characterization and construction of radial basis …...characterization and construction of radial...
TRANSCRIPT
Characterization and construction of radial
basis functions
Robert Schaback and Holger Wendland
March 21, 2000
Abstract
We review characterizations of (conditional) positive definitenessand show how they apply to the theory of radial basis functions. Wethen give complete proofs for the (conditional) positive definitenessof all practically relevant basis functions. Furthermore, we show howsome of these characterizations may lead to construction tools forpositive definite functions. Finally, we give new construction tech-niques based on discrete methods which lead to non-radial, even non-translation invariant, local basis functions.
1 Introduction
Radial basis functions are an efficient tool for solving multivariate scattereddata interpolation problems. To interpolate an unknown function f ∈ C(Ω)whose values on a set X = x1, . . . , xN ⊂ Ω ⊂ IRd are known, a function ofthe form
sf,X(x) =N∑
j=1
αjΦ(x, xj) + p(x) (1)
is chosen, where p is a low degree polynomial and Φ : Ω × Ω → IR is a fixedfunction. The numerical treatment can be simplified in the special situations
1. Φ(x, y) = φ(x − y) with φ : IRd → IR (translation invariance),
2. Φ(x, y) = φ(‖x − y‖2) with φ : [0,∞) → IR (radiality),
and this is how the notion of radial basis functions arose. The most prominentexamples of radial basis functions are:
φ(r) = rβ, β > 0, β 6∈ 2IN,
1
φ(r) = r2k log(r), k ∈ IN (thin-plate splines),
φ(r) = (c2 + r2)β , β < 0, (inverse multiquadrics)
φ(r) = (c2 + r2)β , β > 0, β 6∈ IN (multiquadrics)
φ(r) = e−αr2
, α > 0 (Gaussians),
φ(r) = (1 − r)4+(1 + 4r).
All of these basis functions can be uniformly classified using the concept of(conditionally) positive definite functions:
Definition 1.1 A continuous function Φ : Ω × Ω → C is said to be con-ditionally positive (semi–) definite of degree m on Ω if for all N ∈ IN , alldistinct x1, . . . , xN ∈ Ω, and all α ∈ CN \ 0 satisfying
N∑
j=1
αjp(xj) = 0 (2)
for all polynomials p of degree less than m the quadratic form
N∑
j=1
N∑
k=1
αjαkΦ(xj , xk) (3)
is positive (nonnegative). The function Φ is positive definite if it is condi-tionally positive definite of order m = 0.
Note that in case of a positive definite function the conditions (2) are emptyand hence (3) has to be positive for all α ∈ CN \ 0. Finally, if Φ is asymmetric real-valued function, it is easy to see that it suffices to test onlyreal α.
The use of this concept in the context of multivariate interpolation problemsis explained in the next theorem, which also shows the connection betweenthe degree of the polynomial p in (1) and the order m of conditional positivedefiniteness of the basis function Φ. We will denote the space of d-variatepolynomials of degree at most m by πm(IRd).
Theorem 1.2 Suppose Φ is conditionally positive definite of oder m onΩ ⊆ IRd. Suppose further that the set of centers X = x1, . . . , xN ⊆ Ωis πm−1(IR
d) unisolvent, i.e. the zero polynomial is the only polynomial fromπm−1(IR
d) that vanishes on X. Then for given f1, . . . , fN there is exactlyone function sf,X of the form (1) with a polynomial p ∈ πm−1(IR
d) such thatsf,X(xj) = fj, 1 ≤ j ≤ N and
∑Nj=1 αjq(xj) = 0 for all q ∈ πm−1(IR
d).
It is the goal of this paper to give full proofs for the conditional positivedefiniteness of all aforementioned radial basis functions and to use the ideasbehind these proofs to construct new ones. We only rely on certain analyticaltools that are not directly related to radial basis functions.
2
2 The Schoenberg-Micchelli Characterization
Given a continuous univariate function φ : [0,∞) → IR we can form thefunction Φ(x, y) := φ(‖x − y‖2) on IRd × IRd for arbitrary space dimensiond. Then we can say that φ is conditionally positive definite of order m onIRd, iff Φ is conditionally positive definite of order m on IRd in the sense ofDefinition 1.1.
Taking this point of view, we are immediately led to the question ofwhether a univariate function φ is conditionally positive definite of someorder m on IRd for all d ≥ 1. This question was fully answered in the positivedefinite case by Schoenberg [16] in 1938 in terms of completely monotonefunctions. In the case of conditionally positive definite functions Micchelli[12] generalized the sufficient part of Schoenberg’s result, suspecting that itwas also necessary. This was finally proved by Guo, Hu and Sun [9].
Definition 2.1 A function φ : (0,∞) → IR is said to be completely mono-tone on (0,∞) if φ ∈ C∞(0,∞) and
(−1)ℓφ(ℓ)(r) ≥ 0, ℓ ∈ IN0, r > 0. (4)
A function φ : [0,∞) → IR is said to be completely monotone on [0,∞) if itis completely monotone on (0,∞) and continuous at zero.
Theorem 2.2 (Schoenberg) Suppose φ : [0,∞) → IR is not the constantfunction. Then φ is positive definite on every IRd if and only if the functiont 7→ φ(
√t), t ∈ [0,∞) is completely monotone on [0,∞).
Schoenberg’s characterisation of positive definite functions allows us to provepositive definiteness of Gaussians and inverse multiquadrics without diffi-culty:
Theorem 2.3 The Gaussians φ(r) = e−αr2, α > 0, and the inverse multi-
quadrics φ(r) = (c2 + r2)β, c > 0, β < 0, are positive definite on IRd for alld ≥ 1.
Proof: For the Gaussians note that
f(r) := φ(√
r) = e−αr
satisfies (−1)ℓf (ℓ)(r) = αℓe−αr > 0 for all ℓ ∈ IN0 and α, r > 0. Similarly, forthe inverse multiquadrics we find with f(r) := φ(
√r) = (c2 + r)−|β| that
(−1)ℓf (ℓ)(r) = (−1)2ℓ|β|(|β|+ 1) · . . . · (|β| + ℓ − 1)(r + c2)−|β|−ℓ > 0.
3
Since in both cases φ is not the constant function, the Gaussians and inversemultiquadrics are positive definite. 2
There are several other characterizations of completely monotone func-tions (see [19]), which by Schoenberg’s theorem also apply to positive definitefunctions. The most important is the following one by Bernstein (see Widder[19]). It implies that the proper tool to handle positive definite functions onIRd for all d ≥ 1, is the Laplace transform.
Theorem 2.4 (Bernstein) A function φ is positive definite on IRd for alld ≥ 1, if and only if there exists a nonzero, finite, nonnegative Borel measureµ, not supported in zero, such that φ is of the form
φ(r) =∫ ∞
0e−r2tdµ(t). (5)
Note that the sufficient part of Bernstein’s theorem is easy to prove, if weknow that the Gaussians are positive definite. For every α ∈ IRN \ 0 andevery distinct x1, . . . , xN ∈ IRd the quadratic form is given by
N∑
j,k=1
αjαkφ(‖xj − xk‖2) =∫ ∞
0
∣∣∣∣∣∣
N∑
j=1
αje−t‖xj−xk‖
22
∣∣∣∣∣∣
2
dµ(t).
Another consequence of this theory is the following.
Theorem 2.5 Suppose φ : [0,∞) → IR is positive definite on IRd for alld ≥ 1. Then φ has no zero. In particular, there exists no compactly supportedunivariate function that is positive definite on IRd for all d ≥ 1.
Proof: Since φ is positive definite on IRd for all d ≥ 1, there exists a finite,nonzero, nonnegative Borel measure µ on [0,∞) such that (5) holds. If r0 isa zero of φ this gives
0 =∫ ∞
0e−r2
0tdµ(t).
Since the measure is non-negative and the weight function e−r20t is positive
we find that the measure must be the zero measure. 2
Thus the compactly supported function φ(r) = (1−r)4+(1+4r) given in the
introduction cannot be positive definite on IRd for all d ≥ 1, and it is actuallyonly positive definite on IRd, d ≤ 3. If one is interested in constructing basisfunctions with compact support, one has to take the above negative resultinto account. We shall see in the next section that the Fourier transform is
4
the right tool to handle positive definite translation–invariant functions onIRd with a prescribed d. But before that, let us have a look at conditionallypositive definite functions. We will state only the sufficient part as providedby Micchelli [12].
Theorem 2.6 (Micchelli) Given a function φ ∈ C[0,∞), define f = φ(√·).
If there exists an m ∈ IN0 such that (−1)mf (m) is well–defined and completelymonotone on (0,∞), then φ is conditionally positive semi-definite of orderm on IRd for all d ≥ 1. Furthermore, if f is not a polynomial of degree atmost m, then φ is conditionally positive definite.
This theorem allows us to classify all functions from the introduction,with the sole exception of the compactly supported one. However, to complywith the notion of conditional positive definiteness, we shall have to adjustthe signs properly. To do this we denote the smallest integer greater than orequal to x by ⌈x⌉ .
Theorem 2.7 The multiquadrics φ(r) = (−1)⌈β⌉(c2+r2)β, c, β > 0, β 6∈ IN ,are conditionally positive definite of order m ≥ ⌈β⌉ on IRd for all d ≥ 1.
Proof: If we define fβ(r) = (−1)⌈β⌉(c2 + r)β, we find
f(k)β (r) = (−1)⌈β⌉β(β − 1) · . . . · (β − k + 1)(c2 + r)β−k,
which shows that (−1)⌈β⌉f(⌈β⌉)β (r) = β(β − 1) · . . . · (β −⌈β⌉+ 1)(c2 + r)β−⌈β⌉
is completely monotone, and that m = ⌈β⌉ is the smallest possible choice ofm to make (−1)mf (m) completely monotone. 2
Theorem 2.8 The functions φ(r) = (−1)⌈β/2⌉rβ, β > 0, β 6∈ 2IN , areconditionally positive definite of order m ≥ ⌈β/2⌉ on IRd for all d ≥ 1.
Proof: Define fβ(r) = (−1)⌈β
2⌉r
β
2 to get
f(k)β (r) = (−1)⌈
β
2⌉β
2(β
2− 1) · · · (β
2− k + 1)r
β
2−k.
This shows that (−1)⌈β
2⌉f
(⌈β
2⌉)
β (r) is completely monotone and m = ⌈β2⌉ is the
smallest possible choice. 2
Theorem 2.9 The thin–plate or surface splines φ(r) = (−1)k+1r2k log(r)are conditionally positive definite of order m = k + 1 on every IRd.
5
Proof: Since 2φ(r) = (−1)k+1r2k log(r2) we set fk(r) = (−1)k+1rk log(r).Then it is easy to see that
f(ℓ)k (r) = (−1)k+1k(k − 1) · · · (k − ℓ + 1)rk−ℓ log(r) + pℓ(r), 1 ≤ ℓ ≤ k,
where pℓ is a polyonmial of degree k − ℓ. This means in particular
f(k)k (r) = (−1)k+1k! log(r) + c
and finally (−1)k+1f(k+1)k (r) = k!r−1 which is obviously completely monotone
on (0,∞). 2
3 Bochner’s Characterization
We saw in the last section that the Laplace transform is the right tool foranalyzing positive definiteness of radial functions for all space dimensionsd. However, we did not prove Schoenberg’s and Micchelli’s theorems. Wealso saw that the approach via Laplace transforms excludes functions withcompact support, which are desirable from a numerical point of view. Toovercome this problem and to work around these theorems, we shall nowlook at translation–invariant positive definite functions on IRd for some fixedd. We shall give the famous result of Bochner [2, 3], which characterizestranslation–invariant positive definite functions via Fourier transforms. Inthe next section we generalize this result to handle also translation–invariantconditionally positive definite functions, following an approach of Madychand Nelson [11]. Of course, we define a continuous function Φ : IRd → Cto be a translation–invariant conditionally positive (semi–) definite functionof order m on IRd iff Φ0(x, y) := Φ(x − y) is conditionally positive (semi–)definite of order m on IRd.
Theorem 3.1 (Bochner) A continuous function Φ : IRd → C is a translation–invariant positive semi-definite function if and only if it is the inverse Fouriertransform of a finite non-negative Borel measure µ on IRd, i.e.,
Φ(x) = µ∨(x) = (2π)−d/2∫
IRdeixT ωdµ(ω), x ∈ IRd. (6)
Again, the sufficient part is easy since
N∑
j,k=1
αjαkΦ(xj − xk) =∫
IRd
∣∣∣∣∣∣
N∑
j=1
αjeixT
jω
∣∣∣∣∣∣
2
dµ(ω), (7)
6
and later we shall use this argument repeatedly to prove positive definitenessof certain functions without referring to Bochner’s theorem. In the Fouriertransform setting it is not straightforward to separate positive definite frompositive semi-definite functions as it was in Schoenberg’s characterization.But since the exponentials are linear independent on every open supset ofIRd, we have
Corollary 3.2 Suppose that the carrier of the measure µ of Theorem 3.1contains an open subset of IRd. Then Φ is a translation–invariant positivedefinite function.
For a complete classification of positive definite functions via Bochner’stheorem see [7, 8]. Here, we want to cite a weaker formulation, which weshall not use for proving positive definiteness of special functions. A proofcan be found in [18].
Theorem 3.3 Suppose Φ ∈ L1(IRd) is a continuous function. Then Φ is
a translation–invariant positive definite function if and only if Φ is boundedand its Fourier transform is nonnegative and not identically zero.
Since a non–identically zero function cannot have an identically zeroFourier transform, we see that an integrable, bounded function that is notidentically zero Φ is translation–invariant and positive definite if its Fouriertransform is nonnegative. This can be used to prove the positive definitenessof the Gaussian along the lines of the sufficiency argument for Theorem 3.1.Since this is easily done via (7), we skip over the details and only remarkthat
Φ(x) = e−α‖x‖22
has the Fourier transform
Φ(ω) = (2π)−d/2∫
IRdΦ(x)e−ixT ωdx = (2α)−d/2e−‖ω‖2
2/(4α). (8)
This allows us to circumvent Schoenberg’s and Bochner’s theorem for a directproof of the positive definiteness of the Gaussians (see also Powell [13]).
Now let us have a closer look at the Fourier transform of the inversemultiquadrics. To do this let us recall the definition of the modified Besselfunctions. For z ∈ C with |arg(z)| < π/2 they are given by
Kν(z) :=∫ ∞
0e−z cosh t cosh νtdt.
7
Theorem 3.4 The function Φ(x) = (c2 + ‖x‖22)
β, x ∈ IRd, with c > 0 andβ < −d/2 is a translation–invariant positive definite function with Fouriertransform
Φ(ω) =21+β
Γ(−β)
(‖ω‖2
c
)−β− d2
K d2+β(c‖ω‖2).
Proof: Since β < −d/2 the function Φ is in L1(IRd). From the representation
of the Gamma function for −β > 0 we see that
Γ(−β) =∫ ∞
0t−β−1e−tdt
= s−β∫ ∞
0u−β−1e−sudu
by substituting t = su with s > 0. Setting s = c2 + ‖x‖22 this implies
Φ(x) =1
Γ(−β)
∫ ∞
0u−β−1e−c2ue−‖x‖2
2udu. (9)
Inserting this into the Fourier transform and changing the order of integra-tion, which can be easily justified, leads to
Φ(x) = (2π)−d/2∫
IRdΦ(ω)e−ixT ωdω
= (2π)−d/2 1
Γ(−β)
∫
IRd
∫ ∞
0u−β−1e−c2ue−‖ω‖2
2ue−ixT ωdudω
= (2π)−d/2 1
Γ(−β)
∫ ∞
0u−β−1e−c2u
∫
IRde−‖ω‖2
2ue−ixT ωdωdu
=1
Γ(−β)
∫ ∞
0u−β−1e−c2u(2u)−d/2e−
‖x‖22
4u du
=1
2d/2Γ(−β)
∫ ∞
0u−β− d
2−1e−c2ue−
‖x‖22
4u du, (10)
where we have used (8). On the other hand we can conclude from the defi-nition of the modified Bessel function that for every a > 0
Kν(r) =1
2
∫ ∞
−∞e−r cosh teνtdt
=1
2
∫ ∞
−∞e−
r2(et+e−t)eνtdt
= a−ν 1
2
∫ ∞
0e−
r2( s
a+ a
s)sν−1ds
8
by substituting s = aet. If we now set r = c‖x‖2, a = ‖x‖2/(2c), andν = −β − d/2 we derive
K−β− d2(c‖x‖2) =
1
2
(‖x‖2
2c
) d2+β ∫ ∞
0e−sc2e−
‖x‖224s s−β− d
2−1ds
= 2−β−1Γ(−β)
(‖x‖2
c
)d2+β
Φ(x),
which leads to the stated Fourier transform using K−ν = Kν . Since the modi-fied Bessel function is non-negative and non-vanishing, the proof is complete.2
Note that this result is somewhat weaker than the result given in Theorem2.3, since we require β < −d/2 for integrability reasons. Furthermore, wecan read off from (9) the representing measure for Φ in the sense of Theorem2.4.
4 The Madych–Nelson approach
So far, we have seen that the Schoenberg-Micchelli approach is an elegantway to prove conditional positive definiteness of basis functions for all spacedimensions. But these characterization theorems are rather abstract, hardto prove, and restricted to globally supported and radial basis functions.
On the other hand, Bochner’s characterization provides direct proofs fortranslation–invariant and possibly nonradial functions, but is not applicableto conditionally positive definite functions.
Thus in this section we follow Madych and Nelson [11] to generalize theapproach of Bochner to the case of conditionally positive definite translation–invariant functions. It will turn out that the proof of the basic result isquite easy, but it will be technically difficult to apply the general resultto specific basis functions. But our efforts will pay off by yielding explicitrepresentations of generalized Fourier transforms of the classical radial basisfunctions, and these are important for further study of interpolation errorsand stability results.
Recall that the Schwartz space S consists of all C∞(IRd)-functions thattogether with all their derivatives, decay faster than any polynomial.
Definition 4.1 For m ∈ IN0 the set of all functions γ ∈ S which satisfyγ(ω) = O(‖ω‖2m
2 ) for ‖ω‖2 → 0 will be denoted by Sm.
9
Recall that a function Φ is called slowly increasing if there exists aninteger ℓ ∈ IN0 such that |Φ(ω)| = O(‖ω‖ℓ
2) for ‖ω‖2 → ∞.
Definition 4.2 Suppose Φ : IRd → C is continuous and slowly increasing.A continuous function Φ : IRd \0 → C is said to be the generalized Fouriertransform of Φ if there exists an integer m ∈ IN0 such that
∫
IRdΦ(x)γ(x)dx =
∫
IRdΦ(ω)γ(ω)dω
is satisfied for all γ ∈ Sm. The smallest of such m is called the order of Φ.
We omit the proof that the generalized Fourier transform is uniquely defined,but rather give a first nontrivial example:
Proposition 4.3 Suppose Φ = p is a polynomial of degree less than 2m.Then for every test function γ ∈ Sm we have
∫
IRdΦ(x)γ(x)dx = 0.
Proof: Suppose Φ has the representation Φ(x) =∑
|β|<2m cβxβ. Then
∫
IRdΦ(x)γ(x)dx =
∑
|β|<2m
cβi−|β|∫
IRd(ix)β γ(x)dx
=∑
|β|<2m
cβi−|β|∫
IRd
(∂|β|γ
∂xβ
)∧
dx
= (2π)d/2∑
|β|<2m
cβi−|β|∂
|β|γ
∂xβ(0)
= 0
since γ ∈ Sm. 2
Note that the above result implies that the “inverse” generalized Fouriertransform is not unique, because one can add a polynomial of degree less than2m to a function Φ without changing its generalized Fourier transform. Notefurther that there are other definitions of generalized Fourier transforms, e.g.in the context of tempered distributions.
The next theorem shows that the order of the generalized Fourier trans-form, which is nothing but the order of the singularity of the generalizedFourier transform at the origin, determines the minimal order of a condition-ally positive definite function, provided that the function has a nonnegativeand nonzero generalized Fourier transform. We will state and prove only thesufficient part, but point out that the reverse direction also holds. We needthe following auxiliary result:
10
Lemma 4.4 Suppose that distinct x1, . . . , xN ∈ IRd and α ∈ CN \ 0 aregiven such that (2) is satisfied for all p ∈ πm−1(IR
d). Then
N∑
j=1
αjeixT
jω = O(‖ω‖m
2 )
holds for ‖ω‖2 → 0.
Proof: The expansion of the exponential function leads to
N∑
j=1
αjeixT
jω =
∞∑
k=0
ik
k!
N∑
j=1
αj(xTj ω)k.
For fixed ω ∈ IRd we have pk(x) := (xT ω)k ∈ πk(IRd). Thus (2) ensures that
the first m − 1 terms vanish:N∑
j=1
αjeixT
jω =
∞∑
k=m
ik
k!
N∑
j=1
αj(xTj ω)k,
which yields the stated behaviour. 2
Theorem 4.5 Suppose Φ : IRd → C is continuous, slowly increasing, andpossesses a generalized Fourier transform Φ of order m which is non-negativeand non-vanishing. Then Φ is a translation–invariant conditionally positivedefinite function of order m.
Proof: Suppose that distinct x1, . . . , xN ∈ IRd and α ∈ CN \ 0 satisfy (2)for all p ∈ πm−1(IR
d). Define
f(x) :=N∑
j,k=1
αjαkΦ(x + (xj − xk))
and
γℓ(x) =
∣∣∣∣∣∣
N∑
j=1
αjeixT xj
∣∣∣∣∣∣
2
gℓ(x) =N∑
j,k=1
αjαkeixT (xj−xk)gℓ(x),
where gℓ(x) = (ℓ/π)d/2e−ℓ‖x‖22 . On account of γℓ ∈ S and Lemma 4.4 we have
γℓ ∈ Sm. Furthermore,
γℓ(x) = (2π)−d/2∫
IRd
N∑
j,k=1
αjαkeiωT (xj−xk)gℓ(ω)e−ixT ωdω
=N∑
j,k=1
αjαk(2π)−d/2∫
IRdgℓ(ω)e−iωT (x−(xj−xk))dω
=N∑
j,k=1
αjαkgℓ(x − (xj − xk)),
11
since gℓ = gℓ. Collecting these facts gives together with Definition 4.2
∫
IRdf(x)gℓ(x)dx =
∫
IRdΦ(x)
N∑
j,k=1
αjαkgℓ(x − (xj − xk))dx
=∫
IRdΦ(x)γℓ(x)dx
=∫
IRdΦ(ω)γℓ(ω)dω
=∫
IRd
∣∣∣∣∣∣
N∑
j=1
αjeiωT xj
∣∣∣∣∣∣
2
gℓ(ω)Φ(ω)dω
≥ 0.
Since Φ is only slowly increasing, we have
N∑
j,k=1
αjαkΦ(xj − xk) = limℓ→∞
∫
IRdf(x)gℓ(x)dx ≥ 0
by means of approximation by convolution. Furthermore, the quantity
∣∣∣∣∣∣
N∑
j=1
αjeiωT xj
∣∣∣∣∣∣
2
gℓ(ω)Φ(ω)
is non-decreasing in ℓ and we already know that the limit
limℓ→∞
∫
IRd
∣∣∣∣∣∣
N∑
j=1
αjeiωT xj
∣∣∣∣∣∣
2
gℓ(ω)Φ(ω)dω
exists. Hence, the limit function (2π)−d/2∣∣∣∑N
j=1 αjeiωT xj
∣∣∣2Φ(ω) is integrable
due to the monotone convergence theorem. Thus we have established theequality
N∑
j,k=1
αjαkΦ(xj − xk) = (2π)−d/2∫
IRd
∣∣∣∣∣∣
N∑
j=1
αjeiωT xj
∣∣∣∣∣∣
2
Φ(ω)dω.
This quadratic form cannot vanish if Φ is non-vanishing, since the exponen-tials are linearly independent. 2
12
5 Classical Radial Basis Functions
In order to use this generalization of the Bochner approach we now computethe generalized Fourier transforms of the most popular translation–invariantor radial basis functions. Since it will turn out that these generalized Fouriertransforms are non-negative and non-vanishing, we can read off the orderof conditional positive definiteness of the functions from the order of thesingularity of their generalized Fourier transforms at the origin.
We start with the positive definite inverse multiquadrics as treated inTheorem 3.4 and use analytic continuation to treat the case of the condition-ally positive definite (non–inverse) multiquadrics. To do this we need tworesults on the modified Bessel functions.
Lemma 5.1 The modified Bessel function Kν, ν ∈ C , has the uniform bound
|Kν(r)| ≤√
2π
re−re
|ℜ(ν)|22r , r > 0 (11)
describing its behaviour for large r.
Proof: With b = |ℜ(ν)| we have
|Kν(r)| ≤ 1
2
∫ ∞
0e−r cosh t|eνt + e−νt|dt
≤ 1
2
∫ ∞
0e−r cosh t(ebt + e−bt)dt
= Kb(r)
Furthermore, from et ≥ cosh t ≥ 1 + t2
2, t ≥ 0, we can conclude
Kb(r) ≤∫ ∞
0e−r(1+ t2
2)ebtdt
≤ e−reb2
2r1√r
∫ ∞
−b√r
e−s2/2ds
≤√
2πe−reb2
2r
√1
r.
2
Lemma 5.2 For ν ∈ C the modified Bessel function Kν satisfies
|Kν(r)| ≤
2|ℜ(ν)|−1Γ(|ℜ(ν)|)r−|ℜ(ν)|, ℜ(ν) 6= 0,1e− log r
2, r < 2,ℜ(ν) = 0.
(12)
for r > 0, describing its behaviour for small r.
13
Proof: Let us first consider the case ℜ(ν) 6= 0. We set again b = |ℜ(ν)| andalready know that |Kν(r)| ≤ Kb(r), from the proof of the preceding lemma.Furthermore, from the proof of Theorem 3.4 we get
Kb(r) =1
2
∫ ∞
0e−
r2( s
a+ a
s)(
s
a
)b ds
s
for every a > 0. By setting a = r/2 we see that
Kb(r) = 2b−1r−b∫ ∞
0e−se−
r2
4s sb−1ds ≤ 2b−1Γ(b)r−b.
For ℜ(ν) = 0 we use cosh t ≥ et/2 to derive
K0(r) =∫ ∞
0e−r cosh tdt
≤∫ ∞
0e−
r2et
dt
=∫ ∞
r2
e−u 1
udu
≤∫ ∞
1e−udu +
∫ 1
r2
1
udu
=1
e− log
r
2.
2
We are now able to compute the generalized Fourier transform of the generalmultiquadrics. The basic idea of the proof goes back to Madych and Nelson[11]. It starts with the classical Fourier transform of the inverse multiquadricsas given in Theorem 3.4, and then uses analytic continuation.
Theorem 5.3 The function Φ(x) = (c2 + ‖x‖22)
β, x ∈ IRd, with c > 0 andβ ∈ IR \ IN0 possesses the (generalized) Fourier transform
Φ(ω) =21+β
Γ(−β)
(‖ω‖2
c
)−β− d2
K d2+β(c‖ω‖2), ω 6= 0, (13)
of order m = max(0, ⌈β⌉).
Proof: Define G = λ ∈ C : ℜ(λ) < m and denote the right–hand side of(13) by ϕβ(ω). We are going to show by analytic continuation that
∫
IRdΦλ(ω)γ(ω)dω =
∫
IRdϕλ(ω)γ(ω)dω, γ ∈ Sm, (14)
14
is valid for all λ ∈ G, where Φλ(ω) = (c2 + ‖ω‖22)
λ. First, note that (14) isvalid for λ ∈ G with λ < −d/2 by Theorem 3.4, and in case m > 0, also forλ = 0, 1, . . . , m − 1, by Proposition 4.3 and the fact that 1/Γ(−λ) is zero inthese cases. Analytic continuation will lead us to our stated result, if we canshow that both sides of (14) exist and are analytic functions in λ. We willdo this only for the right–hand side, since the left–hand side can be handledmore easily. Thus we define
f(λ) =∫
IRdϕλ(ω)γ(ω)dω
and study this function of λ. Suppose C is a closed curve in G. Since ϕλ isan analytic function in λ ∈ G it has the representation
ϕλ(ω) =1
2πi
∫
C
ϕz(ω)
z − λdz
for λ ∈ Int C. Now suppose that we have already shown that the integrandin the definition of f(λ) can be bounded uniformly on C by an integrablefunction. This ensures that f(λ) is well defined in G and by Fubini’s theoremwe can conclude
f(λ) =∫
IRdϕλ(ω)γ(ω)dω
=1
2πi
∫
IRd
∫
C
ϕz(ω)
z − λdzγ(ω)dω
=1
2πi
∫
C
1
z − λ
∫
IRdϕz(ω)γ(ω)dωdz
=1
2πi
∫
C
f(z)
z − λdz
for λ ∈ Int C, which means that f is analytic in G. Thus it remains to boundthe integrand uniformly.
Let us first consider the asymptotic behaviour in a neighbourhood of theorigin, say for ‖ω‖2 < 1/c. If we set b = ℜ(λ) we can use Lemma 5.2 andγ ∈ Sm to get in the case b 6= −d/2:
|ϕλ(ω)γ(ω)| ≤ Cγ2b+|b+d/2|Γ(|b + d/2|)
|Γ(−λ)| cb+d/2−|b+d/2|‖ω‖−b−d/2−|b+d/2|+2m2 ,
and in case b = −d/2:
|ϕλ(ω)γ(ω)| ≤ Cγ21−d/2
|Γ(−λ)|
(1
e− log
c‖ω‖2
2
)
‖ω‖2m2 .
15
Since C is compact and 1/Γ is analytic, this gives for all λ ∈ C
|ϕλ(ω)γ(ω)| ≤ Cγ,m,c,C
(
1 + ‖ω‖−d+2ǫ2 − log
c‖ω‖2
2
)
, ‖ω‖2 ≤ 1/c
with ǫ = m − b > 0. For large arguments, the integrand in the definition off(λ) can be estimated via Lemma 5.1 by
|ϕλ(ω)γ(ω)| ≤ Cγ21+b
√2π
|Γ(−λ)| cb+ d−12 ‖ω‖−b− d+1
22 e−c‖ω‖2e
|b+ d2|2
2c‖ω‖2
using that γ ∈ S is bounded. Since C is compact, this can be boundedindependently of λ ∈ C by
|ϕλ(ω)γ(ω)| ≤ Cγ,C,m,ce−c‖ω‖2/2,
completing the proof. 2
Theorem 5.4 The function Φ(x) = ‖x‖β2 , x ∈ IRd, with β > 0, β 6∈ 2IN has
the generalized Fourier transform
Φ(ω) =2β+ d
2 Γ(d+β2
)
Γ(−β2)
‖ω‖−β−d2 , ω 6= 0,
of order m = ⌈β/2⌉.
Proof: Let us start with the function Φc(x) = (c2 + ‖x‖22)
β
2 , c > 0. Thisfunction possesses a generalized Fourier transform of order m = ⌈β/2⌉ givenby
Φc(ω) = ϕc(ω) =21+β/2
Γ(−β/2)‖ω‖−β−d
2 (c‖ω‖2)β+d
2 Kβ+d
2(c‖ω‖2)
due to Theorem 5.3. Here, we use the subscript c instead of β, since β isfixed and we want to let c go to zero. Moreover, we can conclude from theproof of Theorem 5.3 that for γ ∈ Sm the product can be bounded by
|ϕc(ω)γ(ω)| ≤ Cγ
2β+d/2Γ(β+d2
)
|Γ(−β/2)| ‖ω‖2m−β−d2
for ‖ω‖2 → 0 and by
|ϕc(ω)γ(ω)| ≤ Cγ
2β+d/2Γ(β+d2
)
|Γ(−β/2)| ‖ω‖−β−d2
16
for ‖ω‖2 → ∞ independently of c > 0. Since |Φc(ω)γ(ω)| can also be boundedindependently of c by an integrable function, we can use the convergencetheorem of Lebesgue twice to derive∫
IRd‖x‖β
2 γ(x)dx = limc→0
∫
IRdΦc(x)γ(x)dx
= limc→0
∫
IRdϕc(ω)γ(ω)dx
=21+ β
2
Γ(−β2)
∫
IRd‖ω‖−β−d
2 γ(ω) limc→0
(c‖ω‖2)β+d
2 Kβ+d
2(c‖ω‖2)dω
=2β+d/2Γ(d+β
2)
Γ(−β/2)
∫
IRd‖ω‖−β−d
2 γ(ω)dω
for γ ∈ Sm. The last equality follows from
limr→0
rνKν(r) = limr→0
2ν−1∫ ∞
0e−te−
r2
4t tν−1dt = 2ν−1Γ(ν),
see also the proof of Lemma 5.2. 2
Theorem 5.5 The function Φ(x) = ‖x‖2k2 log ‖x‖2, x ∈ IRd, k ∈ IN , pos-
sesses the generalized Fourier transform
Φ(ω) = (−1)k+122k−1+ d2 Γ(k +
d
2)k!‖ω‖−d−2k
2
of order m = k + 1.
Proof: For fixed r > 0 and β ∈ (2k, 2k + 1) we expand the function β 7→ rβ
using Tayloris theorem to
rβ = r2k + (β − 2k)r2k log r +∫ β
2k(β − t)rt log rdt. (15)
From Theorem 5.4 we know the generalized Fourier transform of the functionx 7→ ‖x‖β
2 of order m = ⌈β/2⌉ = k +1. From Proposition 4.3 we see that thegeneralized Fourier transform of order m of the function x 7→ ‖x‖2k
2 equalszero. Thus we can conclude from (15) for any test function γ ∈ Sm that∫
IRd‖x‖2k
2 log ‖x‖2γ(x)dx =1
β − 2k
∫
IRd
(‖x‖β
2 − ‖x‖2k2
)γ(x)dx
− 1
β − 2k
∫
IRd
∫ β
2k(β − t)‖x‖t
2 log ‖x‖2γ(x)dtdx
=2β+ d
2 Γ(d+β2
)
(β − 2k)Γ(−β2)
∫
IRd‖ω‖−β−d
2 γ(ω)dω
+ O(β − 2k)
17
for β → 2k. Furthermore, we know from the property Γ(z)Γ(1 − z) =π/ sin(πz) that
1
Γ(−β2)(β − 2k)
= −sin(πβ2
)Γ(1 + β2)
π(β − 2k).
Because
limβ→2k
sin(πβ2
)
β − 2k= lim
β→2k
π2
cos(πβ2
)
1=
π
2(−1)k,
we see that
limβ→2k
1
Γ(−β2)(β − 2k)
= (−1)k+1k!/2.
Now we can apply the theorem of dominated convergence to derive
∫
IRd‖x‖2k
2 log ‖x‖2γ(x)dx = 22k+d/2Γ(k + d/2)(−1)k+1k!
2
∫
IRd‖ω‖−d−2k
2 γ(ω)dω
for all γ ∈ Sm, which gives the stated generalized Fourier transform. 2
Now it is easy to decide whether the just investigated functions are con-ditionally positive definite. As mentioned before, we state the minimal m.
Corollary 5.6 The following functions Φ : IRd → IR are conditionally pos-itive definite of order m:
• Φ(x) = (−1)⌈β⌉(c2 + ‖x‖22)
β, β > 0, β 6∈ 2IN , m = ⌈β⌉,
• Φ(x) = (c2 + ‖x‖22)
β, β < 0, m = 0,
• Φ(x) = (−1)⌈β/2⌉‖x‖β2 , β > 0, β 6∈ IN , m = ⌈β/2⌉,
• Φ(x) = (−1)k+1‖x‖2k2 log ‖x‖2, k ∈ IN , m = k + 1.
6 Construction via Dimension Walk
So far we have seen that radial functions that work on IRd for all d ≥ 1,are nicely characterized by the abstract results of Schoenberg and Micchelli,while translation invariant functions for fixed dimensions are best handledvia Fourier transform, yielding explicit results for further use.
Here, we want to investigate radial functions for a fixed space dimension.Thus we have to take the Fourier transform, but we shall make use of radialitythroughout, relying on ideas of Wu and Schaback [20], [15]. Our main goalwill be the construction of compactly supported positive definite radial basisfunctions for fixed space dimensions.
18
Theorem 6.1 Suppose Φ ∈ L1(IRd)∩C(IRd) is radial, i.e., Φ(x) = φ(‖x‖2),
x ∈ IRd. Then its Fourier transform Φ is also radial, i.e., Φ(ω) = Fdφ(‖ω‖2)with
Fdφ(r) = r−d−22
∫ ∞
0φ(t)t
d2 J d−2
2(rt)dt,
and φ satisfies φ(t)td−1 ∈ L1[0,∞).
Proof: The case d = 1 follows immediately from
J−1/2(t) =(
2
πt
)1/2
cos t.
In case d ≥ 2, splitting the Fourier integral, and using the representation∫
Sd−1
eixT ξdS(ξ) = (2π)d/2‖x‖−d−22
2 J d−22
(‖x‖2)
of the classical Bessel function Jν via an integral over the sphere Sd−1 ⊂ IRd
yield
Φ(x) = (2π)−d/2∫
IRdΦ(ω)e−ixT ωdω
= (2π)−d/2∫ ∞
0td−1
∫
Sd−1
φ(t‖ω‖2)e−itxT ωdS(ω)dt
= (2π)−d/2∫ ∞
0φ(t)td−1
∫
Sd−1
e−itxT ωdS(ω)dt
= r−(d−2)/2∫ ∞
0φ(t)td/2J(d−2)/2(rt)dt.
The second assertion of the theorem follows from an inspection of the con-dition Φ ∈ L1(IR
d), using the radiality of Φ. 2
Theorem 6.1 gives us the opportunity to interpret the d–variate Fouriertransform of a radial function via Fd as an operator that maps univariatefunctions to univariate functions.
Now let us have a closer look at this operator with respect to the spacedimension. If we use d
dzzνJν(z) = zνJν−1(z) we get via integration by
parts, for d ≥ 3,
Fdφ(r) = r−d+2∫ ∞
0φ(t)t (rt)
d−22 J d−2
2(rt)dt
= r−d+2(−∫ ∞
tφ(s)sds
)(rt)
d−22 J d−2
2(rt)
∣∣∣∣t=∞
t=0
+ r−d+2∫ ∞
0
(∫ ∞
tφ(s)sds
)r
d2 t
d−22 J d−4
2(rt)dt
= Fd−2
(∫ ∞
•φ(s)sds
)(r)
19
whenever the boundary terms vanish. Thus if we define
Iφ(r) :=∫ ∞
rφ(t)tdt
Dφ(r) := −1
r
d
drφ(r)
we get the following result.
Theorem 6.2 If φ ∈ C[0,∞) satisfies t 7→ φ(t)td−1 ∈ L1[0,∞) for somed ≥ 3, then we have Fd(φ) = Fd−2(Iφ). This means that φ is positive definiteon IRd if and only if Iφ is positive definite on IRd−2. On the other hand, ifφ satisfies t 7→ φ(t)td−1 ∈ L1[0,∞) for some d ≥ 1, φ(t) → 0 for t → ∞,and if the even extension of φ to IR is in C2(IR), then Fd(φ) = Fd+2(Dφ).In this situation, the function φ is positive definite on IRd if and only if Dφis positive definite on IRd+2.
Since both operators I and D are easily computable and satisfy I =D−1 and D = I−1 wherever defined, this gives us a very powerful tool forconstructing positive definite functions. For example, we could start with avery smooth compactly supported function on IR1 and apply the operatorD n-times to get a positive definite and compactly supported function onIR2n+1. Before we give an example, let us remark that it is possible togeneralize the operators Fd, I, D to step through the dimensions one by oneand not two by two [15].
Theorem 6.3 Define φℓ(r) := (1 − r)ℓ+ and φd,k by
φd,k = Ikφ⌊d/2⌋+k+1.
Then φd,k is compactly supported, a polynomial within its support, and posi-tive definite on IRd. In particular, the function 20φ3,1(r) = (1 − r)4
+(4r + 1)is positive definite on IR3.
Proof: Since the operator I respects the polynomial structure and compactsupport, we only have to prove positive definiteness. Due to
Fdφd,k = FdIkφ⌊d/2⌋+k+1 = Fd+2kφ⌊(d+2k)/2⌋+1
it remains to show that Fdφ⌊d/2⌋+1 is nonnegative for every space dimensond. We will follow ideas of Askey [1] to do this. Let us start with an odddimension d = 2n + 1. Then the Fourier transform is given by
r3n+2F2n+1φn+1(r) =∫ r
0(r − s)n+1sn+ 1
2 Jn− 12(s)ds.
20
Defining the right-hand side of the last equation as g(r), we see that g is theconvolution g(r) =
∫ r0 g1(r − s)g2(s)ds of the functions g1(s) := (s)n+1
+ andg2(s) := sn+1/2Jn−1/2(s). Thus its Laplace transform Lg(r) =
∫∞0 g(t)e−rtdt
is the product of the Laplace transforms of g1 and g2. These transforms canbe computed for r > 0 as
Lg1(r) =(n + 1)!
rn+2
and
Lg2(r)n! 2n+1/2r√π (1 + r2)n+1
.
This combines into
Lg(r) =2n+1/2n!(n + 1)!√
π
1
rn+1(1 + r2)n+1.
On the other hand, it is well known that the function 1−cos r has the Laplacetransform 1
r(1+r2). Thus, if p denotes the n-fold convolution of this function
with itself, we get
Lp(r) =1
rn+1(1 + r2)n+1.
By the uniqueness of the Laplace transform this leads to
g(r) =2n+1/2n!(n + 1)!√
πp(r),
which is clearly nonnegative and not identically zero. For even space dimen-sion d = 2n we need only to remark that φ⌊ 2n
2⌋+1 = φ⌊ 2n+1
2⌋+1. Hence φ⌊ 2n
2⌋+1
induces a positive definite function on IR2n+1 and therefore also on IR2n. Thefunction φ(r) = (1 − r)4
+(4r + 1) is nothing but 20φ3,1, and hence positivedefinite on IR3. 2
The parameter k in the last theorem controls the smoothness of the basisfunction. It can be shown [17] that φd,k possesses 2k continuous derivativesas a radial function on IRd and is of minimal degree under all piecewisepolynomial compactly supported functions that are positive definite on IRd
and whose even extensions to IR are in C2k(IR). A different technique forgenerating compactly supported radial basis functions is due to Buhmann[4], [5], [6].
21
7 Construction of general functions
So far, we have only dealt with translation–invariant (conditionally) posi-tive definite functions, and most of our work was even restricted to radialfunctions. As a consequence, we had to work with basis functions that are(conditionally) positive definite on all of IRd. In this section we want tochoose a more general approach which allows us to construct positive defi-nite functions on local domains Ω. Consequently, we have to drop Fourierand Laplace transforms, replacing them by expansions into orthogonal sys-tems. As a byproduct, this technique allows us to construct positive definitefunctions on manifolds, in particular on the sphere.
Theorem 7.1 Suppose Ω ⊆ IRd is measurable. Let ϕ1, ϕ2, . . . be an or-thonormal basis for L2(Ω) consisting of continuous and bounded functions.Suppose that the point evaluation functionals are linearly independent on thespace span ϕj : j ∈ IN. Suppose ρn is a sequence of positive numberssatisfying
∞∑
n=1
ρn‖ϕn‖2L∞(Ω) < ∞. (16)
Then
Φ(x, y) =∞∑
n=1
ρnϕn(x)ϕn(y)
is positive definite on Ω.
Proof: Property (16) ensures that Φ is well–defined and continuous. Fur-thermore, we have for α ∈ CN and distinct x1, . . . , xN ∈ Ω that
N∑
j,k=1
αjαkΦ(xj , xk) =∞∑
n=1
ρn
∣∣∣∣∣∣
N∑
j=1
αjϕn(xj)
∣∣∣∣∣∣
2
≥ 0.
Since the point evaluation functionals are linear independent on the spacespan ϕj : j ∈ IN, the last expression can only vanish for α = 0. 2
Note that the condition on the point evaluation functionals is somewhatunnatural for the space L2(Ω). It would be more natural to define Φ tobe positive definite, iff for every linear independent set Λ = λ1, . . . λN ⊆L2(Ω)∗ and every α ∈ CN \ 0 the quadratic form
N∑
j,k=1
αjαkλxj λ
ykΦ(x, y)
22
is positive. But we do not want to pursue this topic any further. Instead, wewant to use Theorem 7.1 to give an example of a positive definite functionon a restricted domain.
Our example deals with the space L2[0, 2π]2 which has the bounded andcontinuous orthogonal basis φn,k(x1, x2) = ei(nx1+kx2) : n, k ∈ ZZ offunctions with a 2π–periodic extension. Thus condition (16) is satisfied ifthe positive coefficients ρn,k have the property
∞∑
n,k=−∞
ρn,k < ∞.
In particular, the bivariate functions
φ1,ℓ(x) = 1 +∑
(n,k)∈ZZ2\0
1
(n2 + k2)ℓei(nx1+kx2)
and
φ2,ℓ(x) = 1 +∞∑
n=−∞n6=0
1
n2ℓeinx1 +
∞∑
k=−∞k 6=0
1
k2ℓeikx2 +
∞∑
n=−∞n6=0
∞∑
k=−∞k 6=0
1
(nk)2ℓei(nx1+kx2)
generate positive definite 2π–periodic translation–invariant functions Φ(x, y) =φ(x − y) on [0, 2π]2 for sufficiently large ℓ. Because of their tensor productstructure, the latter can be computed directly (see [10]). Some examples are:
φ2,1(x) =2∏
j=1
(6 − π2
6+
1
2(xj − π)2
)
φ2,2(x) =2∏
j=1
(360 − 7π4
360+
π2(xj − π)2
12− (xj − π)4
24
)
.
For more examples see [10]. Of course, this tensor product approach gen-eralizes to arbitrary space dimension, but the basic technique is much moregeneral. See [14] for the relation to positive integral operators.
References
[1] Askey, R., Radial characteristic functions, MRC technical report sum:report no. 1262, University of Wisconsin, 1973.
[2] Bochner, S., Vorlesungen uber Fouriersche Integrale, Akademische Ver-lagsgesellschaft, Leipzig, 1932.
23
[3] Bochner, S., Monotone Funktionen, Stieltjes Integrale und harmonischeAnalyse, Math. Ann. 108 (1933), 378-410.
[4] Buhmann, M.D., Radial Functions on Compact Support, Proceedingsof the Edinburgh Mathematical Society 41 (1998), 33-46.
[5] Buhmann, M.D., A new class of radial basis functions with compactsupport. To appear.
[6] Buhmann, M.D., this volume
[7] Chang, K.F., Strictly positive definite functions, J. Approx. Theory, 87
(1996), 148–158.
[8] Chang, K.F., Strictly positive definite functions II, preprint.
[9] Guo,K., S. Hu, and X. Sun, Conditionally positive definite functions andLaplace-Stieltjes integrals, J. Approx. Theory, 74 (1993), 249-265.
[10] Narcowich, F.J., and J.D. Ward, Wavelets associated with periodic basisfunctions, Appl. Comput. Harmonic Anal. , 3 (1996), 40-56.
[11] Madych, W.R., and S.A. Nelson, Multivariate interpolation: a varia-tional theory, manuscript, 1983.
[12] Micchelli, C.A., Interpolation of scattered data: distance matrices andconditionally positive definite functions, Constr. Approx. 2 (1986), 11-22.
[13] Powell, M.J.D., Radial basis functions for multivariable interpolation: areview, in: J. C. Mason and M. G. Cox (eds.): Algorithms for approxi-mation, Clarendon Press, Oxford 1987 1987, 143–167
[14] Schaback, R., A Unified Theory of Radial Basis Functions (NativeHilbert Spaces for Radial Basis Functions II), preprint Gottingen 1999.
[15] Schaback, R., and Z. Wu, Operators on radial functions, J. Comp. Appl.Math. 73 (1996), 257-270.
[16] Schoenberg, I.J., Metric spaces and completely monotone functions,Ann. of Math., 39 (1938), 811-841.
[17] Wendland, H., Piecewise polynomial, positive definite and compactlysupported radial functions of minimal degree, AiCM 4 (1995), 389-396.
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[18] Wendland, H., On the smoothness of positive definite and radial func-tions, Journal of Computational and Applied Mathematics 101 (1999),177-188.
[19] Widder, D.V., The Laplace Transform, Princeton University Press,Princeton, 1946.
[20] Wu, Z., Multivariate Compactly Supported Positive Definite RadialFunctions, AiCM 4 (1995), 283–292
Author’s addresses:
Institut fur Numerische und Angewandte MathematikUniversitat GottingenLotzestraße 16-18D-37083 Gottingen, [email protected]
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