数理解析研究所講究録1080 巻 1999 年 83-99

83

CONTROLS OF THE OUTPUTSBY MEANS OF INPUTS
Saburou Saitoh
Departlnent of Mathematics, Faculty of Engineering,
Gunma University, Kiryu 376-8515, Japan-mail : ssaitoh@eg.gunma-u.ac jp$\mathrm{e}$$k7\mathbb{R}\text{三河}7(\xi*f_{\text{本}\chi}1)$

be a linear system of
Abstract: Let be a melnber of a Hilbert space
be the output of in the system. We assume that the outputs
andare functions on a same set $E$ . Then we consider the problems :, and etc by means of their
How to find the sum $f_{1}+f_{2}$ , the productinputs ?
The theory of reproducing kernels will give naturalin natural situations for these problems.
Surprising enough, for very
nonlinear system , we will be able todiscuss the similar problems.
$\mathcal{H}_{j},$$\mathrm{f}_{j}$$f_{j}$$\mathcal{H}_{j}$$S_{\mathrm{j}}$$f_{j}$$\mathrm{f}_{j}$$f_{1}f_{2}$$\mathrm{f}_{j}$’$\mathrm{a}’ \mathrm{n}\mathrm{s}\mathrm{W}\mathrm{e}\mathrm{r}\mathrm{s}$$\mathrm{g}\mathrm{e}\mathrm{n}\dot{\mathrm{e}}\mathrm{r}\mathrm{a}\acute{\mathrm{l}}$$S_{j}$AMS Subj. Classification:$30\mathrm{C}40,46\mathrm{E}32,44\mathrm{A}10,$ $.35\mathrm{A}22$.

Key Words: Hilbert space, reproducing kernel, linear transform, nonlinear
transform, convolution, norm inequality, integral equation, nonlinear differential equation, algebraic structure in Hilbert spaces.

1. A General Concept
Following Saitoh [1], we shall introduce a general theory for linear transforms
in the framework of Hilbert spaces.
-dimensional) space. Let $E$ be an abstract
be a Hilbert (possiblyLet
set andbe a Hilbert -valued function on $E$ . Then we shal,1 consider the
linear transform(1.1)$\mathcal{H}$$\mathrm{f}\mathrm{i}\dot{\mathrm{m}}\mathrm{t}\Leftrightarrow$$\mathrm{h}$$\mathcal{H}$$f(p)=(\mathrm{f},\mathrm{h}(p))_{\mathcal{H}},$$\mathrm{f}\in \mathcal{H}$into the linear spacecomprising all the complex valued function
on $E$ . In order to investigate the linear transform (1.1), we form a positive
on $E$ defined bymatrixfrom$\mathcal{H}$$\mathcal{F}(E)$$I\mathrm{i}’(p, q)$$K(p, q)=(\mathrm{h}(q), \mathrm{h}(p))_{\mathcal{H}}$on$E\cross E$

.

(1.2)

84
Then, we obtain the following:(I) The range in the linear

ducing kernel Hilbert spaceis characterized as the reproby
admitting the reproducing kernel $K(p, q)$ .
$\mathcal{H}$$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathfrak{m}(1.1)$$H_{h}\cdot(E)$(II) In general, we have the inequality.

$||f||_{H\kappa 1)}E\leq||\mathrm{f}||_{\mathcal{H}}$

Here, for a member$f$of $H_{K}(E)$ there exists a uniquely determined$\mathrm{f}^{*}\in \mathcal{H}$satisfying$f(p)=(\mathrm{f}\mathrm{x}, \mathrm{h}(P))\mathcal{H}$

on

$E$

and
$||f||_{H_{K}(E})=||\mathrm{f}^{*}||_{\mathcal{H}}$.

(III) In general, we have the inversion fornula in (1.1) in the form

(1.3)

$farrow \mathrm{f}^{*}$. However, thisin (II) by using the reproducing kernel Hilbert space
formllla is, in general, involved and delicate. We need, case by case, arguments.
is established.
In this paper, we assume that the inversion$H_{h}\cdot(E)$$\mathrm{f}_{\mathrm{o}\mathrm{r}\mathfrak{m}\mathrm{u}}1\mathrm{a}(1.3)$(IV) Conversely, we assume that an isometrical mapping from a reproducing
kemel Hilbert space $H_{K}(E)$ admitting the reproducing kernel $K(p, q)$ on $E$ ontoa Hilbert space . Then we have the representation (1.1) by$\tilde{L}$$\mathcal{H}$$\mathrm{h}(p):=\tilde{L}K(.,p)$Furthermore, $\{\mathrm{h}(p);p\in E\}$ is complete in$\mathcal{H}$.

.

Now we shall consider two systems$f_{j}(p)=(\mathrm{f}_{j}, \mathrm{h}_{j(}p))_{\mathcal{H}_{j}}$,$\mathrm{f}_{j}\in \mathcal{H}_{j}$(1.4). Here, we assume that $E$ is a samein the above way by using
and $f_{2}(p)$set for the two systems in order to have the output
on the same set $E$ .
), we can consider the operators
For$\{\mathcal{H}_{j}, E, \mathrm{h}_{j}\}2j=1$$\mathrm{f}_{\mathfrak{U}\mathrm{n}\mathrm{C}}\mathrm{t}\mathrm{i}_{\mathrm{o}\mathrm{n}\mathrm{S}}|f_{1}(p)$$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{n}$$\mathrm{P}^{\mathrm{l}\mathrm{e}}$$f_{1}(p)+f_{2}(p)$and$f_{1}(p)f_{2}(p)$in$\mathcal{F}(E)$. Then, we can consider the following problems: How to represent the85$\mathrm{s}\iota\ln)f_{1}(p)+f_{2}(p)$and the product $f_{1}(p)f_{2}(p)$ on $E$ in terms ofthei.r inputs$\mathrm{f}_{1}$and?
In this paper, we shall show that by using the theory of reproducing kernels
we can give natural answers for these problems. Furthermore, for some very
general nonlinear systems, we shall show that we can consider siInilar problems
and solutions.
After introducing several operators in
anded on the above idea,
we shall give typical and concrete operators, as examples.
$\mathrm{f}_{2}$$\mathcal{H}_{2}\mathrm{b}\mathrm{a}s$$\mathcal{H}_{1}$2. SumBy (I), $f_{1}\in H_{h_{1}’}(E)$ and $f_{2}\in H_{K_{2}}(E)$ , and we note that for the reproducing
kernel Hilbert space $H_{K_{1}K_{2}}+(E)$ admitting the reproducing kernel$K_{1}(p,q)+K_{2}(p, q)$$H_{h_{1}+h_{2}’}.\cdot(E)$on $E$ ,is composed of all functions$f(p)=f_{1}(p)+f_{2}(p)$ ;and its norm$||f||_{H\mathrm{t})}\kappa_{1}+K_{2}E$(2.1)$f_{j}\in H_{K_{j}}(E)$is given by(2.2)$||f||_{H\kappa_{\iota}}^{2}+ \kappa_{2}(E)=\min\{||f_{1}||_{H\kappa_{1}(E)}^{2}+||f_{2}||^{2}H_{K}\mathrm{t}E)2\}$

where the minimum is taken overgeneral, we have the inequality$f_{j}\in H_{K_{\mathrm{j}}}(E)$satisfying (2.1) for . Hence, in$f$$||f_{1}+f_{2}||^{2}H_{K}+\kappa_{2}1E)1\leq||f_{1}||^{2}H\kappa 1(E)+||f_{2}||^{2}H\kappa 2\{E$For the positive matrix$K_{1}+\mathrm{A}_{2}’$

on

$E$

$\mathcal{H}_{S}$(2.3), we assume the expression in the form

$K_{1}(p, q)+\mathrm{A}_{2}(p, q)=(\mathrm{h}_{S}(q), \mathrm{h}_{S}(p))\mathcal{H}s$

with a Hilbert space).

on $E\cross E$(2.4)-valued function on $E$ and further we assume that$\{\mathrm{h}s(p);p\in E\}$.

is complete in$\mathcal{H}_{S}.$(2.5)

Such a representation is, in general, possible (Saitoh [1], page 36 and see chapter
1. \S 5). Then, we can consider the linear mapping
onto$\mathrm{f}\mathrm{r}\mathrm{o}\mathfrak{m}\mathcal{H}_{S}$$f_{S}(p)=(\mathrm{f}s, \mathrm{h}_{S}(p))_{\mathcal{H}s},$$\mathrm{f}_{S}\in \mathcal{H}_{S}$$H_{h_{1}^{-}K_{2}}+(E)$(2.6)

and we obtain the isometrical identity$||f_{S}||_{H1)}K_{\mathrm{l}}+K2E=||\mathrm{f}_{S}||_{\mathcal{H}s}$

.

(2.7)

Hence, for such representations (2.4) with (2.5), we obtain the isometrical mappings among the Hilbert space.
$\mathcal{H}_{S}$86Now, for the$\mathrm{s}\iota \mathrm{l}\mathrm{n}1f_{1}(p)+f_{2}(p)$there exists a uniquely deterInined$\mathrm{f}_{\overline{\mathrm{b}}}.\in \mathcal{H}_{S}$satisfying$f_{1}(p)+f_{2}(p)=(\mathrm{f}s, \mathrm{h}s(p))_{\mathcal{H}_{S}}$Then,will be considered as a sum ofso, we shall introduce the notation$\mathrm{f}_{S}$$\mathrm{f}_{1}$and$\mathrm{f}s=\mathrm{f}1[+]\mathrm{f}_{2}$$\mathrm{f}_{2}$

on E.

(2.8)

through theseand$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}_{0}\mathrm{r}\mathrm{n}\tau \mathrm{S}$

.

(2.9)

This
for the members
is introduced through the threeandtransforms induced by.
andThe operator
is expressible in terms ofby the inversion
andformula(2.10)$\mathrm{f}_{1}\in \mathcal{H}_{1}$$\mathrm{s}\iota 1\mathfrak{m}$$\mathrm{f}_{2}\in \mathcal{H}_{2}$$\{\mathcal{H}_{j}, E, \mathrm{h}_{j}\}(j=1,2)$$\{\mathcal{H}_{s}, E, \mathrm{h}_{S}\}$$\mathrm{f}_{1}[+]\mathrm{f}_{2}$$\mathrm{f}_{2}$$\mathrm{f}_{1}$$(\mathrm{f}_{1}, \mathrm{h}_{1}(p))_{\mathcal{H}_{1}}+(\mathrm{f}_{2}, \mathrm{h}_{2}(p))\mathcal{H}_{2}arrow \mathrm{f}_{1}[+1\mathrm{f}_{2}$in the sense (I1) from

$H_{h_{1}+h_{2}}\cdot(E)$onto$\mathcal{H}_{S}$. Then,$\mathrm{f}\mathrm{r}\mathrm{o}\mathfrak{m}(\mathrm{I}\mathrm{I})$and (2.5) we have

Theorem 2.1. We have a triangle inequality$||\mathrm{f}_{1}[+]\mathrm{f}_{2}||_{\mathcal{H}_{S}}^{2}\leq||\mathrm{f}_{1}||_{\mathcal{H}_{1}}^{2}+||\mathrm{f}_{2}||2\mathcal{H}_{2}$.

(2.11). are
If $\{\mathrm{h}_{j}(p);p\in E\}$ are complete in $\mathcal{H}_{j}(j=1,2)$ , thenandisometrical. By using the isometrical mappings induced by Hilbert space valued
function $\mathrm{h}_{j}(j=1,2)$ and, we can introduce the sum space ofinandthe form$H_{h_{j}}$$\mathcal{H}_{j}$$\mathrm{h}_{S}$$\mathcal{H}_{1}$(2.12)$\mathcal{H}_{1}[+1\mathcal{H}_{2}$through the transforms.
For
if for$\mathrm{e}\mathrm{X}\mathrm{a}\mathrm{I}\mathrm{n}_{\mathrm{P}^{\mathrm{l}\mathrm{e}}}$.

$\mathrm{S}\mathrm{O}\mathfrak{m}\mathrm{e}$positive number$K_{1}<<\gamma^{2}\mathrm{A}_{2}’$that is,$\gamma I\iota_{2}-I\backslash \mathrm{z}\cdot- 1$$\mathcal{H}_{2}$$\gamma$on$E$(2.13)

is a positive matrix on $E$ , we have
$H_{K_{1}}(E)\subset H_{K_{2}}(E)$(2.14)

and$||f_{1}||_{H_{K_{2}}(E)}\leq\gamma||f_{1}||_{H_{K_{1}}E)}\mathrm{t}$for $f_{1}\in H_{K_{1}}(E)$(2.15)(Saitoh [1], page 37). Hence, in this case, we need not to introduce a Hilbert
space
and the linear mapping (2.6) in Theorem (2.1) and we can use the$\mathcal{H}_{S}$linear mapping$(\mathrm{f}_{2}, \mathrm{h}_{2}(p))_{\mathcal{H}_{2}}$instead of (2.6) in$\mathrm{T}\mathrm{h}\mathrm{e}\mathrm{o}\mathrm{r}\mathrm{e}\mathfrak{m}2.1$,$\mathrm{f}_{2}\in \mathcal{H}_{2}$.

3. Product87
is a positive matrix on $E$ and the reproducing kerThe productadmitting the reproducing kernel $K_{1}(p, q)Ii^{r_{2}}(P, q)$
nel Hilbert space.
functionsis composed of$K_{1}(p, q)\mathrm{A}^{r_{2}}(p, q)$$H_{K\iota h_{2}}\cdot(E)$$\sim$’$\mathrm{a}.11$$f(p)= \sum_{n=1}^{\infty}f_{\iota,n}:(p)f_{2}.n(p)*$(3.1)

on $E$ ;$f_{j,n}(p)\in H_{h_{j}}\cdot(E)(j=1,2)$

and the norn) in$H_{K_{1}h}\cdot 2(E)$is given by(3.2)$||f||_{H_{K_{1}\kappa_{2}}E}^{2}1)= \min\sum_{n=1}^{\infty}||f1,n||^{2}H_{K}1\mathrm{t}E)||f_{2.n}||_{H\langle)}.z\kappa_{2}E$

where the minimum is taken over all functions satisfying (3.1) for . In particular, (3.1) converges absolutely on $E$ . Especially we obtain the inequality
$f$

.

(3.3)

$E\mathrm{x}E$$-(3.4)$$||f_{1}f2||H_{K}K_{2}(\mathrm{t})E\leq||f_{1}||_{H}\kappa 1^{E)}1||f2||HK_{2}\mathrm{t}E)$As in the sum, we assume that the representationon$K_{1}(p, q)K_{1}(p, q)=(\mathrm{h}P(q),\mathrm{h}_{P}(p))_{\mathcal{H}_{P}}$with a Hilbert space$\mathcal{H}_{P}$-valued function on $E$ , and we assume that$\{\mathrm{h}_{P}(p);p\in E\}$is complete in$\mathcal{H}_{P}$

.

(3.5)

Then we consider the linear mapping(3.6)$f\rho(p)=(\mathrm{f}_{P}, \mathrm{h}_{P}\phi))_{\mathcal{H}}P’ \mathrm{f}_{P}\in \mathcal{H}_{P}$and we obtain the isometrical identity.

$||f_{P}||_{H_{K}}1\kappa 2(E)=$$||\mathrm{f}_{P}||_{H_{P}}$

(3.7)

.

Hence, for any product $f_{1}(p)f_{2}(p)$ there exists a umiquely determined fp

satisfying$f_{1}(p)f_{2}(p)=(\mathrm{f}_{P}, \mathrm{h}_{P}(p))\mathcal{H}_{P}$Then, fp will be considered as a product ofand so, we shall introduce the notation$\mathrm{f}_{1}$

and

(3.8)

on E.
$\mathrm{f}_{2}$through these transforms(3.9)$\mathrm{f}_{P}=\mathrm{f}_{1}[\cross 1\mathrm{f}2\cdot$

This product for the members
transforms induced by
is expressible in ternxs of$\in \mathcal{H}_{P}$is introduced through the t.hree. The operator
andby the inversion$\mathrm{f}_{j}\in \mathcal{H}_{j}(j=1,2)$$\{\mathcal{H}_{j}, E, \mathrm{h}_{j}\}(j=1,2)$$\mathrm{f}_{1}[\cross]\mathrm{r}_{2}$$\mathrm{f}_{1}$and$\{\mathcal{H}_{P}.

E, \mathrm{h}_{P}\}$$\mathrm{f}_{0\mathrm{r}\mathfrak{m}\mathrm{u}}1\mathrm{a}$$\mathrm{f}_{2}$$(\mathrm{f}_{1}, \mathrm{h}_{\mathrm{I}}(p))_{\mathcal{H}_{1}}(\mathrm{f}_{2}, \mathrm{h}_{2}(p))_{\mathcal{H}_{2}}arrow \mathrm{f}_{1}[\mathrm{x}]\mathrm{f}_{2}$(3.10)

88in the sense (III) from

$H_{h_{1}h_{2}}\cdot(E)$onto$\mathcal{H}_{P}$. Then, we obtain

Theorem 3.1. We have a Schwarz type inequalrty$||\mathrm{f}_{1}[\cross]\mathrm{f}2||_{\mathcal{H}_{P}}\leq||\mathrm{f}_{1}||_{\mathcal{H}_{1}}||\mathrm{f}_{2}||_{\mathcal{H}_{2}}$As in the$\mathrm{s}\mathrm{u}\mathfrak{m}$space$\mathcal{H}_{1}[+1\mathcal{H}_{2}$.

(3.11)

we can introduce the product space(3.12)$\mathcal{H}_{1}[\cross]\mathcal{H}_{2}$through the three transforms under the$\mathcal{H}_{j}(j=1,2)$)$\mathrm{c}\mathrm{o}\mathrm{n}$.

For example, if for a positive$\mathrm{s}\mathrm{u}\mathfrak{m}$as sulnptions of$\mathrm{h}_{j}$in$\gamma$on$K_{1}\mathrm{A}_{2}^{-}<<\gamma^{2}K_{1}$as in the$\mathrm{P}\mathrm{l}\mathrm{e}.\mathrm{t}\mathrm{e}\mathrm{n}\mathrm{e}\mathrm{S}\mathrm{s}$, we can consider the linear$(\mathrm{f}_{1}, \mathrm{h}_{1}(p))_{\mathcal{H}_{1}}$,$E$(3.13),$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\Gamma\ln$$\mathrm{f}_{1}\in \mathcal{H}_{1}$instead of (3.6).

In particular, in the setting in Section 1, we obtain
Corollary 3.1. If $K^{2}<<\gamma^{2}K$ on $E$ for a positive constant and $\{\mathrm{h}(p);p\in E\}$is complete in $H$ , thenis a commutative rrng $u’ rth$ the productthroughthe same three transforms. Furthermore
is a Banach
urith the product.
$\gamma$$\mathcal{H}$$\mathrm{f}[\cross]\mathrm{g}$$\iota f\gamma=1,$$\{\mathcal{H}, E, \mathrm{h}\}$$\mathcal{H}$$\dot{-}ng$

4. Differential
In the setting in Section 1, for simplicity we shall assume that $E$ is an interval
on t.he real line and the reproducing kernel Hilbert spaceis composed$E$of C. -class functions on . This smoothness reflects
the smoothness of the
reproducing kernel $K(p,q)$ that
$H_{h}\cdot(E)$$\iota$$\mathrm{t}\mathrm{o}_{\mathrm{i}}$$K_{1,1}(p, q) \backslash .\cdot=\frac{\partial^{2}\mathrm{A}’(p,q)}{\partial p\partial q}$

and$\mathrm{h}(p)$belongs to $C^{1}(E\cross E)$

is differentiable on $E$ in the space$f’(p)=(\mathrm{f},$$\mathcal{H}$$\frac{\partial \mathrm{h}(p)}{\partial p})$(4.1)

. Furthermore, we have

on

$E$

(4.2)

(Saitoh [1], pages 40-41). For the positive matrix (4.1) we have the expression

89$I_{1_{1.1}(q)}’p,=( \frac{\partial \mathrm{h}(q)}{\partial q},$$\frac{\partial \mathrm{h}(p)}{\partial p})$on $E\cross E$ .

(4.3)

.

(4.4)

Here, we atssume that$\{\frac{\partial \mathrm{h}(p)}{\partial p};p\in E\}$is complete in$\mathcal{H}$

Then. the derived function $f’(p)$ belongs to the reproducing kernel Hilbert space$H_{h_{1.\mathrm{l}}1F_{J})}$

admitting the reproducing kemel $K_{1,1}(p, q)$ and we have the isometrical

identity.

$||f’||_{K}1.1=||\mathrm{f}||_{\mathcal{H}}$(4.5)

However, for the positive matrix $K_{1,1}(p, q)$ if we use a representation$I\iota_{1,1}’(p, q)=(\mathrm{h}_{D}(q), \mathrm{h}_{D}(p))_{\mathcal{H}_{D}}$for a Hilbert space$\mathcal{H}_{D}$there exists a unique vector$\mathrm{f}_{D}\in \mathcal{H}_{D}$is complete in$\mathcal{H}_{D}$,satisfyingon E.

$f’(p)=(\mathrm{f}_{D}, \mathrm{h}_{D}(p))\mathcal{H}_{D}$$\mathrm{f}_{D}$(4.6)-valued function such that$\{\mathrm{h}_{D}(p);p\in E\}$Then,

on $E\cross E$(4.7)

will be considered as a derived vector through the tralusforIns induced
and. So, we shall writefron) $\{\mathcal{H}, E, \mathrm{h}_{D}\}$$\{\mathcal{H}_{D}, E, \mathrm{h}\}$(4.8)$\mathrm{f}_{D}=D\mathrm{f}$and the operator

is expressible in terms of$D\mathrm{f}$$\mathrm{f}$by the inversion formula(4.9)$\frac{\partial}{\partial p}(\mathrm{f}, \mathrm{h}(p))_{\mathcal{H}}arrow D\mathrm{f}$in the sense (III) from.

$H_{h_{11}\langle E)}$

onto$?i_{D}$. Then we have

Theorem 4.1. We have the inequality$||D\mathrm{f}||_{\mathcal{H}_{D}}\leq||\mathrm{f}||_{\mathcal{H}}$.

(4.10)As in the stlm and the product spaces, we can introduce the derived Hilbertspace(4.11)$D\mathcal{H}$through theis conlplete in$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\tau \mathrm{s}$$\mathcal{H}\}$$\prime \mathcal{H}$.

induced by$\{\mathcal{H}, E, \mathrm{h}\}$and$\{\mathcal{H}_{D}, E, \mathrm{h}_{l)}\}$

if$\{\mathrm{h}(p);p\in$90
5. Integral

In the setting in Section 4, we.
in the derived space$\mathrm{c}.\mathrm{a}\mathrm{I}1$

consider an integral of a Hilbert space$\mathcal{H}$as$D\mathcal{H}$$\backslash \mathrm{t}’’ \mathrm{e}$assume that$K^{1,1}(p.q):= \int_{p_{0}}^{\rho}\int_{l0}^{q}‘ K(\tilde{p}, q)\sim d\tilde{p}dq\sim$

on $E\cross E$(5.1)

exists, and $K^{1,1}(p.q)$ is expressible in the form
$K^{1,1}(p, q):=(\mathrm{h}_{t}(q), \mathrm{h}_{I}(p))_{\mathcal{H}}$for a Hilbert space$\mathcal{H}$;-valued function

, on

(5.2)

on $E$ such that$\mathrm{h}_{T}$is complete in$\{\mathrm{h}r(p);p\in E\}$$E\cross E$$\mathcal{H}_{T}$.

Then, as in the derived vector we can introduce the integrated vector(5.3)$\int_{p_{0}}^{p}\mathrm{f}$and the space$\int_{p_{0}}^{p}\mathcal{H}$The vector (5.3) is expressible in ternus of

.

(5.4)
$\mathrm{f}$by the inversion foumula$\int_{p_{0}}p(\mathrm{f}, \mathrm{h}(p))\mathcal{H}dparrow\int_{\rho 0}^{\rho}\mathrm{f}$in the sense (III) from

$H_{h^{1.1}}\cdot(E)$onto$\mathcal{H}_{T}$(5.5). Then we have

Theorem 5.1. We have the inequality$|| \int_{p_{0}}^{\rho}\mathrm{f}||\mathcal{H},$$\leq||f||_{\mathcal{H}}$

.

(5.6)

6. Integral of Hilbert spaces

In the setting in Section 2, we shall consider systems with a continuous paramas follows:eter on an index set
$t$

$T$

{

$\mathcal{H}_{\iota}$,E.

$\mathrm{h}_{t}$}, $t\in T$.

We assume that the associated reproducing kernel

(6.1)

91

is integrable on

$T$

(6.2)

on $E\cross E$ .

$K_{\mathrm{t}}(p,q)=(\mathrm{h}_{\mathrm{t}}(q), \mathrm{h}p\langle p))_{\mathcal{H}_{t}}$with respect to a -finite positive$\sigma$$\mathrm{A}_{T}’(p, q)=\int_{T}K_{t}(p, q)d\sigma(t)$$\mathrm{m}\mathrm{e}\mathrm{a}\mathrm{e}\iota \mathrm{r}\mathrm{e}$da on $T$ andon $E\cross E$ .

(6.3)

As a generalization of the sum of reproducing kernels, note that the reproducis composed of all functions $f(p)$ which are
ing kernel Hilbert
expressible in the form$\mathrm{s}\mathrm{p}\mathrm{a}\mathrm{C}\mathrm{e}H_{h_{T}^{-}}(E)$$f(p, t)\in H_{h_{C}}\cdot(E)$(6.4)$||f||_{H\{}^{2} \kappa_{T}E)=\min\int_{T}||f(p, t)||2d\sigma(H_{K}Et\tau^{()})$(6.5)$f(p)= \int_{T}f(p, t)d\sigma(t)$and the norm$||f||_{H\kappa}\tau(E)$is given by

is takenwhere theWe shall assume the expression$\mathrm{o}\mathrm{v}\mathrm{e}\mathrm{r}\mathrm{d}1\backslash _{l}$$\mathrm{m}\mathrm{i}\mathrm{n}\mathrm{i}\mathfrak{m}\mathrm{u}\mathrm{I}\mathrm{n}$,the expressions (6.4) for$I\mathrm{i}’\tau(p, q)=(\mathrm{h}\tau(q), \mathrm{h}_{T(}p))\mathcal{H}\tau$$\mathrm{h}_{T}$.

on $E\cross E$

on $E$ and-valued function
by a Hilbert space. Then, we can introduce the integral ofin$\mathcal{H}_{T}$$f$$\{\mathrm{h}_{T}(p);p\in E\}$

(6.6)

is complete$\mathrm{f}_{\ell}$$\mathcal{H}_{T}$(6.7)$\int_{T}\mathrm{f}_{t}$

and
(6.8)

$\int_{T}\mathcal{H}_{\iota}$

similarly. The integral (6.7) is expressible in terms of$\mathrm{f}_{l}$$\int_{T}(\mathrm{f}t, \mathrm{h}_{\ell}(p))_{\mathcal{H}c}d\sigma(t)arrow\int_{T}\mathrm{f}_{t}$from$H_{h_{T}}\cdot(E)$onto$\mathcal{H}_{T}$by the inversion formula(6.9). Then we have

Theorem 6.1. We have the inequality$|| \int_{T}\mathrm{f}p||_{\mathcal{H}_{T}}^{2}\leq\int_{T}||\mathrm{f}_{t}||_{\mathcal{H}_{T}}^{2}d\sigma(t)$.

(6.10)As shown in Appendix 1 in Saitoh [1], for very general nonlinear trallsforms ofa reproducing kernel Hilbert space, their ranges belong to naturally determined
reproducing kernel Hilbert spaces and norm inequalities hold in the nonlinear

92
transfornls. Therefore, for very general nonlinear
we can obtain
results as in the linear transforms. In order to reduce this paper ina reasonable size. we shall present their exact formulations and applications inpapers. In thepart, we shall give a concrete
as a prototype
result.
Our background idea comes
and for our conthe idea ofcrete applications we can give various solution of integral equations. In fact, theproduct
will be regarded as a convolution of. As a first step
paper in our new concept, in the sequel we shall present typical concrete examples in the general and abstract operators. It seems that to exalnine concrete)
operators in various concrete transforms is to rich our.
$\mathrm{t}_{\Gamma \mathrm{a}\mathrm{n}}\mathrm{S}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{l}\mathrm{n}\mathrm{S}$$\sin\dot{\mathrm{u}}\mathrm{l}\mathrm{a}\mathrm{l}$$\mathrm{a}\mathrm{I}\mathrm{l}\mathrm{o}\mathrm{t}\mathrm{l}\mathrm{l}\mathrm{e}\mathrm{r}$$1\mathrm{a}s\mathrm{t}$$\mathrm{e}\mathrm{x}\mathrm{a}\mathrm{n}\mathrm{l}\mathrm{p}\mathrm{l}\mathrm{e}$$\mathrm{f}\mathrm{r}\mathrm{o}\mathfrak{m}$$” \mathrm{c}\mathrm{o}\mathrm{n}\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}^{r}$$\mathrm{f}_{1}[\mathrm{x}]\mathrm{f}_{2}$$\mathrm{f}_{1}\mathrm{a}\mathrm{l}\tau \mathrm{d}\mathrm{f}_{2}$$\mathrm{m}\mathrm{a}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{n}$$\mathrm{a}\mathrm{t}\mathrm{i}_{\mathrm{C}_{\iota}\mathrm{s}}$7. Examples of Operators
7.1. We shall consider two linear transforms

where$\rho_{j}\mathrm{a}\mathrm{l}\cdot \mathrm{e}$positive continuous functions on$T$(7.1.1)$p\in E$$f_{j}(p)= \int_{T}F_{j}(t)\overline{h(\mathrm{f},p)}\rho j(t)dm(t),$,$\int_{T}|h(t,p)|^{2}\rho_{j(t})dm(t)<\infty$

on $p\in E$(7.1.2).

(7.1.3)and$\int_{T}|F_{j}(t)|2\rho j(t)dm(t)<\infty$

We aesunle that $\{h(t,p);P\in E\}$ is complete in the spaces satisfying (7.1.3).
Then, we consider the associated reproducing kemel on $E$$\mathrm{A}_{j}’(p,q)=\int_{T}h(t,p)\overline{h(t,p)}\rho j(t)dm(\mathrm{f})$and. for$\mathrm{e}\mathrm{X}\mathrm{a}\mathrm{I}\mathrm{n}_{\mathrm{P}^{\mathrm{l}\mathrm{e}}}$we consider the expression$K_{1}(p.q)+ \mathrm{A}_{2}’(p,q)=\int_{T}h(t,p)\overline{h(t,p)}(\rho 1(t)+\rho_{2}(t))dm(t)$.

(7.1.4)

So. we can consider the linear transform$f(p)=I_{\tau}^{F}(t)\overline{h(t,p)}(\rho 1(t)+\rho 2(t))dm(f)$for$\mathrm{f}\backslash _{1\mathrm{n}\mathrm{c}}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{S}F$

satisfying$\int_{T}|F(t)|2(\rho_{1}(t)+\rho 2(\iota))dm(t)<\infty$.

(7.1.5)

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Hence, thro$\iota \mathrm{g}\mathrm{h}$the three transforms (7...1.1) and (7.1.5) we have the sum$(F_{\iota}[+1^{p}2)(t)= \frac{F_{1}(t)\rho_{1}(t)+F2(t)\rho 2(t)}{\rho_{1}(t)+\rho_{2}(t)}$.

(7.1.6)7.2. We shall consider two linear transforms(7.2.1)$f_{\mathrm{j}}(x)= \int_{-\infty}^{\infty}e^{ix}F\ell(jt)\rho j(t)dt$

for positive$L_{1}$$(-\infty, \infty)$

integrable functions$\rho_{\mathrm{j}}$and for funtions$\int_{-\infty}^{\infty}|F_{j}(t)|2\rho j(t)dt<\infty$.

$F_{j}$

satisfying(7.2.2)

We consider the associated reproducing kernels$K_{j}(x,y)= \int_{-\infty}^{\infty}e^{i\mathrm{t}}t-iy\ell_{\rho}ej(t)dt$

and, for example we have the expression$K_{1}(x, y)K2(x,y)= \int_{-\infty}^{\infty}ee-xiyt(i\ell\rho_{1}*\rho 2)(t)dt$(7..2.3)

and the induced linear transform(7.2.4)$f(x.)= \int_{-\infty}^{\infty}F(t)e(ix\ell\rho_{\iota}*\rho_{2})(t)dt$

for functions $F$ satisfying$\int_{-\infty}^{\infty}|F(t)\mathfrak{j}^{2}(\rho\iota*\rho_{2})(t)dt<\infty$$1_{\mathrm{A}}\prime \mathrm{f}\mathrm{e}\mathrm{a}\mathrm{n}\mathrm{W}\mathrm{h}\mathrm{i}\mathrm{l}\mathrm{e}$.

(7.2.5), we have the expression from (7.2.1)$f_{1}(x)f_{2}(X)= \int_{-\infty}^{\infty}e^{ixt}(F_{\iota}\rho 1)*(F_{2}\rho_{2})(t)dt$

We thtls have the product$F_{\iota}[\cross]F2$.

(7.2.6)through the transforms (7.2.1) { $j=1,2)$ and(7.2.4)$(F_{1}[ \cross 1^{F}2)(t)=\frac{(F_{1}\rho_{1})(F_{2}\rho_{2})(t)}{(\rho_{1}*\rho_{2})(t)}$.

In particular, we obtain the inequality$\int_{-\infty}^{\infty}\frac{|\int_{-\infty}^{\infty}F_{1}(\xi)\rho_{1}(\xi)F2(t-\epsilon)\rho_{2}(t-\epsilon)d\epsilon|2}{\int_{-\infty^{\rho_{\mathrm{t}}}}^{\infty}(\xi)\rho_{2}(t-\xi)d\epsilon}dt$(7.2.7)

94$\leq\int_{-\infty}^{\infty}|F_{1}(t)|^{2}\rho 1(t)dt\int_{-\infty}^{\infty}1^{F_{2}}(t)|2\rho 2(t)dt$

7.3. Let$\mathrm{A}_{j}’(z,\overline{u})$.

(7.2.8)be two reproducing kernels on $\{|z|<r_{j}\}$ defined by the

expansions$h_{j}^{-}(Z, \overline{u})=\sum_{n=0}C^{\mathrm{t}}j)Z^{nn}\infty n\overline{u}$Then, the reproducing kernel Hilbert spaces
funtions $f_{j}(z)$ defined by$f_{j}(z)= \sum^{\infty}azn=0(jn)n$,$(C_{n}^{1j})>0)$$H_{K_{j}}$.

(7.3.1)are composed of all analytic

on $\{|z|<r_{j}\}$(7.3.2)

with finite norms$||f_{j}||_{H_{K_{j}}}^{2}= \sum_{n=0}^{\infty}\frac{|a_{n}^{\mathrm{t}j)}|^{2}}{C_{\eta}^{(j)}}$(7.3.3),

as we see easily, respectively. We have the expressions”$K_{1}(_{\sim}$$\overline{u})\mathrm{A}_{2}-(z,\overline{u})=\sum_{=n0}\infty(_{\nu+\mu=n}\sum C11)C^{(2})\mathrm{I}\nu\mu z^{n}\overline{u}^{n}$,(7.3.4)and$f_{1}(Z)f2(Z)= \sum_{=n0}^{\infty}(_{\nu+\mu=}\sum_{n}o_{\nu}O\mathrm{t}1)\mathrm{t}2)\mu)z^{n}$Hence, we have the sum$\mathrm{a}^{\langle 1)}\iota+1^{\mathrm{a}^{(2}}$).

$(\mathrm{a}^{(j)}=(a_{0}^{(j)}, a_{1}^{\langle j\}}, \cdots))$(7.3.5)satisfying (7.3.3)

for $j=1,2$(7.3.6)$\{\sum_{\mu\nu+=n}aa\nu\mu 11)\mathrm{t}2)\}_{n=0}^{\infty}$

through the two transforms (7.3.2) satisfying (7.3.3) and the$f(z)= \sum_{\mathrm{n}}\infty=0a_{n^{Z}}n$satisfying$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\mathfrak{m}$(7.3.7)95.

(7.3.8)$\sum_{n=0}^{\infty}\frac{|a_{n}|^{2}}{\sum_{\nu+\mu}\mu=nC_{\nu}^{\mathrm{t}1)(}c2)}<\infty$

In particular, we have the inequality$\sum_{n=0}^{\infty}\frac{|\sum_{\nu+\mu=n}a\nu a|^{2}(1)\langle 2)\mu}{\sum_{\nu+\mu=n}c_{\nu}^{(}1)C^{(2)}\mu}<(\sum_{n=0}^{\infty}\frac{|a_{n}^{11})|^{2}}{C_{n}^{(1)}}\mathrm{I}(\mathrm{n}\sum_{0=}^{\infty}\frac{|a_{n}^{(2\mathrm{I}}|^{2}}{C_{n}^{(2)}})$.

(7.3.9)7.4. Recall that
$K(x,y)= \frac{1}{2}e-|x-y|=\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{ix\iota_{e}-i}y^{\ell}}{t^{2}+1}dt$(7.4.1). comprising all absolutelyis the reproducing kemel for the Sobolov spacewith finite norms
continuous function $f(x)$ on$H_{h}$$(-\infty, \infty)$$||f||^{2}H_{K}= \int_{-\infty}^{\infty}(|f(x)|^{2}+|f’(X)|2)dx$.

(7.4.2)

Then,$K(x, y)^{2}= \frac{1}{4}e^{-2|x-y|}$$= \frac{1}{2}\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{e^{i2x\iota i2}e^{-}y^{p}}{t^{2}+1}dt$(7.4.3).

$= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{eeix\mathrm{t}-iyt}{t^{2}+4}dt$

Hence,$K(x, y)^{2}<<K(x, y)$ on $(-\infty, \infty)$ .

We consider the linear$\mathrm{t}_{\mathrm{f}\mathrm{a}\mathrm{n}}\mathrm{S}\mathrm{f}\mathrm{o}\mathrm{r}\mathrm{n}\mathrm{u}$induced from (7.4.1)$f_{j}(x)= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{F_{j}(t)ei\mathrm{r}l}{t^{2}+1}dt$

for$\mathrm{f}_{\mathrm{l}\mathrm{m}\mathrm{C}}\mathrm{t}\mathrm{i}_{0}\mathrm{n}\mathrm{s}F_{j}$(7.4.5)satisfying.

$\frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{|F_{j}(t)|^{2}}{t^{2}+1}dt<\infty$

Then, we have

(7.4.4)(7.4.6)96$f_{1}(x)f_{2}(x)= \frac{1}{4\pi^{2}}\int_{-\infty}^{\infty}\frac{1}{t^{2}+1}((\frac{F_{1}(t))}{t^{2}+1})*(\frac{F_{2}(t))}{t^{2}+1}))(t)(t^{2}+1)e^{ix\iota_{dt}}$.

$= \frac{1}{2\pi}\int_{-\infty}^{\infty}\frac{F(t)e^{ix}\mathrm{t}}{t^{2}+1}dt$(7.4.7)

Hence, through the same three transforms (7.4.5) we have the product$(F_{\mathrm{l}}[ \cross 1F2)(t)=\frac{1}{2\pi}\{(\frac{F_{1}(t))}{t^{2}+1})*(\frac{F_{2}(t))}{t^{2}+1})\}(t)(t^{2}+1)$$\mathrm{B}_{\sim}\mathrm{v}$.

(7.4.8)Corollary 3.1, the space (7.4.6) is a Banach ring under the product (7.4.8).

7.5. For any separable Hilbert space $H$ and its complete orthonormal system$\{\mathrm{f}_{n}\}_{rl=}^{\infty}\mathrm{l}$’we shall consider the linear transform from $H$ onto$a_{n}=(\mathrm{f}, \mathrm{f}_{n})_{H}$Then, the reproducing kernel for$l^{2}$.

(7.5.1)is Kronecker’s$\delta_{nm}^{2}<<\delta_{nm}$$l^{2}$$\delta_{nm}$, and of course.

(7.5.2)Hence for$\mathrm{f}=\sum_{n=1}^{\infty}a_{n}\mathrm{f}n\in H$

and$\mathrm{g}=\sum_{n=1}^{\infty}b_{n}\mathrm{f}n\in H$

through the three transforms (7.5.1), we have the product$\mathrm{f}[\mathrm{x}]\mathrm{g}=\sum_{n=1}^{\infty}a_{nn}b\mathrm{f}_{n}$.

(7.5.3)

Under this product, $H$ is a Banach ring.

7.6. For$q> \frac{1}{2}$,$I \mathfrak{i}_{q}(Z,\overline{u})=,\frac{\Gamma(2q)}{(_{\sim}+\overline{u})^{2q}}$$= \int_{-\infty}^{\infty}e^{-ztt}e^{-\overline{u}}t^{2}q-1dt$(7.6.1)

97
on the comis the reproducing kernel for the Bergman-Selbert spaceplex half plane $R^{+}=\{Rez>0\}$ comprising all analytic functions $f(z)$ on
with finite$H_{h_{q}}\cdot(R^{+})$$R^{+}$$\mathrm{n}\mathrm{o}\mathrm{r}\mathrm{n}\iota \mathrm{s}$$||f||_{H}^{2} \kappa\tau=\frac{1}{\pi\Gamma(2q-1)}\int\int_{R^{+}}|f(z)|2[2ReZ]2q-2dxdy$.

(7.6.2)

Note that(7.6.3)$K_{1}(z, \overline{u})=\int_{0}^{\infty}e^{-z\ell}e-\overline{u}ttdt$

and$\frac{\partial^{2}\mathrm{A}_{1}’(_{\sim},\overline{u})}{\partial z\partial\overline{u}},=\frac{6}{(z+\overline{u})^{4}}$$=I_{0}^{\infty}e^{-}ez\ell-\overline{u}t3tdt$$=\mathrm{A}_{2}’(Z,\overline{u})$.

(7.6.4)In the transform induced from (7.6.3)$f(z)= \int_{0}^{\infty}e^{-z}F\iota(\mathrm{f})tdt$(7.6.5)$\int_{0}^{\infty}|F(t)|2td\mathrm{f}<\infty$(7.6.6)for functions $F$ satisfying

we have$f’(Z)= \int \mathrm{o}te^{-z\ell}F()(-t^{2})dt\infty$

By using the$\mathrm{t}\mathrm{r}\mathrm{a}\mathrm{n}\mathrm{s}\mathrm{f}\mathrm{o}\mathrm{r}\iota \mathrm{n}$$G$(7.6.7)

induced from (7.6.4)$f’(z)= \int_{0}^{\infty}e^{-z\iota}G(t)(t^{3})d\mathrm{f}$for a function.

(7.6.8)satisfying$\int_{0}^{\infty}|G(t)|2t^{3}dt<\infty$

we have the derived vector of,(7.6.9)$F$$DF=- \frac{F(t)}{t}$(7.6.10)

98
through the transforms (7.6.5) and (7.6.8).
4

and by using the correto, by integrating (7.6.4) fronl
sponding transfornls to (7.6.8) and (7.6.5) we have the integrated vector of $F$$+\infty$$\mathrm{F}\mathrm{u}\mathrm{r}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathfrak{m}\mathrm{o}\mathrm{r}\mathrm{e}$$z$

as follows:(7.6.11)$\int F=-tF(t)$ .

7.7. Note that for example, for the nonlinear transform$d_{1}f’’+d_{2}f’f+d_{3}f^{3}$($d_{j}$(7.7.1): constants)

for $f\in H_{h_{1}}$. in 7.6, it has a specially simple struct,ure and it belongs to the. Furthermore we have the inequalityspace$H_{h_{3}}\cdot$$\frac{120}{127}||d_{1}f/\prime d2f+\prime f+d_{3}f3||^{\mathrm{z}}HK3$(7.7.2)$\leq||f||_{H_{K_{1}}}^{2}(\frac{1}{120}|d_{1}|2\frac{1}{6}+|d_{2}|^{2}||f||2H_{K_{1}}+|d_{3}|^{2}||f||_{H}4\kappa_{\iota})$(Saitoh [1], Appendix 2).

In the transform (7.6.5), we have$f”(z)= \int_{0}^{\infty}e^{-zt}F(t)t^{3}dt$,$f’( \approx)f(Z)=\int_{0}^{\infty}e^{-z\iota}((F(t)(-t)2)*(F(t)(t)))(t)dt$

and$f(z)^{3}= \int_{0}^{\infty}e^{-z\iota_{(}}F(t)t)\propto 3(t)dt$

Here,

$($

$)^{*3}$.

means the three times convolution product. Hence we have theexpression$d_{1}f^{\prime/}(_{Z)+f}d2/(Z)f(_{Z})+d_{3}f(z)3= \int_{0}^{\infty}e^{-\mathrm{z}t}\{d_{\iota}t^{-}F(2t)+d_{2}t^{-}\mathrm{s}$$((F(t)(-t^{2}))*(F(t)t))(t)+ds^{t^{-}}(\mathrm{s}F(t)\mathrm{f})*.\cdot,$ $(t)\}t^{\mathrm{s}_{d}}t$.

(7.7.3)Hence, by using the transform$f(z)= \int_{0}^{\infty}e^{-z}G\mathrm{t}(t)\mathrm{f}^{\mathrm{s}}.dt$(7.7.4)

99
for functions $G$ satisfying(7.7.5)$\int_{0}^{\infty}|G(t)|2\mathrm{S}tdt<\infty$

induced from the expression (7.6.1)$\mathrm{f}\dot{\mathrm{o}}\mathrm{r}q=3$, we have$G_{}(t)=d_{1}t^{-2}F(t)+d_{2}t^{-\mathrm{S}}((F(t)(-t^{2}))$$*(F(t)t))(t)+d3t-5(F(t)\mathrm{f})*3(t)$.

(7.7.6)Note that the nonlinear transform (7.7.1) is transformed to the form (7.7.6)
containing convolutions.
$,\mathfrak{i}$Acknowlegment
This research is partially supported by the Japanese Ministry of Education,, 10304009.
Science, Sports an.d Culture, Grant-in-Aid, Kibankenkyu$\mathrm{A}(1)$Reference[1]

S. Saitoh, Integral Transforms, Reproducing Kemels and their $Appli_{C}ati_{or}lS$,
Pitman Research Notes in Mathematics Series, 369, Addison Wesley Longman, UK, 1998.