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Approximation by Translates of a Positive Definite Function
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.
201, 631]641 Ž1996.
0278
Approximation by Translates of a Positive Definite Function S. J. Kilmer Department of Mathematics, Southwest Missouri State Uni¨ ersity, Springfield, Missouri 65804
W. A. Light Department of Mathematics and Computer Science, Uni¨ ersity of Leicester, Uni¨ ersity Road, Leicester LE1 7RH, England
and X. Sun and X. M. Yu Department of Mathematics, Southwest Missouri State Uni¨ ersity, Springfield, Missouri 65804 Submitted by E. W. Cheney Received December 16, 1994
1. INTRODUCTION There are now many papers dealing with approximation of real-valued functions by linear combinations of a single function. At its simplest, this approach consists of selecting a function h: R n ª R in some appropriate function class and points a1 , . . . , a m g R n. The approximating subspace is then H s span h Ž ?y a i . : i s 1, . . . , m4 . This subspace may now be used for the purposes of approximation and interpolation either over the whole of R n, or over some subset V of R n. Let us now make the discussion more concrete by assuming V is a compact subset of R n, and we are interested in approximation in the space C Ž V .. One of the first questions which arises is that of density. If the number of points is allowed to grow, and if these points somehow fill out 631 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
632
KILMER ET AL.
the whole of V, then can we approximate any function f g C Ž V . to arbitrary accuracy by functions in H ? A neat way to pose this problem is to set H s span h Ž ?y y . : y g V 4 , and to ask whether H is dense in C Ž V ., where the density is measured in the supremum norm. An important special case of the above is as follows. Let 5 ? 5 2 denote the usual Euclidean norm in R n. Let f g C ŽR., and set hŽ x . s f Ž5 x 5 2 . for x g R n. Then h g C ŽR n . and is a radial function. The translates of such a function have very good properties with regard to interpolation ŽMicchelli w4x.. They also have good properties with regard to approximation ŽBrown w1x.. It is perhaps remarkable that both the interpolation and the density results depend on the function f possessing the same property of conditional positi¨ e definiteness. For a unified treatment of these two results, see Light w3x. Our purpose here is to investigate density properties of translates of a single function, but our density will be measured with a Sobolev norm. Let V be a compact subset of R n. Let r g Zq and define C r Ž V . s f : V ª R : D a f g C Ž V . whenever < a < F r 4 . There are several equivalent ways of imposing a norm on this space; we will use 5f5s
Ý
5 D a f 5` ,
f g CrŽ V. .
n agZ q
< a
The situation considered by Brown can be regarded as studying the r s 0 case. Our results will show that simultaneous approximation of the function and its derivatives is also possible. We conclude this section with some notation and a few elementary observations. We will use standard multi-index notation throughout. The set of all positive definite functions on R n will be denoted by P. Thus f g P if the expression m
Ý
ci c j f Ž x i y x j . G 0
i , js1
whenever x 1 , . . . , x m g R n and c1 , . . . , c m g R. Let M Ž V . denote the set of all complex, regular Borel measures on V, and M the corresponding measures on the whole of R n. For f g L1 ŽR n .,
APPROXIMATION OF FUNCTIONS
633
the Fourier transform of f, denote by fˆor F Ž f ., is the continuous function on R n defined by fˆŽ x . s
yi x t
HR f Ž t . e n
x g R n.
dt,
The inverse Fourier transform fˇ of f is defined by fˇŽ x . s Ž 2p .
yn
HR f Ž t . e
i xt
n
dt,
x g R n.
We alert the reader that both the Fourier transform and its inverse are to be interpreted in the distributional sense whenever necessary. We use S to denote the space of infinitely differentiable and rapidly decreasing functions, and S X to denote the space of tempered distributions Žcontinuous, linear functionals on S .. It is well known that the Fourier transform is a one-to-one, continuous operator from S X onto S X . Bochner’s theorem w2x asserts that a continuous function h is positive definite on R n if and only if h is the Fourier transform of a unique positive measure m g M, that is, hŽ x . s
yi x t
HR e n
dm Ž t . ,
x g R n.
The correspondence between m and h will sometimes be made explicit by the notation m h . The Lebesgue decomposition theorem tells us that each m g M can be written in a unique way as
m s ma q ms , where m a is absolutely continuous, and m s is singular with respect to Lebesgue measure on R n. Let X be a normed linear space, and A a subset of X. If the linear span of A is dense in X, we say that A is fundamental in X. Part of our work depends on a representation theorem for linear functionals on C r Ž V .. Let l s a a g Z n : a G 0 and < a < F r 4 , and let X be the l-fold Cartesian product of C Ž V . with itself. Define 5F5 s
Ý
aG0 < a
5 Fa 5 ` ,
where f s Ž Fa . aG0, < a
There is an obvious isometric embedding of C r Ž V . into X, which associates f g C r Ž V . with F g X via the rule Fa s D a f. Given a continuous,
634
KILMER ET AL.
linear functional r on C r Ž V ., this functional can be extended to a continuous, linear functional on X by the Hahn]Banach theorem. Using the Riesz representation theorem in a coordinate fashion, we can find measures Ž ma .a G 0, < a < F r such that ma g M Ž V ., and so that the action of r on any element x g X is
rŽ x. s
Ý
Hx
a )0 V < a
d ma .
a
This establishes the following result. THEOREM 1.1. Let r be a continuous, linear functional on C r Ž V ., where V is a compact subset of R n. Then for each a g Z n, with a G 0 and < a < F r there exist ma g M Ž V ., such that
rŽ f . s
Ý
a G0 < a
HVD
a
f d ma ,
f g CrŽ V. .
2. RESULTS We begin by examining some properties of derivatives of positive definite functions. We say that a multi-index a s Ž a 1 , . . . , a n . is e¨ en if each a j is a positive even integer, or is zero. THEOREM 2.1. Let r s 0, 2, 4, . . . , and let h g P l C r ŽR n .. For a g Z n, a G 0, let Va be the function defined by Va Ž t . s t a , t g R n. Then Va g L1 Ž m h . for each a with < a < F r. Proof. Let f be the Gauss kernel which is defined by
f Ž x . s Ž 2p .
yn r2
exp y5 x 5 22r2 4 ,
x g R n.
For each m g N and x g R n, define fmŽ x . s m nf Ž mx .. Since m h is a $ tempered distribution, and h s m h, it follows that D a h is a tempered n distribution for each a g Zq . Let ² ? , ? : denote the duality bracket X between S and S . For < a < F r, we have ² D a h, fm : s Ž y1 . < a < ² h, D afm : $
s Ž y1 .
²m h , D afm :
s Ž y1 .
² m h , D afm :
s Ž y1 .
² m h , i < a < Vafm :
s Ž y1 .
$
$
HR t n
a
$
fm d m h .
Ž 1.
635
APPROXIMATION OF FUNCTIONS
It is easy to see that lim m ª` fm s d in the topology of tempered distributions, where d is the Dirac distribution. Thus, by continuity, we have lim ² D a h, fm : s Ž D a h . Ž 0 . .
Ž 2.
mª`
At this juncture, it is convenient to break the proof into two cases. Case 1. The Multi-index$ a Is E¨ en In this case, we have Vafm Va . Hence, by the Monotone Convergence theorem, $
H Vf mª` R lim
n
a
m
du h s
HR
n
lim Va fm d m h s
mª`
dmh .
HR V
a
n
Therefore, by Ž1. and Ž2., we have
HR V n
a
$
d m h s lim
H Vf mª` R n
a
m
d mh s i < a < Ž D a h . Ž 0. .
This implies that Va g L1 Ž m h .. Case 2. The Multi-index a Is Not E¨ en In this case we write a kq 1 ??? t na n . Va Ž t . s t 1a i ??? t ka k t kq1
Without loss of generality, we assume that a 1 , . . . , a k are odd and a kq 1 , . . . , a n are even or zero. We may further assume that a 1 F a j for all 1 F j F k. Under these assumptions, Va Ž t . s Vb Ž t . Ž t 1 . . . t k .
a1
,
where b is even multi-index defined by
b s Ž 0, a 2 y a 1 , . . . , a k y a 1 , a kq1 , . . . , a n . . Again, it is helpful to distinguished two subcases. Subcase 1.
< a < is even. Then k is even also, and we have < Ž t1 . . . t k .
a1
a
< s < Ž t1 t 2 . 1 Ž t 3 t4 . F
ž
t 12 q t 22 2
a1
. . . Ž t ky1 t k .
a1
???
/ ž
2 t ky1 q t k2
2
a1
<
a1
/
.
Therefore, < Va Ž t . < F Vb Ž t . ? s
1
ž / 2
ž
t 12 q t 22 2
k a 1 r2
Ý tg , g
a1
/ ž ...
2 t ky1 q t k2
2
a1
/
636
KILMER ET AL.
where the summation is taken over a finite number of g for which easy g is even and satisfies < g < s < a < F r. It follows from the result in Case 1 that Va g L1 Ž m h .. Subcase 2.
< a < is odd. Then k is odd also, and so
< Ž t1 . . . t k .
a1
ž
t 12 q t 22 2 1
a1
???
/ ž
Ž ky1 . a 1 r2
ž / 2
2 2 t ky2 q t ky1
2
a1
/
Ž 1 q t ka q1 . 1
Ý td, d
where the summation over d is finite, each d is even, and satisfies < d < F < a < q 1. Since r is even, < a < is odd, and < a < F r we have < d < F r. Again, Case 1 may be used to show that Va g L1 Ž m h .. Let h g P l C m ŽR n ., where m s 2 r. For each a with
LEMMA 2.2. < a < F m,
a
Ž D a h . Ž x . s H neyi x t Ž yit . d m h Ž t . , R
x g R n.
Proof. We will calculate D a h, where a s Ž1, 0, . . . , 0., since all the calculations needed for the other values of a with < a < s 1 are similar, and a straightforward induction argument gives the result for 1 - < a < F m. Since h is positive definite, there exists a positive measure m h in M such that hŽ x . s
yi x t
HR e n
dmh Ž t . ,
x g R n.
Set a s Ž1, 0, . . . , 0.. By the Mean-Value theorem, 1 k
exp yi Ž x 1 q k, x 2 , . . . , x n . t 4 y exp yixt 4 s yit exp yi Ž j , x 2 , . . . , x n . t 4 s < t < < a < ,
637
APPROXIMATION OF FUNCTIONS
where j lies between x 1 and x 1 q k. By Theorem 2.1, t a g L1 Ž m h . and so, by the Lebesgue Dominated Convergence theorem, we obtain 1
Ž D a h . Ž x . s lim
½H
k
kª0
exp yi Ž x 1 q k, x 2 , . . . , x n . t 4 d m h Ž t .
Rn
y
yi x t
HR e
s
HR
s
HR e
s
HR e
n
1
lim kª0
k
Ž yit1 . d m h Ž t .
yi x t
Ž yit . d m h Ž t . .
n
dmh Ž t .
5
exp yi Ž x 1 q k, x 2 , . . . , x n . t 4 y eyi x t 4 d m h Ž t .
yi x t
n
n
a
For each bounded, linear functional r on C r Ž V ., it follows from Theorem 1.1 that there exists ma g M Ž V ., < a < F r, such that
rŽ f . s
Ý
HD
a
< a
f d ma ,
f g CrŽ V. .
We may regard r as a tempered distribution and calculate its Fourier transform. LEMMA 2.3. For each bounded, linear functional r on C r Ž V . with associated measures ma g M Ž V ., < a < F r, its Fourier transform rˆ is gi¨ en by
rˆ Ž t . s
a Ý Ž yit . H eyi x t d ma Ž x . ,
V
< a
t g R n.
Ž 3.
Moreo¨ er, rˆ is entire. Proof. Let S denote the set of all infinitely differentiable, rapidly decreasing functions on R n. For each f g S , we have, by Theorem 1.1, ² rˆ, f : s ² r , fˆ: s
HV Ž D fˆ . d m
Ý
HVHR
< a
s
a
Ý
< a
a
a
n
eyi x t Ž yit . f Ž t . dt d ma Ž x . .
638
KILMER ET AL.
Using Fubini’s theorem to interchange the order of integration, we obtain
² rˆ, f : s
Ž yit .
HR Ý n
a
yi x t
HVe
< a
d ma Ž x . f Ž t . dt.
This shows that Ž3. is true. Since r is a compactly supported distribution, it follows from the Paley]Wiener theorem that rˆ is entire. The next lemma contains the heart of the result we are aiming for. LEMMA 2.4. Let r be a bounded, linear functional on C r Ž V .. Let h g P l C m Ž V ., where m s 2 r. For each y g R n, define h y Ž x . s hŽ x y y ., x g R n. If r Ž h y . s 0 for all y g V, then rˆŽ x . s 0 for m h-almost all x g R n. Proof. It will be helpful throughout this proof to label the differential operator D with a suffix to indicate the variable with respect to which the differentiation is taking place. For example, Dx indicates a differentiation with respect to the x variable. Suppose r Ž h y . s 0 for all y g R n. Then, using Theorem 1.1 to identify r with the set of measures Ž ma ., 0 s rŽhy. s
Ý
a G0 < a
HVD
a x hy
d ma Ž x . .
Differentiating this equation with respect to y and applying Lemma 2.2, we get 0 s Dyb Ž r Ž h y . . s
Ý
Dyb
Ý
Dyb
a G0 < a
s
a G0 < a
s
HVD
a x hy
d ma Ž x .
yiŽ xyy .t
HVHR e n
< < Ý Ž y1. a H H
a G0 < a
V R
n
a
Ž yit . d m h Ž t . d ma Ž x .
eyiŽ xyy .t Ž it .
aq b
d m h Ž t . d ma Ž x . .
639
APPROXIMATION OF FUNCTIONS
Integrating with respect to mb for b G 0, < b < F r and then summing over this same set of indices gives < < Ý Ý Ž y1. a H H H
b G0 a G0 < b
V V Rn
eyiŽ xyy .t Ž it .
aq b
d m h Ž t . d ma Ž x . dmb Ž y . s 0.
Ž 4. Now, using Fubini’s theorem and Lemma 2.3,
HR < rˆŽ t . < n
2
dmh Ž t . s s
HR rˆŽ t . rˆŽ t . d m Ž t . h
n
½
Ž yit .
HR Ý n
¡ ¢
a G0 < a
~Ý
=
Ž it .
b G0 < b
s
b
a
yi x t
HVe
HVe
i yt
d ma Ž x .
5
¦¥ §
dmb Ž y . d m h Ž t .
< < Ý Ý Ž y1. a
aG0 b G0 < a
=
yiŽ xyy .t
HVHVHR e n
Ž it .
aq b
d m h Ž t . d ma Ž x . d mb Ž y .
s 0. Ž . by 4 . Since m h is a positive Borel measure, rˆŽ t . s 0 for m h-almost all t in R n. THEOREM 2.5. Let h g C m ŽR n . be a positi¨ e definite function on R n, with associated Borel measure m h . Let A s h Ž ?y y . : y g V 4 , and suppose r g Zq is such that 2 r F m. If the support of m h does not lie entirely within a set of analytic ¨ ariety, then the set A is fundamental in C r Ž V .. Proof. Suppose that r is a bounded, linear functional on C r Ž V . such that r Ž h y . s 0 for all y g V, where h y is defined by h y Ž x . s hŽ x y y ., x g R n. By Lemma 2.4, rˆŽ t . s 0 for m h-almost all t. Since the support of m h does not lie entirely within an analytic variety and Žby Lemma 2.3. rˆ is entire, we have rˆŽ t . s 0 for all t g R n. It follows that r is the trivial functional. Thus the set A is fundamental in C r Ž V ..
640
KILMER ET AL.
COROLLARY 2.6. Let h g P l C m ŽR n . with associated Borel measure m h . Let m h s m ha q m hs be the Lebesgue decomposition of m h into its absolutely continuous and singular parts, respecti¨ ely. Suppose m ha is not the zero measure and 2 r F m. Then the set A s hŽ?y y . : y g V 4 is fundamental in C r Ž V .. Proof. Let f be the Radon]Nikodym derivative of m ha . Then f g L R n ., and f Ž x . G 0 almost everywhere with respect to Lebesgue measure. Since m ha is not the zero measure, f is not the trivial element in L1 ŽR n .. Therefore, there exists a measurable set E in R n with positive Lebesgue measure such that f Ž x . ) 0 for all x g E. Since any analytic variety must have Lebesgue measure zero, it follows that m ha is not concentrated on an analytic variety. Thus, by Theorem 2.5, A is fundamental in C r Ž V .. 1Ž
COROLLARY 2.7. Let h g P l C m ŽR n .. Assume that ˆ h is a locally integrable function, and that there is a set E of positi¨ e Lebesgue measure such that ˆ hŽ x . / 0 for all x g E. Then the set hŽ?y y . : y g V 4 is fundamental in C r Ž V ., where 2 r F m. Proof. Since h g P, h is a bounded function, and hence a tempered distribution. Let h1 be the function defined by h1Ž x . s hŽyx ., for x g R n. $ $ Then in the distributional sense, h1s ˇ h s Žm h.ˇs m h . Hence, $
HR h Ž x . f Ž x . dx s HR f Ž x . d m Ž x . , n
1
n
h
for each f g S .
$
Thus h1 is the Radon]Nikodym derivative of the measure m h . Since $ h1Ž x . / 0 on the set E with positive Lebesgue measure, m ha Ž' m h . is not the zero measure. It follows from Corollary 2.6 that the set Ž h y : y g V 4 is fundamental in C r Ž V .. COROLLARY 2.8. Let h g P l C m ŽR n .. Assume that h is also in L p ŽR n . for some p with 1 F p F 2, and that 5 h 5 p ) 0. Then the set hŽ?y y . : y g V 4 is fundamental in C r Ž V ., for 2 r F m. Proof. If h g L p ŽR n . and if 5 h 5 p ) 0, then by the Hausdorff]Young theorem ŽRudin w5, p. 261x., ˆ h g Lq ŽR n ., where py1 q qy1 s 1, and there exists a set E of positive Lebesgue measure such that ˆ hŽ x . / 0 for x g E. The desired result now follows from Corollary 2.7.
APPROXIMATION OF FUNCTIONS
641
REFERENCES 1. A. L. Brown, Uniform approximation by radial basis functions, in ‘‘Advances in Numerical Analysis. Vol. 2. Wavelets, Subdivision Algorithms and Radial Basis Function’’ ŽW. A. Light, Ed.., Chap. 3, Appendix B, pp. 103]206, Oxford Univ. Press, London, 1992. 2. S. Bochner, ‘‘Vorlesungen ¨ uber Fouriersche Integrale,’’ Akademische Verlagsgesellschaft, Liepzig, 1932. 3. W. A Light, Some aspects of radial basis function approximation, in ‘‘Approximation Theory, Spline Functions and Applications’’ ŽS. P. Singh, Ed.., pp. 163]190, Kluwer Academic, Dordrecht, 1992. 4. C. A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx. 2 Ž1986., 11]22. 5. W. Rudin, ‘‘Functional Analysis,’’ 2nd ed., McGraw]Hill, New York, 1973.
201, 631]641 Ž1996.
0278
Approximation by Translates of a Positive Definite Function S. J. Kilmer Department of Mathematics, Southwest Missouri State Uni¨ ersity, Springfield, Missouri 65804
W. A. Light Department of Mathematics and Computer Science, Uni¨ ersity of Leicester, Uni¨ ersity Road, Leicester LE1 7RH, England
and X. Sun and X. M. Yu Department of Mathematics, Southwest Missouri State Uni¨ ersity, Springfield, Missouri 65804 Submitted by E. W. Cheney Received December 16, 1994
1. INTRODUCTION There are now many papers dealing with approximation of real-valued functions by linear combinations of a single function. At its simplest, this approach consists of selecting a function h: R n ª R in some appropriate function class and points a1 , . . . , a m g R n. The approximating subspace is then H s span h Ž ?y a i . : i s 1, . . . , m4 . This subspace may now be used for the purposes of approximation and interpolation either over the whole of R n, or over some subset V of R n. Let us now make the discussion more concrete by assuming V is a compact subset of R n, and we are interested in approximation in the space C Ž V .. One of the first questions which arises is that of density. If the number of points is allowed to grow, and if these points somehow fill out 631 0022-247Xr96 $18.00 Copyright Q 1996 by Academic Press, Inc. All rights of reproduction in any form reserved.
632
KILMER ET AL.
the whole of V, then can we approximate any function f g C Ž V . to arbitrary accuracy by functions in H ? A neat way to pose this problem is to set H s span h Ž ?y y . : y g V 4 , and to ask whether H is dense in C Ž V ., where the density is measured in the supremum norm. An important special case of the above is as follows. Let 5 ? 5 2 denote the usual Euclidean norm in R n. Let f g C ŽR., and set hŽ x . s f Ž5 x 5 2 . for x g R n. Then h g C ŽR n . and is a radial function. The translates of such a function have very good properties with regard to interpolation ŽMicchelli w4x.. They also have good properties with regard to approximation ŽBrown w1x.. It is perhaps remarkable that both the interpolation and the density results depend on the function f possessing the same property of conditional positi¨ e definiteness. For a unified treatment of these two results, see Light w3x. Our purpose here is to investigate density properties of translates of a single function, but our density will be measured with a Sobolev norm. Let V be a compact subset of R n. Let r g Zq and define C r Ž V . s f : V ª R : D a f g C Ž V . whenever < a < F r 4 . There are several equivalent ways of imposing a norm on this space; we will use 5f5s
Ý
5 D a f 5` ,
f g CrŽ V. .
n agZ q
< a
The situation considered by Brown can be regarded as studying the r s 0 case. Our results will show that simultaneous approximation of the function and its derivatives is also possible. We conclude this section with some notation and a few elementary observations. We will use standard multi-index notation throughout. The set of all positive definite functions on R n will be denoted by P. Thus f g P if the expression m
Ý
ci c j f Ž x i y x j . G 0
i , js1
whenever x 1 , . . . , x m g R n and c1 , . . . , c m g R. Let M Ž V . denote the set of all complex, regular Borel measures on V, and M the corresponding measures on the whole of R n. For f g L1 ŽR n .,
APPROXIMATION OF FUNCTIONS
633
the Fourier transform of f, denote by fˆor F Ž f ., is the continuous function on R n defined by fˆŽ x . s
yi x t
HR f Ž t . e n
x g R n.
dt,
The inverse Fourier transform fˇ of f is defined by fˇŽ x . s Ž 2p .
yn
HR f Ž t . e
i xt
n
dt,
x g R n.
We alert the reader that both the Fourier transform and its inverse are to be interpreted in the distributional sense whenever necessary. We use S to denote the space of infinitely differentiable and rapidly decreasing functions, and S X to denote the space of tempered distributions Žcontinuous, linear functionals on S .. It is well known that the Fourier transform is a one-to-one, continuous operator from S X onto S X . Bochner’s theorem w2x asserts that a continuous function h is positive definite on R n if and only if h is the Fourier transform of a unique positive measure m g M, that is, hŽ x . s
yi x t
HR e n
dm Ž t . ,
x g R n.
The correspondence between m and h will sometimes be made explicit by the notation m h . The Lebesgue decomposition theorem tells us that each m g M can be written in a unique way as
m s ma q ms , where m a is absolutely continuous, and m s is singular with respect to Lebesgue measure on R n. Let X be a normed linear space, and A a subset of X. If the linear span of A is dense in X, we say that A is fundamental in X. Part of our work depends on a representation theorem for linear functionals on C r Ž V .. Let l s a a g Z n : a G 0 and < a < F r 4 , and let X be the l-fold Cartesian product of C Ž V . with itself. Define 5F5 s
Ý
aG0 < a
5 Fa 5 ` ,
where f s Ž Fa . aG0, < a
There is an obvious isometric embedding of C r Ž V . into X, which associates f g C r Ž V . with F g X via the rule Fa s D a f. Given a continuous,
634
KILMER ET AL.
linear functional r on C r Ž V ., this functional can be extended to a continuous, linear functional on X by the Hahn]Banach theorem. Using the Riesz representation theorem in a coordinate fashion, we can find measures Ž ma .a G 0, < a < F r such that ma g M Ž V ., and so that the action of r on any element x g X is
rŽ x. s
Ý
Hx
a )0 V < a
d ma .
a
This establishes the following result. THEOREM 1.1. Let r be a continuous, linear functional on C r Ž V ., where V is a compact subset of R n. Then for each a g Z n, with a G 0 and < a < F r there exist ma g M Ž V ., such that
rŽ f . s
Ý
a G0 < a
HVD
a
f d ma ,
f g CrŽ V. .
2. RESULTS We begin by examining some properties of derivatives of positive definite functions. We say that a multi-index a s Ž a 1 , . . . , a n . is e¨ en if each a j is a positive even integer, or is zero. THEOREM 2.1. Let r s 0, 2, 4, . . . , and let h g P l C r ŽR n .. For a g Z n, a G 0, let Va be the function defined by Va Ž t . s t a , t g R n. Then Va g L1 Ž m h . for each a with < a < F r. Proof. Let f be the Gauss kernel which is defined by
f Ž x . s Ž 2p .
yn r2
exp y5 x 5 22r2 4 ,
x g R n.
For each m g N and x g R n, define fmŽ x . s m nf Ž mx .. Since m h is a $ tempered distribution, and h s m h, it follows that D a h is a tempered n distribution for each a g Zq . Let ² ? , ? : denote the duality bracket X between S and S . For < a < F r, we have ² D a h, fm : s Ž y1 . < a < ² h, D afm : $
s Ž y1 .
²m h , D afm :
s Ž y1 .
² m h , D afm :
s Ž y1 .
² m h , i < a < Vafm :
s Ž y1 .
$
$
HR t n
a
$
fm d m h .
Ž 1.
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APPROXIMATION OF FUNCTIONS
It is easy to see that lim m ª` fm s d in the topology of tempered distributions, where d is the Dirac distribution. Thus, by continuity, we have lim ² D a h, fm : s Ž D a h . Ž 0 . .
Ž 2.
mª`
At this juncture, it is convenient to break the proof into two cases. Case 1. The Multi-index$ a Is E¨ en In this case, we have Vafm Va . Hence, by the Monotone Convergence theorem, $
H Vf mª` R lim
n
a
m
du h s
HR
n
lim Va fm d m h s
mª`
dmh .
HR V
a
n
Therefore, by Ž1. and Ž2., we have
HR V n
a
$
d m h s lim
H Vf mª` R n
a
m
d mh s i < a < Ž D a h . Ž 0. .
This implies that Va g L1 Ž m h .. Case 2. The Multi-index a Is Not E¨ en In this case we write a kq 1 ??? t na n . Va Ž t . s t 1a i ??? t ka k t kq1
Without loss of generality, we assume that a 1 , . . . , a k are odd and a kq 1 , . . . , a n are even or zero. We may further assume that a 1 F a j for all 1 F j F k. Under these assumptions, Va Ž t . s Vb Ž t . Ž t 1 . . . t k .
a1
,
where b is even multi-index defined by
b s Ž 0, a 2 y a 1 , . . . , a k y a 1 , a kq1 , . . . , a n . . Again, it is helpful to distinguished two subcases. Subcase 1.
< a < is even. Then k is even also, and we have < Ž t1 . . . t k .
a1
a
< s < Ž t1 t 2 . 1 Ž t 3 t4 . F
ž
t 12 q t 22 2
a1
. . . Ž t ky1 t k .
a1
???
/ ž
2 t ky1 q t k2
2
a1
<
a1
/
.
Therefore, < Va Ž t . < F Vb Ž t . ? s
1
ž / 2
ž
t 12 q t 22 2
k a 1 r2
Ý tg , g
a1
/ ž ...
2 t ky1 q t k2
2
a1
/
636
KILMER ET AL.
where the summation is taken over a finite number of g for which easy g is even and satisfies < g < s < a < F r. It follows from the result in Case 1 that Va g L1 Ž m h .. Subcase 2.
< a < is odd. Then k is odd also, and so
< Ž t1 . . . t k .
a1
ž
t 12 q t 22 2 1
a1
???
/ ž
Ž ky1 . a 1 r2
ž / 2
2 2 t ky2 q t ky1
2
a1
/
Ž 1 q t ka q1 . 1
Ý td, d
where the summation over d is finite, each d is even, and satisfies < d < F < a < q 1. Since r is even, < a < is odd, and < a < F r we have < d < F r. Again, Case 1 may be used to show that Va g L1 Ž m h .. Let h g P l C m ŽR n ., where m s 2 r. For each a with
LEMMA 2.2. < a < F m,
a
Ž D a h . Ž x . s H neyi x t Ž yit . d m h Ž t . , R
x g R n.
Proof. We will calculate D a h, where a s Ž1, 0, . . . , 0., since all the calculations needed for the other values of a with < a < s 1 are similar, and a straightforward induction argument gives the result for 1 - < a < F m. Since h is positive definite, there exists a positive measure m h in M such that hŽ x . s
yi x t
HR e n
dmh Ž t . ,
x g R n.
Set a s Ž1, 0, . . . , 0.. By the Mean-Value theorem, 1 k
exp yi Ž x 1 q k, x 2 , . . . , x n . t 4 y exp yixt 4 s yit exp yi Ž j , x 2 , . . . , x n . t 4 s < t < < a < ,
637
APPROXIMATION OF FUNCTIONS
where j lies between x 1 and x 1 q k. By Theorem 2.1, t a g L1 Ž m h . and so, by the Lebesgue Dominated Convergence theorem, we obtain 1
Ž D a h . Ž x . s lim
½H
k
kª0
exp yi Ž x 1 q k, x 2 , . . . , x n . t 4 d m h Ž t .
Rn
y
yi x t
HR e
s
HR
s
HR e
s
HR e
n
1
lim kª0
k
Ž yit1 . d m h Ž t .
yi x t
Ž yit . d m h Ž t . .
n
dmh Ž t .
5
exp yi Ž x 1 q k, x 2 , . . . , x n . t 4 y eyi x t 4 d m h Ž t .
yi x t
n
n
a
For each bounded, linear functional r on C r Ž V ., it follows from Theorem 1.1 that there exists ma g M Ž V ., < a < F r, such that
rŽ f . s
Ý
HD
a
< a
f d ma ,
f g CrŽ V. .
We may regard r as a tempered distribution and calculate its Fourier transform. LEMMA 2.3. For each bounded, linear functional r on C r Ž V . with associated measures ma g M Ž V ., < a < F r, its Fourier transform rˆ is gi¨ en by
rˆ Ž t . s
a Ý Ž yit . H eyi x t d ma Ž x . ,
V
< a
t g R n.
Ž 3.
Moreo¨ er, rˆ is entire. Proof. Let S denote the set of all infinitely differentiable, rapidly decreasing functions on R n. For each f g S , we have, by Theorem 1.1, ² rˆ, f : s ² r , fˆ: s
HV Ž D fˆ . d m
Ý
HVHR
< a
s
a
Ý
< a
a
a
n
eyi x t Ž yit . f Ž t . dt d ma Ž x . .
638
KILMER ET AL.
Using Fubini’s theorem to interchange the order of integration, we obtain
² rˆ, f : s
Ž yit .
HR Ý n
a
yi x t
HVe
< a
d ma Ž x . f Ž t . dt.
This shows that Ž3. is true. Since r is a compactly supported distribution, it follows from the Paley]Wiener theorem that rˆ is entire. The next lemma contains the heart of the result we are aiming for. LEMMA 2.4. Let r be a bounded, linear functional on C r Ž V .. Let h g P l C m Ž V ., where m s 2 r. For each y g R n, define h y Ž x . s hŽ x y y ., x g R n. If r Ž h y . s 0 for all y g V, then rˆŽ x . s 0 for m h-almost all x g R n. Proof. It will be helpful throughout this proof to label the differential operator D with a suffix to indicate the variable with respect to which the differentiation is taking place. For example, Dx indicates a differentiation with respect to the x variable. Suppose r Ž h y . s 0 for all y g R n. Then, using Theorem 1.1 to identify r with the set of measures Ž ma ., 0 s rŽhy. s
Ý
a G0 < a
HVD
a x hy
d ma Ž x . .
Differentiating this equation with respect to y and applying Lemma 2.2, we get 0 s Dyb Ž r Ž h y . . s
Ý
Dyb
Ý
Dyb
a G0 < a
s
a G0 < a
s
HVD
a x hy
d ma Ž x .
yiŽ xyy .t
HVHR e n
< < Ý Ž y1. a H H
a G0 < a
V R
n
a
Ž yit . d m h Ž t . d ma Ž x .
eyiŽ xyy .t Ž it .
aq b
d m h Ž t . d ma Ž x . .
639
APPROXIMATION OF FUNCTIONS
Integrating with respect to mb for b G 0, < b < F r and then summing over this same set of indices gives < < Ý Ý Ž y1. a H H H
b G0 a G0 < b
V V Rn
eyiŽ xyy .t Ž it .
aq b
d m h Ž t . d ma Ž x . dmb Ž y . s 0.
Ž 4. Now, using Fubini’s theorem and Lemma 2.3,
HR < rˆŽ t . < n
2
dmh Ž t . s s
HR rˆŽ t . rˆŽ t . d m Ž t . h
n
½
Ž yit .
HR Ý n
¡ ¢
a G0 < a
~Ý
=
Ž it .
b G0 < b
s
b
a
yi x t
HVe
HVe
i yt
d ma Ž x .
5
¦¥ §
dmb Ž y . d m h Ž t .
< < Ý Ý Ž y1. a
aG0 b G0 < a
=
yiŽ xyy .t
HVHVHR e n
Ž it .
aq b
d m h Ž t . d ma Ž x . d mb Ž y .
s 0. Ž . by 4 . Since m h is a positive Borel measure, rˆŽ t . s 0 for m h-almost all t in R n. THEOREM 2.5. Let h g C m ŽR n . be a positi¨ e definite function on R n, with associated Borel measure m h . Let A s h Ž ?y y . : y g V 4 , and suppose r g Zq is such that 2 r F m. If the support of m h does not lie entirely within a set of analytic ¨ ariety, then the set A is fundamental in C r Ž V .. Proof. Suppose that r is a bounded, linear functional on C r Ž V . such that r Ž h y . s 0 for all y g V, where h y is defined by h y Ž x . s hŽ x y y ., x g R n. By Lemma 2.4, rˆŽ t . s 0 for m h-almost all t. Since the support of m h does not lie entirely within an analytic variety and Žby Lemma 2.3. rˆ is entire, we have rˆŽ t . s 0 for all t g R n. It follows that r is the trivial functional. Thus the set A is fundamental in C r Ž V ..
640
KILMER ET AL.
COROLLARY 2.6. Let h g P l C m ŽR n . with associated Borel measure m h . Let m h s m ha q m hs be the Lebesgue decomposition of m h into its absolutely continuous and singular parts, respecti¨ ely. Suppose m ha is not the zero measure and 2 r F m. Then the set A s hŽ?y y . : y g V 4 is fundamental in C r Ž V .. Proof. Let f be the Radon]Nikodym derivative of m ha . Then f g L R n ., and f Ž x . G 0 almost everywhere with respect to Lebesgue measure. Since m ha is not the zero measure, f is not the trivial element in L1 ŽR n .. Therefore, there exists a measurable set E in R n with positive Lebesgue measure such that f Ž x . ) 0 for all x g E. Since any analytic variety must have Lebesgue measure zero, it follows that m ha is not concentrated on an analytic variety. Thus, by Theorem 2.5, A is fundamental in C r Ž V .. 1Ž
COROLLARY 2.7. Let h g P l C m ŽR n .. Assume that ˆ h is a locally integrable function, and that there is a set E of positi¨ e Lebesgue measure such that ˆ hŽ x . / 0 for all x g E. Then the set hŽ?y y . : y g V 4 is fundamental in C r Ž V ., where 2 r F m. Proof. Since h g P, h is a bounded function, and hence a tempered distribution. Let h1 be the function defined by h1Ž x . s hŽyx ., for x g R n. $ $ Then in the distributional sense, h1s ˇ h s Žm h.ˇs m h . Hence, $
HR h Ž x . f Ž x . dx s HR f Ž x . d m Ž x . , n
1
n
h
for each f g S .
$
Thus h1 is the Radon]Nikodym derivative of the measure m h . Since $ h1Ž x . / 0 on the set E with positive Lebesgue measure, m ha Ž' m h . is not the zero measure. It follows from Corollary 2.6 that the set Ž h y : y g V 4 is fundamental in C r Ž V .. COROLLARY 2.8. Let h g P l C m ŽR n .. Assume that h is also in L p ŽR n . for some p with 1 F p F 2, and that 5 h 5 p ) 0. Then the set hŽ?y y . : y g V 4 is fundamental in C r Ž V ., for 2 r F m. Proof. If h g L p ŽR n . and if 5 h 5 p ) 0, then by the Hausdorff]Young theorem ŽRudin w5, p. 261x., ˆ h g Lq ŽR n ., where py1 q qy1 s 1, and there exists a set E of positive Lebesgue measure such that ˆ hŽ x . / 0 for x g E. The desired result now follows from Corollary 2.7.
APPROXIMATION OF FUNCTIONS
641
REFERENCES 1. A. L. Brown, Uniform approximation by radial basis functions, in ‘‘Advances in Numerical Analysis. Vol. 2. Wavelets, Subdivision Algorithms and Radial Basis Function’’ ŽW. A. Light, Ed.., Chap. 3, Appendix B, pp. 103]206, Oxford Univ. Press, London, 1992. 2. S. Bochner, ‘‘Vorlesungen ¨ uber Fouriersche Integrale,’’ Akademische Verlagsgesellschaft, Liepzig, 1932. 3. W. A Light, Some aspects of radial basis function approximation, in ‘‘Approximation Theory, Spline Functions and Applications’’ ŽS. P. Singh, Ed.., pp. 163]190, Kluwer Academic, Dordrecht, 1992. 4. C. A. Micchelli, Interpolation of scattered data: Distance matrices and conditionally positive definite functions, Constr. Approx. 2 Ž1986., 11]22. 5. W. Rudin, ‘‘Functional Analysis,’’ 2nd ed., McGraw]Hill, New York, 1973.