数理解析研究所講究録 1211 巻 2001 年 166-174

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A new proof of Carlson’s theorem by P1ana’ssummation formula
邦生
将範

吉野諏訪

(Kunio Yoshino)(Masanori Suwa)

Department of Mathematics, Sophia University
Abstract
In this paper we will prove Carlson’s theorem by using Plana’ssummation formula.
In \S 1, we will recall the definitions and transformations of analytic
functionals with unbounded carrier.
In \S 2, we will give aproof of Carlson’s theorem by Plana’s summation formula.

1The definitions of analytic functionals withunbounded carrier and their transformations.
Let$L$and$L_{\epsilon}$

be following strip regions:
$L$

For

For

$\epsilon>0$,$\epsilon’>0$$Q_{b}(L_{\epsilon} :$:=$$\epsilon$$>0$ ,

and$L_{\epsilon}$,$=$$(-\infty, a]+\iota[-b, b]$$=$$(-\infty, a+\epsilon]+\iota[-b-\epsilon, b+\epsilon]$.

, we put$k’\in \mathbb{R}$k’+\epsilon’)$$\{f(\zeta)\in H(L_{\epsilon})\cap C\circ(L_{\epsilon})$:$\sup_{\zeta\in L_{\epsilon}}|f(\zeta)e^{k’\xi+\epsilon’|\xi|}|<\infty$,$\zeta=\xi+\iota\eta\}$

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is the space of holomorphic functions defined on , (interior of L.).
$C(L.)$ is the space of continuous functions defined on. We put$7\mathrm{f}(L.)$$L_{g)}$$L_{e}$$Q(L : k’)= \lim_{\epsilonarrow\vec{0,\epsilon’}arrow 0}Q_{b}(L_{\epsilon} :k’+\epsilon’)$,means inductive limit. If, then the functionof belongs to $Q(L:k’)$ . We denote by $Q’(L : k’)$ the dual space of $Q(L : k’)$ .
where$z\in(-k’, \infty)+\iota \mathbb{R}$$\lim_{arrow}$$e^{\zeta z}$$\langle$

The elements of $Q’(L:k’)$ is called analytic functional with carrier andof type .
We define the Fourier-Borel transformof $T\in Q’(L : k’)$ as follows:$L$$\mathrm{s}$$k’$$\tilde{T}(z)$$\tilde{T}(z)=<T_{\zeta}$,$e^{\zeta z}>$.

is holomorphic function on the right half plane $(-k’, \infty)+i\mathbb{R}$ and satisfies following estimate :$\tilde{T}(z)$$\forall\epsilon>0$,$\epsilon’>0$,$\exists C_{\epsilon,\epsilon’}\geq 0$such that$|\tilde{T}(z)|\leq C_{\epsilon,\epsilon’}e^{ax+b|y|+\epsilon|z|}$,$({\rm Re} z\geq-k’+\epsilon’, z=x+\iota y)$.

(1): ) denotes the space of holomorphic functions definedon the right half plane $(-k’, \infty)+$-ill and satisfy the estimates (1). Following
theorem characterizes the Fourier-Borel transform of $Q’(L:k’)$ .
$\mathrm{E}\mathrm{x}\mathrm{p}((-k’, \infty)+i\mathbb{R}$$L$

Theorem 1.1 $([2],[9])$ . Fourier-Borel transfom is a linear topologictil is0morphism from $Q’(L:k’)$ onto.
$\mathrm{E}\mathrm{x}\mathrm{p}((-k’, \infty)+i\mathbb{R}:L)$

Definition 1.2.
follows :$\epsilon’$-Cauchy transform$\check{T}(w, \epsilon’)$$=$$\check{T}(w, \epsilon’)$$\frac{-1}{2\pi \mathrm{z}}\langle T_{\zeta}$,of $T\in Q’(L : k’)$ is defined as$\frac{e^{(-k’+\epsilon’)(\zeta-w)}}{\zeta-w}\rangle$.

Proposition 1.3 $([2],[9])$ . The following integral representation holds :$<T$ ,Put$\varphi>=\frac{1}{2\pi i}\int_{\partial L_{6}}\check{T}(\zeta, \epsilon’)\varphi(\zeta)d\zeta$$\varphi(\zeta)=e^{\zeta z}$,$(\forall\varphi\in Q_{b}(L_{\epsilon} :, then toe have.

$\tilde{T}(z)=\frac{1}{2\pi i}\int_{\partial L_{\epsilon}}\check{T}(\zeta, \epsilon’)e^{\zeta z}d\zeta$k’+\epsilon_{1}’), \epsilon’<\epsilon_{1}’)$.

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Conversely, we can express$\check{T}$by$\tilde{T}$in as follows :.

$\check{T}(w, \epsilon’)=\int_{-k’+\epsilon’}^{\infty}\tilde{T}(z)e^{-wz}dz$

Definition 1.4. For
define$S\in \mathrm{H}’(\mathrm{K})$, $K$ is acompact set and$T\in \mathrm{Q}’(\mathrm{L}:k’)$, we$\mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}*\mathrm{b}\mathrm{y}$$T*S=S*T=<T_{\zeta},$ $<S_{\tau}$ , $\varphi(\tau+\zeta)>>$ ,$\varphi\in Q’(L+K : k’)$.

Proposition 1.5. We have the following equalities:(i) $T*S\in Q’(L+K : k’)$ ,(ii) $(\overline{T*S})(z)=\tilde{T}(z)*\tilde{S}(z)$ .

In [15] we derived Plana’s summation formula for holomorphic functionsof exponential type by using the theory of analytic functional with unbounded carrier.
$\mathrm{s}$Proposition 1.6 (Plana’s summation formula [6], [13],[15]). Let $T\in$$Q’(L:k’)$ . If $k’>0$ , $0\leq b<2\pi$ , ${\rm Res}>a$ and $|{\rm Im} s|+b<2\pi$ , then we have$\sum_{n=0}^{\infty}\tilde{T}(n)e^{-\epsilon n}$$= \frac{1}{2}\tilde{T}(0)+\int_{0}^{\infty}\tilde{T}(x)e^{-\epsilon x}dx+\iota\int_{0}^{\infty}\frac{\tilde{T}(\iota x)e^{-lsx}-\tilde{T}(-\iota x)e^{\prime sx}}{e^{2\pi x}-1}dx$.

2Applications.
In this section we determine the form of holomorphic functions $f(z)$ of.exponential type with{0}.
$f(\mathrm{N})=$

Lemma 2.1. Suppose that

, $L=(-\infty,\alpha]+\dot{x}[-b, b]$ , $0<k’<$$1,0\leq b<2\pi$ . If
, $(n=0,1,2, \cdots)$ , then there eist holomorphic$a(s)$$b(s)$
andsuch thatfunctions$T\in \mathrm{Q}’(\mathrm{L}:\mathrm{k}’)$$\tilde{T}(n)=0$$\check{T}(s, \epsilon’)=a(s)+b(s)$,where $a(s)\in H(\{s\in \mathbb{C}:{\rm Im} s<2\pi-b\})$ , $b(s)\in H(\{s\in \mathbb{C} : {\rm Im} s>b-2\pi\})$and satisfy the following estimate :$|a(s)|$$\leq$$|b(s)|$$\leq$$C_{\epsilon’}e^{(k’-\epsilon’){\rm Res}}$,$C_{\epsilon’}e^{(k’-\epsilon’){\rm Res}}$.

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Corollary 2.2. Let $T\in Q’(L : k’)$ . If $\tilde{T}(n)=0(n=0,1,2, \cdots)$ , theiis holomorphic in $|{\rm Im} s|<2\pi-b$ , and.
$|\check{T}(s, \epsilon’)|\leq C_{\epsilon’}e^{(k’-\epsilon’){\rm Res}}$$\check{T}(s, \epsilon’)$

Proof of lemma 2.1.
$\tilde{T}(n)=0$0

By Plana’s summation formula and assumptioi, $(n=0,1,2, \cdots)$ , we have$=$$\int_{0}^{\infty}\tilde{T}(x)e^{-xs}dx+i\int_{0}^{\infty}\frac{\tilde{T}(ix)e^{-\iota xs}-\tilde{T}(-ix)e^{\iota xs}}{e^{2\pi x}-1}dx$$=$$\int_{0}^{\infty}\tilde{T}(x)e^{-xs}dx+i\int_{0}^{\infty}\frac{\tilde{T}(ix)e^{-\iota xs}}{e^{2\pi x}-1}dx-i\int_{0}^{\infty}\frac{\tilde{T}(-ix)e^{\iota xs}}{e^{2\pi x}-1}dx$$=$$\int_{0}^{\infty}\tilde{T}(x)e^{-xs}dx+\int_{0}^{l\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz$$- \int_{0}^{-\iota\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz-\int_{0}^{-\iota\infty}\tilde{T}(z)e^{-sz}dz$$=$$\int_{0}^{\infty}\tilde{T}(x)e^{-xs}dx+\int_{-k+\epsilon}^{0},,\tilde{T}(x)e^{-xs}dx+\int_{0}^{l\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi lz}-1}dz$$- \int_{0}^{-\iota\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz-\int_{0}^{-\iota\infty}\tilde{T}(z)e^{-sz}dz-\int_{-k+\epsilon}^{0},,\tilde{T}(x)e^{-xs}dx$$=$$\int_{-k’+\epsilon’}^{\infty}\tilde{T}(x)e^{-xs}dx+\int_{-k+\epsilon’}^{\iota\infty},\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz$$- \int_{-k’+\epsilon’}^{-\iota\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz-\int_{-k’+\epsilon’}^{-\iota\infty}\tilde{T}(z)e^{-sz}dz$

.

Hence
$\int_{-k’+\epsilon’}^{\infty}\tilde{T}(x)e^{-xs}dx$$=$$- \int_{-k+\epsilon’}^{l\infty},\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz+\int_{-k’+\epsilon’}^{-\iota\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz+\int_{-k’+\epsilon’}^{-\iota\infty}\tilde{T}(z)e^{-sz}dz$

Put,$a(s)$$=$$- \int_{-k+\epsilon’}^{\iota\infty},\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz$$b(s)$$=$$\int_{-k’+\epsilon’}^{-t\infty}\frac{\tilde{T}(z)e^{-sz}}{e^{-2\pi\iota z}-1}dz+\int_{-k’+\epsilon’}^{-\iota\infty}\tilde{T}(z)e^{-sz}dz$

.

.

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Since by proposition 1.3,

$T \vee(s, \epsilon’)=\int_{-k+\epsilon}^{\infty},,\tilde{T}(z)e^{-sz}dz$, we have$\check{T}(s, \epsilon’)=$$a(s)+b(s)$ . Then $a(s)$ and $b(s)$ satisfy the conditions in lemma 2.1.

1

Proposition 2.3. Suppose that $T\in Q’(L : k’),$ $L=(-\infty, a]+i[-b, b]\subset$, $0<k’<1.\tilde{T}(n)=0$ , $(n=0,1,2, \cdots)$ .
$\{\zeta\in \mathbb{C} :|{\rm Im}\zeta|<2\pi\}$(i) If $0\leq b<\pi$ , Then $T\equiv 0$ .
(ii) $Ifrr\leq b<2\pi$ , Then iAere exists an analytic functional with unbounded
carrier such that
$T=(\delta_{l\pi}-\delta_{-l\pi})*S$ , $S\in Q’((-\infty, a]+\iota[\pi-b, b-\pi]$ :), $where*is$convolution.
$S$

$k’$

Proof,$<T$,$\varphi>$

(i)
$=$

$\frac{1}{2\pi\iota}\int_{\partial L_{e}}\check{T}(\zeta, \epsilon’)\varphi(\zeta)d\zeta$,$(\varphi\in Q_{b}(L_{\epsilon} :. By lemma 2.1,k’+\epsilon_{1}’), \epsilon’<\epsilon_{1}’)$({$2\pi-b\})$ . Prom the assumption $0\leq b<\pi$ ,is contained in {$2\pi-b\}$ and. Hence $<T$, $\varphi>=0$ .

and$\check{T}(\zeta, \epsilon’)\in H(\mathbb{C}\backslash L)$$\check{T}(\zeta, \epsilon’)\in H$$\zeta\in \mathbb{C}$$L$:$\zeta\in \mathbb{C}$,$|{\rm Im}\zeta|<$:$|{\rm Im}\zeta|<$$|\check{T}(\zeta, \epsilon’)|\leq Ce^{(k’-\epsilon’)\zeta}$

In this case$\iota[-b, b-2\pi]\})$ . We define analytic functionals with unbounded carrier Si,
as follows :(ii)$\check{T}(\zeta, \epsilon’)\in H(\mathbb{C}\backslash \{(-\infty, a]+\iota[2\pi-b, b]\cup(-\infty, a]+$$<S_{1}$,$<S_{2}$,$\varphi>$$=$$\frac{1}{2\pi\iota}\int_{\partial L_{\acute{\epsilon}}}\check{T}(\zeta+\pi\iota)\varphi(\zeta)d\zeta$$\varphi>$$=$$\frac{1}{2\pi\iota}\int_{\partial L_{\acute{e}}}\check{T}(\zeta-\pi\iota)\varphi(\zeta)d\zeta$$S_{2}$,,), $L_{\epsilon}’=(-\infty, a+\epsilon]+\iota[\pi-b-$
where $\varphi\in Q_{b}((-\infty, a]+\iota[\pi-b, b-\pi]$ :, $b-\pi+\epsilon]$ . By lemma 2.1, Si, $S_{2}\in Q’((-\infty, a]+\iota[\pi-b, b-\pi]$ : ) and we. Prom the assumption $\tilde{T}(n)=0$ , $(n=0,1,2, \cdots)$ ,have$k’+\epsilon’$$k’$$\epsilon$$T=S_{1}*\delta_{\pi},+S_{2}*\delta_{-m}$0$=$$\tilde{T}(n)$$=$$\tilde{S}_{1}(n)e^{n\pi\iota}+\tilde{S}_{2}(n)e^{-n\pi l}$.

. By $b-\pi|$ $-(\pi-b)=2(b-\pi)<2\pi$ and (i) in
Therefore$S_{1}=-S_{2}$.
this proposition,$\tilde{S}_{1}(n)=-\tilde{S}_{2}(n)$171Corollary 2.4. Let $f(z)$ E
0, (n 0,1,2, \cdots ). Thenoo)$\mathrm{E}\mathrm{x}\mathrm{p}((-\mathrm{A}’,$$+\mathrm{z}\mathrm{R}\ovalbox{\tt\small REJECT}$L), $0<k’<1$ ,$f^{\ovalbox{\tt\small REJECT}}(n)\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$(i) $0\leq b<\pi\Rightarrow f(z)=0$ .
(ii) $\pi\leq b<2\pi\Rightarrow f(z)=\sin\pi z\tilde{S}(z)$ ,, $b-\pi]$ : $k’$ ).

uthere $S\in Q’((-\infty, a]+\iota[\pi-$$b$

Remark 2.5. Corollary 2.4(i) are called “Carlson’s theorem”.([3])

Corollary 2.6.
(i)$0\leq b<\pi\Rightarrow\{e^{n\zeta}\}_{n=0}^{\infty}$(ii) $\pi\leq b<2\pi--$,2.1

is total in $Q(L:k’)$ .

$\{e^{m\zeta}, \zeta e^{n\zeta}\}_{m}^{\infty}n=0=0$,

is total in $Q(L:k’)$ .

ExamplesCarlson’s theorem is used in several branches of mathematical physics. For
example,1. Auniqueness of analytic continuation of scattering amplitude to complex angular momentum plane ([7], [11],[12]).
2. Dyson Conjecture in statistical mechanics ([5], [8]).

3. Calculation of Selberg Integral $([1],[8])$ .
4. Auniqueness of Ramanujan Resummation ([4]).
For the history of Carlson’s theorem, we refer the reader to [14]. Here wehave an example of proposition 2.3 (ii) :Example 2.7 (Functional equation of Riemann zeta function, $[6],[13]$ ).
We define an analytic functional $T$ as follows :
$<T$,

,

$\varphi>=\frac{1}{2\pi\iota}\int_{\partial L_{\epsilon}}\frac{1}{2(e^{-e^{-\mathrm{z}^{\zeta}}}-1)1}\varphi(\zeta)d\zeta$$\varphi\in Q(L:k’)$

,(2)

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where $\yen^{-\mathrm{v}\mathrm{r}-\mathit{6},\tau T}+e$ ],t eA\langle and. Then we have$\mathit{6}>0$. In (2), we put$e^{\ovalbox{\tt\small REJECT}}$$\mathrm{t}(0\ovalbox{\tt\small REJECT}$$\ovalbox{\tt\small REJECT}$$\tilde{T}(z)$Remark that,$e^{\zeta z}>$$=$$<T_{\zeta}$$=$$- \frac{1}{2\pi\iota}\int_{(+\infty)}^{(+0)}\frac{(-t)^{-2z-1}}{e^{t}-1}dt$$=$$\frac{\zeta(-2z)}{\Gamma(2z+1)}$$\tilde{T}(n)=\frac{\zeta(-2n)}{\Gamma(2n+1)}=0$.

(3). On the other hand, we have$| \frac{1}{2\pi\iota}\int_{\partial L_{e}}\frac{e^{\zeta z}}{2(e^{-e^{-\}\zeta}}-1)}d\zeta|arrow 0$,$(\xiarrow-\infty, {\rm Re} z>0)$.

Therefore by residue theorem,$\frac{1}{2\pi\iota}\int_{\partial L_{e}}\frac{e^{\zeta z}}{2(e^{-e^{-:\zeta}}-1)}d\zeta$

$=$

$=$

$- \sum_{n=0}^{\infty}\frac{(2\pi n)^{-2z}e^{\pi lz}}{2\pi\iota n}+\sum_{n=0}^{\infty}\frac{(2\pi n)^{-2z}e^{-\pi\iota z}}{2\pi\iota n}$$- \frac{(2\pi)^{-2z}}{\pi}\sin\pi z\zeta(2z+1)$.

(4)

Therefore by (3) and (4), we have$\frac{\zeta(-2z)}{\Gamma(2z+1)}$$=$$- \frac{(2\pi)^{-2z}}{\pi}\sin\pi z\zeta(2z+1)$.

This means $\tilde{T}(z)=\sin\pi z\tilde{S}(z),\tilde{S}=-\frac{1}{\pi}(2\pi)^{-2z}\zeta(2z+1)$ . Namely, $T=$
, where $S=$ . Now we replace to. Then$(\delta_{\pi l}-\delta_{-m})*S$$z$$\frac{\dot{\zeta}(1-z)}{\Gamma(z)}$$\zeta(1-z)$$\frac{z-1}{2}$$=$$- \frac{(2\pi)^{-z+1}}{\pi}\sin(\frac{\pi}{2}z-\frac{\pi}{2})\zeta(z)$$=$$2(2 \pi)^{-z}\cos(\frac{\pi}{2}z)\zeta(z)$$=$$2(2 \pi)^{-z}\cos(\frac{\pi}{2}z)\Gamma(z)\zeta(z)$.

Since (5), we obtain “Functional equation of Riemann zeta function”

(5)

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References[1]

AomotO-Kita :Theory of Hypergeometrr.c Functions, Appendix, (Selberg Integral and BC type Hypergeometric Functions), (in Japanese),
Springer-Verlag.

[2]

C.A.Berenstein and R.Gay :Complex Analysis and Special Topics inHarmonic Analysis, Springer-Verlag (1991).

[3]

R.P.Boas :Entire Functions, Academic Press, (1954).

[4]

B.Candelpergher, M.A.Coppo and E.Delabaere :La sommation de Ramanujan, L’Enseignement Math\’ematique, (1997), 93-132.

[5]

F.Dyson :Statistical theory of the Energy level of complex system I,J.Math.Phy. Vo1.3, (1962), 140-175.

[6]

A.Erdely: Higher Trancsendental Funtions Vol.l, Mcgrawhill, (1953).

[7]

N.N.Khuri :Regge Pole, Power series, Physical Review, Vo1.132, N0.2,(1963), 914-926.

[8]

M.L.Mehta :Random Matr ices, Academic Press (1990).

[9]M.Morimoto :Analytic functional with non-compact car riers, Tokyo
J.Math. VoI.I, (1978), 77-103.

[10] M.Morimoto and K.Yoshino :A uniqueness theorem for holomorphic
functions of exponential type, Hokkaido Math.J. Vo1.7, (1978), 259-270.
[11] H.M.Nussentzweig :Causality and Dispersion Relation;. Academic Press(1972).
[12] T.Regge :Introduction to Complex Orbital Mom enta: Nuovo Cimento,
Vo1.14, (1959), 951-976.

[13] E.T.Whittaker and G.N.Watson :Moderm Analysis, Cambridge University Press, (1927).
[14] K.Yoshino: On Carlson’s theorem for holomorphic functions, Algebraic
Analysis (Edited by M.Kashiwara and T.Kawai), Vo1.2, Academic Press,(1988), 943-950

174[15] K.Yoshino and M.Suwa :Plana’s summation for mula for holomorphic
functions of exponential type, S\^uriken K\^oky\^uroku, N0.1158, (2000), 180-189.