for all $z$ such that $Im(z)\geq 0$ and $c,\lambda,\delta>0$ constants, with the condition that $|z|=x+N+1/2$, where $N-$natural number saficiently large. Then the integral
$$
\int^{\infty}_{-\infty}f(t)M\Psi(x+it)dt
$$
converges absolutely, the series
$$
\sum^{\infty}_{m=0}\frac{\Psi^{(m)}(0)}{m!}f(i(x+m))
$$
converges in Abel sence and
$$
\int^{\infty}_{-\infty}f(t)(M\Psi)(x+it)dt=2\pi\lim_{r\rightarrow 1}\sum^{\infty}_{m=0}\frac{\Psi^{(m)}(0)}{m!}f(i(x+m))r^m.\tag 2
$$
Moreover if
$$
\left|\frac{\Psi^{(m)}(0)}{m!}f(i(x+m))\right|\leq \frac{C'}{m+1},
$$
then (2) converges.
As application of the above theorem we have
Theorem 3. Let $f$ be analytic in the upper half plane $Im(z)>0$ and continuous in $Im(z)\geq 0$ and such that
$$
|f(z)|\leq C (1+|z|)^{\rho}\left(\frac{|z|}{e}\right)^{Im(z)}e^{b|Re(z)|},
$$
in $Im(z)\geq 0$, where $0\leq b\leq \pi/2$. Then for $x>0$ we have
$$
\int^{\infty}_{-\infty}f(t)\Gamma(x+it)dt=2\pi\lim_{r\rightarrow1}\sum^{\infty}_{m=0}\frac{(-1)^m}{m!}f(i(x+m))r^m
$$
Theorem 3 is Ramanujan's Master Theorem.
I have full proofs of that in my PhD thesis but there are written in Greek (see Researchgate: Nikos Bagis).
A: The shift operator $e^{mt}$ directly multiplies the Laplace transformed coeficient function $\hat{a}(t)$.  Compare equations (D.4) and (L.3)  This shift property provides a proof.
$$ .. $$
Given:
\begin{align}
 I(m)=\int_{0}^{\infty} z^{m-1} \ f(z) \ dz \tag{G.1} \\ 
 f(z)= \sum_{s=0}^{\infty} \ \frac {\phi (s) \ (-z)^{s}}{s!} \tag{G.2}  \\
 Domain \ \phi (s)= \{ s\in R : s\geq -m \} \tag{G.3}  \\
 e^{-u}= \sum_{s=0}^{\infty} \ \frac{(-u)^{s}} {s!} \tag{G.4} \\
 \Gamma (m) = \int_{0}^{\infty} u^{m-1} \ e^{-u} \ du \tag{G.5}
\end{align}
Define:
\begin{align}
a(s)=\phi (s-m) \tag{D.1} \\
\hat{a}(t)= \int_{0}^{\infty}  e^{-ts} a(s) \ ds \tag{D.2} \\
\hat{a}(\lambda + i \omega) = \int_{0}^{\infty} e^{- i \omega s} e^{- \lambda s} a(s) \ ds \tag{D.3} \\
a(s)=  \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} e^{st} \ \hat{a}(t) \ dt \tag{D.4} \\
v=t; \ u= e^{t}*z \tag{D.5} \\
dv \wedge du = e^{t} \ dt \wedge dz \tag{D.6} 
\end{align}
Assumptions:
\begin{align}
\exists \lambda \ for \ \int_{0}^{\infty} e^{- i \omega} e^{- \lambda s} a(s) \ ds < \infty \tag{A.1}  \\
\exists \lambda \ for \ a(s)=  \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} e^{st} \ \hat{a}(t) \ dt \tag{A.2}  
\end{align}
Lemmas:
\begin{align}
a(s+m)= \phi(s) \tag{L.1}   \\
a(0)= \phi(-m) \tag{L.2}   \\
a(s+m)=  \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} e^{mt} e^{st} \ \hat{a}(t) \ dt 
\tag{L.3}   \\
a(0)=  \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} \hat{a}(t) \ dt \tag{L.4}
\end{align}
Lemma proofs: L.1: rewrite using D.1; L.2: rewrite using D.1; L.3: rewrite using D.4; L.4: rewrite using D.4.
$$ .. $$
Proof:
\begin{align}
I(m)= & \int_{0}^{\infty} z^{m-1} \ f(z) \ dz \tag{P.1} \\
& = \int_{0}^{\infty} z^{m-1} \ \sum_{s=0}^{\infty} \ \frac {\phi (s) \ (-z)^{s}}{s!} \ dz \tag{P.2} \\
& = \int_{0}^{\infty} z^{m-1} \ \sum_{s=0}^{\infty} \ \frac {a(s+m) \ (-z)^{s}}{s!} \ dz  \tag{P.3} \\
& = \int_{0}^{\infty} \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} z^{m-1}            \ \sum_{s=0}^{\infty} \ \frac {e^{mt} e^{st} \hat{a}(t) (-z)^{s}}{s!} \ dt \ dz \tag{P.4}  \\
& = \int_{0}^{\infty} \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} \ \hat{a} (t)  \sum_{s=0}^{\infty} \ \frac{ (e^{t}z)^{m-1} (- e^{t} z)^{s}}{s!} \ (e^{t} dt \ dz) \tag{P.5}  \\
& = \int_{0}^{\infty} \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} \ \hat{a}(v)  \sum_{s=0}^{\infty} \ \frac {(u)^{m-1} (- u)^{s}}{s!} \ dv \ du \tag{P.6}  \\
& = \left( \frac{1}{2 \pi i} \int_{\lambda - i \infty}^{\lambda + i \infty} \ \hat{a}(v) \ dv \right) \left( \int_{0}^{\infty} 
\sum_{s=0}^{\infty} \ \frac{ (u)^{m-1} (- u)^{s}}{s!} \ du \right) \tag{P.7}  \\
& = a(0) \left( \int_{0}^{\infty} \sum_{s=0}^{\infty} \ \frac{ (u)^{m-1} (- u)^{s}}{s!} \ du \right) \tag{P.8}  \\
& = \phi(-m) \left( \int_{0}^{\infty} \sum_{s=0}^{\infty} \ \frac{ (u)^{m-1} (- u)^{s}}{s!} \ du \right) \tag{P.9}  \\
& = \phi(-m) \left( \int_{0}^{\infty}  u^{m-1} e^{-u} \ du \right) \tag{P.10} \\
& = \phi(-m)\Gamma(m) \tag{P.11} 
\end{align}
Proof steps: P.2: rewrite using G.2; P.3: rewrite using L.1; P.4: rewrite using L.3; P.5: reorder the factors; P.6: rewrite using D.5 and D.6; P.7: reorder the factors; P.8: rewrite using L.4; P.9: rewrite using L.2; P.10: rewrite using G.4, P.11: rewrite using G.5.
$$..$$
Comments: For equation (A.1), $\lambda$ must be large enough to make the integral convergent.  For equation (A.2), $\lambda$ must be to the right of all singularities of $\hat{a}(t)$ in the complex plane.  If we interpret equation (D.3) as a  Fourier transform, compute the inverse Fourier transform, we obtain equation (D.4).  For detailed proof and discussion of equation (D.4) see: Bromwich integral (Inverse Laplace Transform) page 696 in MATHEMATICAL METHODS IN THE PHYSICAL SCIENCES Third Edition MARY L. BOAS.
