数理解析研究所講究録
第 1719 巻 2010 年 142-154

142

DISCRETIZATIONBY THE THEORY OF REPRODUCINGKERNELS
Luis P. Castro , Dep. of Math. University of Aveiro
Saburou Saitoh , Dep. of Math. University of Aveiro
Yoshihiro Sawano , Dep. of Math. Kyoto University
andAlberto M. Simoes , Dep. of Math. University of Beira Interior

March 15, 2010

1

Introduction

The main purpose of this work is to analyse linear partial differential
equations and corresponding inverse source problems via a reproducing
kernel Hilbert spaces approach. The method here proposed is built ina very general framework. Detailed applications of the present method,e.g. to the most known basic linear partial differential equations (suchthat the heat equation, the wave equation and the Laplace equation), areexpected to appear in forthcoming works by using the present approachas a starting point.
Assuming to be any complex valued
function, we shall first givevery simple approximate solutions to the partial differential equation,$L_{2}$$g$$L(D)u=g$on$\mathbb{R}^{n}$

,

(1.1)

in the class of the functions of the order Sobolev Hilbert space $H^{s}(s>$on the whole real space , with $n\geq 1,$ $s\geq m\geq 1$ and $s>n/2$ andwhere $m$ denotes the order of the nontrivial general linear partial differential operator $L(D)$ with complex constant coefficients on. That is,$s$$\frac{n}{2})$$\mathbb{R}^{n}$$\mathbb{R}^{n}$143
we are given a linear partial differential operator$L(D)= \sum_{|\alpha|\leq m}a_{\alpha}(\frac{\partial}{\partial x})^{\alpha}$,ofwhere the ’s are complex numbers and there exists a multi-index
length $m$ such that
. Anyway, many constant coefficient partial
differential equations fall under the scope of our main results in case ofbeing inhomogeneous linear partial differential equations with complex.
constant coefficients of all types on the whole space
For simplicity, we write$\alpha_{0}$$a_{\alpha}$$a_{\alpha}\neq 0$$\mathbb{R}^{n}$$L(D)e^{ix\cdot\xi}=L(\xi)e^{ix\cdot\xi}$by using a complex polynomial L.
Theorem 1. Let $n\geq 1,$ $s\geq m\geq 1$ and $s>n/2$ .

(i) For any complex valued function $g\in L_{2}$ and for any$\lambda>0$(1.2)$\inf_{F\in H^{s}}\{\lambda\Vert F\Vert_{H^{S}}^{2}+\Vert g-L(D)F\Vert_{L_{2}}^{2}\}$is attained by a unique function$F_{\lambda,s,g}^{*}$,.

(ii) Let us write$Q_{\lambda,s}( \eta)=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{\overline{L(p)}e^{-ip\cdot\eta}dp}{\lambda(|p|^{2}+1)^{s}+|L(p)|^{2}}$Then, the extremal function$F_{\lambda,s,g}^{*}$

.

(1.3)

is represented by$F_{\lambda,s,g}^{*}(x)= \int_{\mathbb{R}^{n}}g(\xi)Q_{\lambda,s}(\xi-x)d\xi$

.

(1.4)

(iii) If is expressed as $g=L(D)F_{f}$ for a function $F\in H^{s}$ , then we
have the favourable result: as$g$$\lambdaarrow 0$$F_{\lambda,s,g}^{*}arrow F$

,

(1.5)

uniformly.

As first examples that we envisage to consider within the basis of
the present work, include the differential equations characterized by thefollowing operators:

144
1. The -operator$\overline{\partial}$$\frac{\partial}{F\overline{z}}=\frac{1}{2}(\frac{\partial}{\partial x}-i\frac{\partial}{\partial y})$

.

(1.6)

2. The heat operator

on$\partial_{t}u(x, t)-c^{2}\triangle_{x}u(x, t)$

where$\Delta_{x}$denotes the Laplacian on$\mathbb{R}^{n-1}$

,

(1.7)

$(c>0)$ .

(1.8)$\mathbb{R}^{n-1}\cross \mathbb{R}$.

3. The wave operator$\partial_{t}^{2}u(x, t)-c^{2}\Delta_{x}u(x, t)$on$\mathbb{R}^{n-1}\cross \mathbb{R}$Note that although there are not global solutions to (1.7), our main result
is still applicable.
From concrete examples we see that the representations (1.4) are
effectively computable by using computers and the approximate solutions(1.4) converge to the true solutions of (1.1) as in (1.5) (cf. [1] and [6]).
When the practical data contain error or noises, we need error
estimates for the approximate solution (1.4). As an improved version in[4], we can establish the following result.
$g$

Theorem 2. For$g,$ $g_{\delta}\in L^{2}(\mathbb{R}^{n})$we have$\Vert F_{\lambda,s,g}^{*}-F_{\lambda,s,g_{\delta}}^{*}\Vert_{H^{s}}\leq\frac{1}{2\sqrt{\lambda}}\Vert g-g_{\delta}\Vert_{L_{2}}$

.

(1.9)

In this note, we shall first give the corresponding results for the general partial differential equation (1.1). Furthermore, we shall establish
error estimates for our solutions because practical data contain error
and noises. In the second part, we shall construct the solutions for discrete differential equations, following the recent general idea ([2]) andfurthermore, we shall give their inversions. That is, we shall give the corresponding construction of the solutions of the inverse source problems.

2

Preliminaries on reproducing kernel Hilbert
spaces2.1

Sobolev reproducing kernel Hilbert spaces

We shall use the Sobolev reproducing kernel Hilbert spaces.

145
Let $m$ be an integer such that
. Then, by
the Sobolev Hilbert space whose norm is given by$m> \frac{n}{2}$$H^{m}$

we shall denote$\Vert F\Vert_{H^{m}}=[\sum_{\nu=0}^{m}{}_{m}C_{\nu}(\sum_{\alpha\in \mathbb{Z}_{+}^{n},|\alpha|\leq\nu}\frac{\nu!}{\alpha!}\int_{\mathbb{R}^{n}}|\frac{\partial^{\alpha}F(x)}{\partial x^{\alpha}}|^{2}dx)]^{\frac{1}{2}}$(2.1)

Observe that the corresponding inner product of this space is given by$\langle F,$$G \rangle_{H^{m}}=\sum_{\nu=0}^{m}{}_{m}C_{\nu}(\sum_{\alpha\in \mathbb{Z}_{+}^{n},|\alpha|\leq\nu}\frac{\nu!}{\alpha!}\int_{\mathbb{R}^{n}}\frac{\partial^{\alpha}F(x)}{\partial x^{\alpha}}\overline{\frac{\partial^{\alpha}G(x)}{\partial x^{\alpha}}}dx)$(2.2)

Here we have denoted$x=(x_{1}, x_{2}, \ldots, x_{n})$,$\xi=(\xi_{1}, \xi_{2}, \ldots, \xi_{n})$,$d\xi=d\xi_{1}\cdot d\xi_{2}\cdots\cdot\cdot d\xi_{n}$(2.3)

as usual. If we define the Fourier transform$( \mathcal{F}F)(\xi)=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}F(x)e^{-ix\cdot\xi}dx$

(2.4)

,

, then we
is a standard inner product in
where
have, by virtue of the Fourier transform and the multinomial expansion$\mathbb{R}^{n}$$x\cdot\xi=x_{1}\xi_{1}+\cdots\cdot\cdot+x_{n}\xi_{n}$$\langle F,$$G \rangle_{H^{m}}=\sum_{\nu=0}^{m}{}_{m}C_{\nu}(\sum_{\alpha\in \mathbb{Z}_{+}^{n},|\alpha|\leq\nu}\frac{\nu!}{\alpha!}\int_{\mathbb{R}^{n}}\xi^{2\alpha}(\mathcal{F}F)(\xi)\overline{(\mathcal{F}G)(\xi)}d\xi)$$= \sum_{\nu=0}^{m}{}_{m}C_{\nu}\int_{\mathbb{R}^{n}}|\xi|^{2\nu}(\mathcal{F}F)(\xi)\overline{(\mathcal{F}G)(\xi)}d\xi$$= \int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{m}(\mathcal{F}F)(\xi)\overline{(\mathcal{F}G)(\xi)}d\xi$.

If we set
$K_{m}(x, y)= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{e^{i(x-y)\xi}}{(1+|\xi|^{2})^{m}}d\xi$

,

(2.5)

then we have$\langle F,$$K_{m}( \cdot, x)\rangle_{H^{m}}=\frac{1}{(2\pi)^{\frac{n}{2}}}\int_{\mathbb{R}^{n}}(\mathcal{F}F)(\xi)e^{ix\cdot\xi}d\xi=F(x)$(2.6)

is a reproducing kernel Hilbert space whose
for all $F\in H^{m}$ . Hence.
reproducing kernel is given by$H^{m}$$K_{m}$

146
We have an analogue to the above model case when a real number
satisfies. If defining$s$$s> \frac{n}{2}$$K_{s}(x, y)= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{-s}e^{i(x-y)\cdot\xi}d\xi$

,

(2.7)

then it follows$H_{K_{S}}=H^{s}$, that is, $F(x)=\langle F,$$K_{s}(\cdot, x)\rangle_{H^{S}},$$F\in H^{s}$

,

(2.8)

and the norm is given by(2.9)$\Vert f\Vert_{H^{s}}:=(\int_{\mathbb{R}^{n}}(1+|\xi|^{2})^{s}|(\mathcal{F}f)(\xi)|^{2}d\xi)^{\frac{1}{2}}$Observe that$|F(x)| \leq\Vert F\Vert_{H^{S}}(\int_{\mathbb{R}^{n}}\frac{d\xi}{(1+|\xi|^{2})^{2s}}I$(2.10),

if $s>:.$ So the convergence in
implies the uniform convergence. We
refer e.g. to [10] for additional details on this. Here we content ourselveswith mentioning thatis called the potential space.
$H^{s}$$H^{s}$

2.2

Paley-Wiener reproducing kernel Hilbert spaces

We shall also use the Paley-Wiener reproducing kernel. See J. R. Higgins[3] and F. Stenger[ll] for the details in connection with the sampling
theory.
Let $h>0$ be fixed. Let us consider the following integral transform:.

$g \in L_{2}((-\pi/h, +\pi/h)^{n})\mapsto f(z)=\frac{1}{(2\pi)^{n}}\int_{(-\pi/h,+\pi/h)^{n}}g(t)e^{-iz}.{}^{t}dt$(2.11)

Here,$\chi_{h}(t)=\prod_{\nu=1}^{n}\chi_{(-\pi/h,+\pi/h)}(t_{\nu})$

stands for the characteristic functionof $(-\pi/h, +\pi/h)^{n}$ . In order to identify the image space by means of thetheory of reproducing kernels, we consider the reproducing kernel$K_{h}(z, \overline{u})=\frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\chi_{h}(t)e^{-iz}\overline{{}^{t}e^{-iu}}{}^{t}dt$$= \prod_{\nu=1}^{n}\frac{1}{\pi(z_{\nu}-\overline{u}_{\nu})}\sin\frac{\pi}{h}(z_{\nu}-\overline{u}_{\nu}),$$z,$$u\in \mathbb{C}^{n}$.

147
The image space of (2.11) is called the Paley-Wiener spacecomprised of all analytic functions of exponential type satisfying,for each , for some constant$W( \frac{\pi}{h})(:=$$W_{h})$$C_{\nu}$$\nu$$|f(z_{1}, z_{2}, \ldots, z_{\nu}, z_{\nu+I}, \ldots, z_{n})|\leq C_{\nu}\exp(\frac{\pi|z_{\nu}|}{h}),$$z=(z_{1}, z_{2}, \ldots, z_{n})\in \mathbb{C}^{n}$and$\int_{\mathbb{R}^{n}}|f(x)|^{2}dx<\infty$.

Let denote the Kronecker delta. For multi-indices $j=(j_{1}, j_{2}, \cdots, j_{n})$and $j’=(j_{1}’,j_{2}’, \cdots,j_{n}’)$ , from the identity$\delta$$s$$K_{h}(jh,j’h)= \prod_{\nu=1}^{n}\frac{1}{h}\delta(j_{\nu},j_{\nu}’)=\{\begin{array}{l}h^{-n}, j=j’,0, otherwise\end{array}$(for each ), taking into account thatis the reproducing kernel
for the Hilbert space , and using the general theory (see [9]) of integral
transforms and Parseval identity, we have the isometric identities inconnection to (2.11)$\delta(j_{\nu}, j_{\nu}’)$$\nu$$l^{2}$$s$$\frac{1}{(2\pi)^{n}}\int_{(-\pi/h,\pi/h)^{n}}|g(t)|^{2}dt=h^{n}\sum_{j}|f(jh)|^{2}=\int_{\mathbb{R}^{n}}|f(x)|^{2}dx$.

withis charThat is, the reproducing kernel Hilbert space
and with theacterized as a space comprising the Paley-Wiener space
norms above in the senses of both discrete version and continuous veris a uniquenesssion. Here we invoked the well-known result that, in the sense that if $f(jh)=0$ for all
set for the Paley-Wiener spacestates thatthen $f\equiv 0$ . Therefore, the reproducing property of$K_{h}(z, \overline{u})$$H_{K_{h}}$$W_{h}$$\{jh\}_{j}$$j$$W_{h}$$K_{h}(z, \overline{u})$$f(x)$

$=$

$=$

$=$

$(f(\cdot), K_{h}(\cdot, x))_{H_{K_{h}}}$$h^{n} \sum_{j}f(jh)K_{h}(jh, x)$$\int_{\mathbb{R}^{n}}f(\xi)K_{h}(\xi, x)d\xi$.

This representation is the sampling theorem which represents the whole
data $f(x)$ in terms of the discrete data $\{f(jh)\}_{j}$ . Furthermore, we referto [10] for a general theory about sampling theory and error estimates.
for finite points$\{hj\}_{j}$

148

3

Tikhonov regularization

Let $E$ be an arbitrary set, and let
be a reproducing kernel Hilbert
space (RKHS) admitting the reproducing kernel $K(p, q)$ on $E$ . For any
Hilbert space we consider a bounded linear operator frominto
. We are generally interesting in the best approximate problem$H_{K}$$L$$\mathcal{H}$$H_{K}$$\mathcal{H}$(3.1)$\inf_{f\in H_{K}}\Vert Lf-d\Vert_{\mathcal{H}}$for any vector in . However, this extremal problem is involved in
the both senses of the existence of the extremal functions in (3.1) andtheir representations. See [10] for the details. So, we shall consider its
Tikhonov regularization.
,
We set, for any fixed positive$d$$\mathcal{H}$$\lambda>0$$K_{L}( \cdot,p;\lambda)=\frac{1}{L^{*}L+\lambda I}K(\cdot,p)=(L^{*}L+\lambda I)^{-1}K(\cdot,p)$

wheredenotes the adjoint operator ofinner product$L^{*}$$L$,. Then, by introducing the$(f, g)_{H_{K}(L;\lambda)}=\lambda(f, g)_{H_{K}}+(Lf, Lg)_{\mathcal{H}}$

,

(3.2)

we shall construct the Hilbert spacecomprising functions of. It is clear that this space admits a reproducing kernel.
$H_{K}(L;\lambda)$$H_{K}$

Proposition 1.

(i) The extremal function$f_{d,\lambda}(p)$in the Tikhonov regularization(3.3)$\inf_{f\in H_{K}}\{\lambda\Vert f\Vert_{H_{K}}^{2}+\Vert d-Lf\Vert_{\mathcal{H}}^{2}\}$

exists uniquely and it is represented in terms of the kemel$K_{L}(p, q;\lambda)$

as follows:(3.4)$f_{d,\lambda}(p)=(d, LK_{L}(\cdot,p;\lambda))_{\mathcal{H}}$

where the kemel
space.

$K_{L}(p, q;\lambda)$

is the reproducing kemel for the Hilbert$H_{K}(L;\lambda)$(ii)$K_{L}(p, q;\lambda)$

is determined as the unique solution$\tilde{K}(p, q;\lambda)$

equation:$\tilde{K}(p, q;\lambda)+\frac{1}{\lambda}(L\tilde{K}_{q}, LK_{p})_{\mathcal{H}}=\frac{1}{\lambda}K(p, q)$of the
(3.5)

149
with

for

$\tilde{K}_{q}=\tilde{K}(\cdot, q;\lambda)\in H_{K}$$q\in E$

(3.6)

,

and
$K_{p}=K(\cdot,p)\in H_{K}$for $p\in E$ .

(3.7)

In (3.4), when contains error or noises, we need its error estimate.
For this, we can obtain the general result:$d$Proposition 2. In (3.4), we have the estimate$|f_{d,\lambda}(p)$I$\leq\frac{1}{2\sqrt{\lambda}}\sqrt{K(p,p)}\Vert d\Vert_{\mathcal{H}}$

(3.8)

.

We would like to point out that in a corresponding estimate ofin [4], the factor 2 is missing when compared with (3.8) above. That is,andthe estimate (3.8) is improved here. As the example of$L=I$ shows, non-trivial equality (3.8) is attained.
$|f_{d,\lambda}(p)|$$\mathcal{H}=H_{K}$Proposition 3. Suppose thatis a reproducing kemel Hilbert space$K$with kemel
is a Hilbert space. If we are given a surjective
and that
we can findthen forcontinuous linear tmnsform$f\in H_{K}$ so as to minimize. Such belongs to the mnge of$H_{K}$$\mathcal{H}$$L:H_{K}arrow \mathcal{H}_{f}$$L^{*}$$f$$\Vert f\Vert_{H_{K}}$4$g\in \mathcal{H}$

Discrete differential equation

The representation (1.4) may not yield any solution of the discrete differential equation (1.1) for a given finite number of the data and so
we will be interested in such discrete differential equation, because we
obtain only a finite number of values as observation data.
$g$

$g$

4.1

Paley-Wiener type space, the Paley-Wiener reIn the first half of this section, we work on
producing kernel Hilbert space described in Section 2. Recall that any, we shall consider
is smooth. For different points
element in
into :
the bounded linear operators from the RKHS
$H_{K_{h}}$$\{x_{j}\}_{j=1}^{N}$$H_{K_{h}}$$H_{K_{h}}$$H_{K_{h}}\ni F\mapsto(LF)(x_{j}):=(L(D)F)(x_{j})$;$j=1,2,$$\mathbb{R}$$\ldots$

, N.

(4.1)

150
Needless to say,
We define$K_{h}$

being nice, the mapping $F\mapsto L(D)F$ is continuous.

$L:H_{K_{h}}\ni F\mapsto((LF)(x_{1}), (LF)(x_{2}), \ldots, (LF)(x_{N}))\in \mathbb{R}^{N}$

We shall use a standard orthonormal system
Then, we see.

in the space$\{e_{j}\}_{j=1}^{N}$$(LF)(x_{j})=(LF, e_{j})_{\mathbb{R}^{N}}=(F, L^{*}e_{j})_{H_{K_{h}}}$

.

(4.2)
$\mathbb{R}^{N}$

.

(4.3)

Let us set$A:=\{a_{jj’}\}_{j,j’=1,2,\cdots,N}:=\{(L^{*}e_{j’}, L^{*}e_{j})_{H_{K_{h}}}\}_{j,j’=1,2,\cdots,N}$

Here note that$a_{jj’}$

.

(4.4)

are calculated as follows:$(L^{*}e_{j})(x)=((L^{*}e_{j})(\cdot), K_{h}(\cdot, x))_{H_{K_{h}}}$$=(e_{j}, LK_{h}(\cdot, x))_{\mathbb{R}^{N}}$$=L[K_{h}(\cdot, x)](x_{j})$$= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\chi_{h}(\eta)L(-\eta)e^{-i\eta\cdot(x-x_{j})}d\eta$.

Hence, it follows that$a_{jj’}= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\chi_{h}(\eta)|L(-\eta)|^{2}e^{-i\eta\cdot(x_{j}-x_{j’})}d\eta$

.

(4.5)

In particular, we conclude from this that the matrix $A=\{a_{jj’}\}_{j,j’=1,2,\cdots,N}$is strictly positive definite.
Denote by
the image of . Since the matrix
isinvertible, then, the norm in
is given by$L$$H_{A}$$\{a_{jj’}\}_{j,j’=1,2,\cdots,N}$$H_{A}$$\Vert LF\Vert_{H_{A}}^{2}=(LF)^{*}A^{*}(LF)$

(4.6)

,

(see [10]. p. 250).
where
We thus obtain the desired result from Proposition 1.
$A^{*}=\overline{A^{-1}}=\{\tilde{a}_{jj’}\}_{j,j’=1,2,\cdots,N}$

Theorem 3. For any given $N$ values
functions $F$ taking the values$d=\{g(x_{j})\}_{j=1}^{N}\in \mathbb{R}^{n}$, among the$H_{K_{h}}$$(LF)(x_{j})=g(x_{j})$$j=1,2,$$N$$\ldots,$

,

(4.7)

151
the function
with the minimum norm
mined and it is represented as follows:$F_{d}^{*}(x)$is uniquely deter-$\Vert F\Vert_{H_{K_{h}}}$$F_{d}^{*}(x)= \sum_{j,j=1}^{N}g(x_{j})\tilde{\alpha}_{jj’}(L^{*}e_{j’})(x)$

.

(4.8)

Proof. The function given by (4.8) belongs to the range ofwith at$j=1,2,$of Proposition 3.
$g$$x_{j},$$N$$\cdots,$

and agrees. So, the result is an immediate consequence$L^{*}$$\square$

Theorem 3 can now be directly applied. In connection with this, werefer to [5] for a quite simple algorithm for inversion of any matrix (withseveral lines) using the theory of reproducing kernels and the Tikhonov
regularization.
As we see from the isometrical relation between the data (4.1) and, the observation data and thethe inverse with the minimum norm
corresponding inverse are rigid, and our inversion formula is given in thesense of well-posed problem.
$F_{d}^{*}(x)$4.2

Sobolev space

In the Sobolev space
formulae:$H^{s}$

of $s\geq m$ and$s> \frac{n}{2}$, we have the corresponding$(L^{*}e_{j})(x)=((L^{*}e_{j})(\cdot), K_{s}(\cdot, x))_{H_{Ks}}$$=(e_{j}, LK_{s}(\cdot, x))_{\mathbb{R}^{N}}$$= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{\overline L(\eta)e^{-i\eta(x-x)}j}{(1+|\eta|^{2})^{s}}d\eta$and$a_{jj’}= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{|L(\eta)|^{2}e^{i\eta(x_{j}-x_{j},)}}{(1+|\eta|^{2})^{s}}d\eta$

5

.

(4.9)

Inverse source problem

We shall consider the inversion problem for (4.1) and (4.2). That is,from a finite number of observation data $u(x_{j}),$ $j=1,2,$ $N$ , we wish
to determine the optimal source in (1.1). For this purpose, we shall
$..,$

$g$

152
use the integral representation (1.4); that is, for
we shall use theapproximation data
. So, we shall consider the bounded linear
operators from the RKHS
into :$u(x_{j})$$F_{\lambda,s,g}^{*}(x_{j})$$H_{K_{h}}$$\mathbb{R}$$H_{K_{h}}\ni g\mapsto(Mg)(x_{j}):=F_{\lambda,s,g}^{*}(x_{j})$;$j=1,2,$$\ldots$, $N$ .

(5.1)

We set$M:H_{K_{h}}\ni g\mapsto((Mg)(x_{1}), (Mg)(x_{2}), \ldots, (Mg)(x_{N}))\in \mathbb{R}^{N}$

.

(5.2)

Then, we see$(Mg)(x_{j})=(Mg, e_{j})_{\mathbb{R}^{N}}=(g, M^{*}e_{j})_{H_{K_{h}}}$

.

(5.3)

Let us set$B:=\{b_{jj’}\}_{j,j’=1,2,\cdots,N}:=\{(M^{*}e_{j’}, M^{*}e_{j})_{H_{K_{h}}}\}_{j,j’=1,2,\cdots,N}$

Here note that$b_{jj’}$

.

(5.4)

are calculated as follows:$(M^{*}e_{j})(x)=((M^{*}e_{j})(\cdot), K_{h}(\cdot, x))_{H_{K_{h}}}$$=(e_{j}, MK_{h}(\cdot, x))_{\mathbb{R}^{N}}$$=M[K_{h}(\cdot, x)](x_{j})$.

$= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{\chi_{h}(\eta)L(-\eta)e^{-i\eta(x-x_{j})}}{\lambda(|\eta|^{2}+1)^{s}+|L(\eta)|^{2}}d\eta$Hence, it follows that$b_{jj’}= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{\chi_{h}(\eta)|L(-\eta)|^{2}e^{-i\eta\cdot(x_{j}-x_{j};)}}{(\lambda(|\eta|^{2}+1)^{s}+|L(\eta)|^{2})^{2}}d\eta$

.

(5.5)

In particular, we conclude from this that the matrix $B=\{b_{jj’}\}_{j,j’=1,2,\cdots,N}$is strictly positive definite.
We thus obtain the desired result from Proposition 1.
Theorem 4. For any given $N$ values
thefunctions taking the values$H_{K_{h}}$$d=\{F_{\lambda,s,g}^{*}(x_{j})\}_{j=1}^{N}\in \mathbb{R}^{n}$, among$g$$F_{\lambda,s,g}^{*}(x_{j})=(Mg)(x_{j})$$j=1,2,$

with the minimum norm
the function
mined and it is represented as follows:$g_{d}^{*}(x)$$N$$\ldots,$(5.6)

is uniquely deter-$\Vert g\Vert_{H_{K_{h}}}$$g_{d}^{*}(x)= \sum_{j,j=1}^{N}F_{\lambda,s,g}^{*}(x_{j})\tilde{b}_{jj’}(L^{*}e_{j’})(x)$

,

.

(5.7)

153
In the Sobolev space of $s\geq m$ andformulae:$s> \frac{n}{2}$, we have the corresponding$(M^{*}e_{j})(x)=((M^{*}e_{j})(\cdot), K_{s}(\cdot, x))_{H_{K_{8}}}$$=(e_{j}, MK_{s}(\cdot, x))_{\mathbb{R}^{N}}$$= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{\overline{L(\eta)}e^{-i\eta\cdot(x-x_{j})}}{(1+|\eta|^{2})^{s}(\lambda(|\eta|^{2}+1)^{s}+|L(\eta)|^{2})}d\eta$and.

$b_{jj’}= \frac{1}{(2\pi)^{n}}\int_{\mathbb{R}^{n}}\frac{|L(\eta)|^{2}e^{i\eta\cdot(x-x_{j’})}j}{(1+|\eta|^{2})^{s}(\lambda(|\eta|^{2}+1)^{s}+|L(\eta)|^{2})^{2}}d\eta$(5.8)

Acknowledgements. This work was supported in part by Center ofResearch and Development in Mathematics and Applications, Universityof Aveiro, Portugal, through FCT-Portuguese Foundation for Scienceand Technology. The second author was supported in part by the Grantin-Aid for the Scientific Research (C)(2)(No. 21540111) from the Japan
Society for the Promotion Science, and the third author was supported
by Grant-in-Aid for Young Scientists (B) (No.21740104) Japan Society
for the Promotion of Science.

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[10] S. Saitoh, Integml Transforms, Repmducing Kemels and their Applications, Pitman Research Notes in Mathematics Series, 369(1997). Addison Wesley Longman, UK.

[11] F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer Series in Computational Mathematics, 20, 1993.
e-mail: castro@ua.pt; saburou.saitoh@gmail.com; yosihiro@math.kyotou.ac.jp; asimoes@mat.ubi.pt