Multivariate Compactly Supported Positive Definite Radial Functions

Zongmin Wu ^{1}

Department of Mathematics, Fudan University 200433 Shanghai, China

Abbreviated Title: Compactly Supported Radial Functions

Author’s Address: Before Sept. 30, 1994 After Oct. 1, 1994

Wu Zongmin Wu Zongmin Institut für Numerische Department of Mathematics und Angewanddte Mathematik Fudan University Lotze Str. 16-18 200433 Shanghai China 37083 Göttingen Germany

1The work was done during the author’s visit to Göttingen under support of a DAAD-Wong fellowship.

Abstract: We provide criteria for positive definiteness of radial functions with compact support. Based on these criteria we will produce a series of positive definite and compactly supported radial functions, which will be very useful in applications. The simplest ones are cut-off polynomials, which consist of a single polynomial piece on [0, 1] and vanish on [1, ∞). More precisely, for any given dimension n and prescibed C^{k} smoothness, there is a function in C^{k}(IR^{n}), which is a positive definite radial function with compact support and is a cut-off polynomial as a function of Euclidean distance. Another example is derived from odd-degree B -splines.

AMS classifications: 41A15, 41A25, 41A30, 41A63, 65D10, 65D17 .

Keywords: Compact support, Positive definite, Radial function, Scattered data interpolation, Approximation.

1 Introduction

Multivariate functions Φ : IR^{n} → IR can be efficiently evaluated if they are expressible as univariate functions Φ(·) = ϕ(||·||) of the Euclidean norm ||·|| in IR^{n}. Such functions are called radial, and a typical application is the construction of interpolants

S_{f}(x) = ∑_{j=1}^{N} λ_{j}ϕ(||x − x_{j}||)  (1.1)

for scattered data {x_{j}, f_{j}} ∈ IR^{n} × IR, 1 ≤ j ≤ N, with a set X = {x_{1},…, x_{N}} of pairwise distinct centers [3] [4] [5] [9] [10] [11]. A major feature of (1.1) is the independence of the solution on Euclidean transformations of the data set.

It is easy to see that the interpolation problem is uniquely solvable, if the coefficient matrix

A = (ϕ(||x_{j} − x_{k}||))_{N×N}  (1.2)

is positive definite for any set X ⊂ IR^{n}, |X| = N. Thus a radial function ϕ(r) is called positive definite on IR^{n}, in short: ϕ ∈ PD_{n} , if for any set X = {x_{1},…, x_{N}} of N distinct points in IR^{n} the quadratic form

λ^{T} Aλ = ∑_{j,k=1}^{N} λ_{j}λ_{k}ϕ(||x_{j} − x_{k}||) λ = (λ_{1},…, λ_{N}) ∈ IR^{n},

is positive definite. There are lots of positive definite (or conditional positive definite, see [4]) functions in the literature, such as the inverse multiquadrics (c^{2} +r^{2})^{β} with β < 0, the Gaussian e^{−αr^{2}}, (or the thin-plate splines r^{β} and r^{β} log r, respectively). But none of these is compactly supported. Thus the matrix (1.2) is not sparse, and the evaluation of the (1.1) runs over the whole set of N summands. There would be a major inprovement, if there were radial basis functions of compact support.

However these were not available so far. This seems to be a consequence of the fact that the fundamental paper [4] of C.A. Micchelli does not cover positive definite functions of compact support. In fact, it shows that univariate completely monotonic functions F (·) yield radial functions ϕ(||r||) = F (r^{2}) that are in PD_{n} for all n. The converse of this was proved by Schoenberg [8] and for conditionally positive definite function recently by Guo, Hu and Sun [1]. Since completely monotonic functions can not have any zeros, there are no functions ϕ of compact support (we write ϕ ∈ CS for this) that are in PD_{n} for all n. However, there may be functions in CS ∩ PD_{n} for a restricted range of values of n and these can be extremely useful in practice, because the sum in (1.1) can be efficently evaluated and because the matrix A in (1.2) is sparse.

Based on a characterization of (conditional) PD_{n} functions for fixed n by (generalized) Fourier transforms in the dissertation of A. Iske [2], the first functions in CS ∩ PD_{n} were constructed by Schaback & Wendland [7]. They use a radialized tensor product of univariate odd degree B -splines. But the radialization is not very easy, since it involves a recursion for explicit calculation of an n-dimensional integral over trigonometric functions. But they opened the search for more simple compactly supported positive definite radial functions.

In this paper we will first derive some general criteria for CS ∩ PD_{n} functions in section 3. ¿From the criteria we can construct a series of examples defined by simple mathematical expressions and having good properties. The simplest consist of a single polynomial piece of some order k on [0, 1] (a cut-off polynomial, or in short, a CP_{k} function). More precisely, for any given dimension n and any prescibed C^{ℓ} smoothness, there is a cut-off polynomial function in C^{ℓ} ∩ CS ∩ CP_{4⌈ \frac{ℓ+1}{2} ⌉+2⌈ \frac{n}{2} ⌉+1} ∩ PD_{n} , which is also in PD_{k} for all k ≤ n. Another example will be derived from odd degree B -splines. The approximation order of such radial basis functions and the condition of the matrices A of (1.2) can be controlled, since both depend on the smoothness order ℓ [6][11].

2 Definitions and Lemmas

At first we recall our abbreviations. We write ϕ ∈ CS if the function ϕ has compact support and we always assume the support to be contained in [−1, 1]. For other cases we can use ϕ(\frac{r}{a}) to get a function with support of radius a. We write ϕ ∈ PD_{n} if the function Φ(·) = ϕ(|| · ||) is positive definite in IR^{n}. This means that for any pairwise distinct points {x_{1},…, x_{N}} ∈ IR^{n} the matrix A of (1.2) is positive definite.

A function ϕ is in PD_{n} iff the n-dimensional Fourier transform of ϕ (i.e. the Fourier transform of Φ(·) = ϕ(||·||) is nonnegative on IR^{n} and positive in some open subset. We denote this by

Φ̂(ω) = ∫_{IR^{n}} e^{i<x,ω>}ϕ(||x||)dx≠^{≥}0

See Iske [2] for an extension of this fact for conditionally positive definite functions on IR^{n}. For the discussion of functions in CS ∩ PD_{n} we state some easy facts as lemmas.

Since a univariate radial function is symmetric, we have Lemma 2.1 A univariate function ϕ is in CS ∩ PD_{1} iff

∫_{0}^{1} ϕ(x) cos(xω)dx≠^{≥}0

A simple example is ϕ(x) = B^{2ℓ−1}(ℓx) where B^{2ℓ−1} is the B-spline of order 2ℓ with knots

ZZ ∩ [−ℓ, ℓ], since the Fourier transform of B^{2l−1}(ℓx) is sinc^{2ℓ}(\frac{t}{2ℓ})≠^{≥}0.

Since the convolution of a positive function with a positive function is positive, we have

Lemma 2.2 If ψ_{1}(x) and ψ_{2}(x) are in CS∩PD_{k}, then the function ϕ = ψ_{1}ψ_{2} is in CS∩PD_{k}.

By simple scaling arguments we get

Lemma 2.3 ϕ(x) = ψ(\frac{x}{λ}) is in CS ∩ PD_{k}, if ψ is in CS ∩ PD_{k} and 0 < λ < 1. In fact, if ψ is in CS ∩ PD_{k}, then the function ϕ is in PD_{k} and compactly sopported with a support of radius λ.

Lemma 2.4 ϕ(x) = ∫_{0}^{1} ψ(\frac{x}{λ})g(λ)dλ is in CS ∩ PD_{k}, if ψ is in CS ∩ PD_{k} and g(λ) > 0.

3 Multivariate positive definite radial functions with compact support

We already mentioned in the introduction that there are no compactly supported radial functions which are positive definite in all dimensions. Specifically if the function ϕ(x) is positive definite in all dimensions, then from Schoenberg’s classical paper [8] the function F (x) = ϕ(\sqrt{x}) must be completely monotonic and can be expressed as a Laplace-Stieltjes integral

ϕ(x) = F (x^{2}) = ∫ e^{−λx^{2}}dσ(λ)

with a nonnegative Borel measure σ(λ). But then the function ϕ can not be compactly supported, it can not even vanish somewhere. However, the following simple lemma is useful:

Lemma 3.1 Because IR^{k} can be imbedded in to IR^{n}, ϕ(r) ∈ PD_{n} implies ϕ(r) ∈ PD_{k} for k ≤ n.

This invites us to find some radial functions in CS ∩ PD_{n} for use in IR^{k} for k ≤ n. The most interesting cases for applications are n = 3 or n = 4.

We now proceed to characterize PD_{n} functions, and use ≐ to mean equality up to a positive constant factor. The radial function is written as Φ(x) = ϕ(||x||) = F (||x||^{2}) for x ∈ IR^{n}. If we introduce r = ||x||_{2} and t = ||ω||_{2}, the Fourier transform of the function Φ can be calculated in polar coordinates as

Φ̂(ω) = ∫_{IR^{n}} F (||x||_{2}^{2})e^{ixω}dx
= ∫_{0}^{∞} ∫_{0}^{π} area(S^{n−2}(r sin θ))F (r^{2})e^{irt cos θ}rdθdr (3.1)

where S^{n−2}(s) is the surface of (n−1)-dimensional ball with radius s. We denote it by S^{n−2} because it is an (n − 2)-dimensional manifold. Then

Φ̂(ω) ≐ ∫_{0}^{∞} ∫_{0}^{π} F (r^{2})e^{irt cos θ}r^{n−1} sin^{n−2} θdθdr
= ∫_{−∞}^{∞} ∫_{0}^{∞} F (x^{2} + y^{2})e^{ixt}y^{n−2}dydx
≐ ∫_{0}^{∞} ∫_{0}^{∞} F (x^{2} + y^{2}) cos(xt)y^{n−2}dydx

(3.2)

if we return to 2-dimensional Euclidean space with x = r cos θ and y = r sin θ. This yields our first criterion:

Theorem 3.1 Φ(x) = ϕ(||x||) = F (||x||^{2}) ∈ PD_{n} iff

ψ(x) = ∫_{0}^{∞} F (x^{2} + y^{2})y^{n−2}dy ∈ PD_{1}.  (3.3)

Furthermore, if F is compactly supported on [−1, 1], then Φ(x) = ϕ(||x||) = F (||x||^{2}) ∈ CS ∩ PD_{n} iff

∫_{0}^{1} F (x^{2} + y^{2})y^{n−2}dy ∈ CS ∩ PD_{1}  (3.4)

or

∫_{0}^{1} ∫_{0}^{1} F (x^{2} + y^{2})y^{n−2} cos(xt)dydx≠^{≥}0.  (3.5)

This result reduces a multivariate problem to a univariate problem. Now we consider a univariate continuous compactly supported function ϕ with ϕ(1) = 0. Assume

F (r) = ∫_{r}^{1} G(a)da = ∫_{0}^{1} χ(\frac{r}{a})G(a)da

where χ = χ_{[0,1]} is the characteristic function of the interval [0, 1]. If F ∈ C^{1} or G ∈ C^{0}, we have G = −F′, but this is not required. Then

F (x^{2} + y^{2}) = ∫_{0}^{1} χ(\frac{x^{2} + y^{2}}{a})G(a)da

and if we keep in mind that χ(\frac{x^{2}+y^{2}}{a}) = 0, if x^{2} + y^{2} > a, then

∫_{0}^{1} ∫_{0}^{1} F (x^{2} + y^{2})y^{n−2} cos(xt)dydx

= ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} χ(\frac{x^{2}+y^{2}}{a})G(a)y^{n−2} cos(xt)dadydx
= ∫_{0}^{1} ∫_{0}^{1} ∫_{0}^{1} χ(\frac{x^{2}+y^{2}}{a})G(a)y^{n−2} cos(xt)dxdady
= ∫_{0}^{1} ∫_{y^{2}}^{1} ^{\frac{sin(t\sqrt{a−y^{2}})}{t}}G(a)y^{n−2}dady
≐ ∫_{0}^{1} ∫_{y^{2}}^{1} ^{\frac{cos(t\sqrt{a−y^{2}})}{\sqrt{a−y^{2}}}}F (a)y^{n−2}dady
= ∫_{0}^{1} ∫_{0}^{√\overline{a}} ^{\frac{cos(t\sqrt{a−y^{2}})}{\sqrt{a−y^{2}}}}F (a)y^{n−2}dyda.

(3.6)

Now define

I_{n}(a, t) = ∫_{0}^{√\overline{a}} \frac{cos(t\sqrt{a − y^{2}})}{\sqrt{a − y^{2}}}y^{n−2}dy

and get the second criterion: Theorem 3.2 Φ(x) = ϕ(||x||_{2}) = F (||x||_{2}^{2}) ∈ CS ∩ PD_{n} iff

∫_{0}^{1} F (a)I_{n}(a, t)da≠^{≥}0.

For the function I_{n}(a, t) we integrate by parts

\begin{aligned}I_{n}(a, t) & = & ∫_{0}^{√\overline{a}} ^{\frac{cos(t\sqrt{a−y^{2}})}{\sqrt{a−y^{2}}}}y^{n−2}dy \\ & = & \frac{n−3}{t} ∫_{0}^{√\overline{a}} sin(t\sqrt{a − y^{2}})y^{n−4}dy \end{aligned}  (3.7)

and take derivatives:

\begin{aligned}\frac{∂I_{n}(a,t)}{∂a} & = & \frac{n−3}{2} ∫_{0}^{√\overline{a}} ^{\frac{cos(t\sqrt{a−y^{2}})}{\sqrt{a−y^{2}}}}y^{n−4}dy \\ & = & \frac{n−3}{2}I_{n−2}(a, t). \end{aligned}  (3.8)

If we take I_{n}(0, t) = 0 in to account, we have deduced a recursive formula

I_{n}(a, t) = \frac{n − 3}{2} ∫_{0}^{a} I_{n−2}(b, t)db.  (3.9)

We will point out here that

I_{3}(a, t) = ∫_{0}^{√\overline{a}} \frac{cos(t\sqrt{a − y^{2}})}{\sqrt{a − y^{2}}}ydy = \frac{sin(t\sqrt{a})}{t},

while there is no readily available expression for I_{2}(a, t). ¿From Theorem 3.2 we have Φ(x) = ϕ(||x||) = F (||x||^{2}) ∈ CS ∩ PD_{3} iff

0 ≠^{≤} ∫_{0}^{1} F (a)\frac{sin(t\sqrt{a})}{t}da
≐ ∫_{0}^{1} F (b^{2})\frac{sin(bt)}{t}bdb
= ∫_{0}^{1} ϕ_{1}(a) cos(at)da

(3.10)

where ϕ_{1}(a) = ∫_{a}^{1} F (b^{2})bdb. Using Lemma 2.1 this is a nonnegative function if, for instance, ϕ_{1}(a) = B^{2ℓ−1}(ℓa) and

ϕ(r) = F (r^{2}) = −\frac{[B^{2ℓ−1}(ℓr)]′}{r} .

This furnishes our

Example 1: If B^{2ℓ−1} is the B- spline of order 2ℓ with knots ZZ ∩ [−ℓ, ℓ], then the piecewise polynomial functions

ϕ(r) = F (r^{2}) = −\frac{[B^{2ℓ−1}(ℓr)]′}{r}

are in CS∩PD_{3} for all ℓ > 1. Plenty of other functions can be derived from this case, applying lemmas 2.2, 2.3 and 2.4 to it. We finally note that ϕ(r) = F (r^{2}) = −\frac{[B^{2ℓ−1}(ℓr)]′}{r} ∈ C^{2ℓ−4} at least. □

For the n-dimensional problem we use the recursive relation (3.9) of I_{n}(a, t) in the form

∫_{0}^{1} F (a)I_{n}(a, t)da ≐ ∫_{0}^{1} G(a)I_{n−2}(a, t)da  (3.11)

where G(a) := ∫_{a}^{1} F (b)db, since I_{n}(0, t) = 0 = G(1). Together with Theorem 3.2 this yields:

Theorem 3.3 A function ϕ(r) = F (r^{2}) is in CS ∩ PD_{n}, n ≥ 3 iff there is a function

G ∈ C^{1}[0, ∞), such that −G′ = F or in other words ϕ = −\frac{ψ′}{r} = (−\frac{1}{r} \frac{d}{dr})ψ and ψ(r) = G(r^{2}) is in CS ∩ PD_{n−2}.

¿From the theorem a series of corollaries can be derived.

Corollary 3.1 A function ϕ(r) = F (r^{2}) is in CS ∩ PD_{n} iff there is a radial function G ∈ C^{k}[0, ∞), F = (−1)^{k}G^{(k)} such that ψ(r) = G(r^{2}) is in CS ∩ PD_{n−2k} for n > 2k. This is

equivalent to ϕ = (−\frac{1}{r} \frac{d}{dr})^{k}ψ.

Corollary 3.2 If a function ψ(r) = G(r^{2}) is in CS∩PD_{1} and G ∈ C^{k}[0, ∞) then ϕ(r) = F (r^{2})

with F = (−1)^{k}G^{(k)} is in CS ∩ PD_{2k+1}. This is equivalent to ϕ = (−\frac{1}{r} \frac{d}{dr})^{k}ψ.

These two corollaries are trivial but very useful. Introducing the differential operator D with

D = −\frac{1}{r} \frac{d}{dr}, the results get a very simple form:

Corollary 3.3 We have ψ(r) ∈ CS ∩ PD_{n} for n ≥ 3 iff there is function ψ_{1}(r) ∈ CS ∩ PD_{n−2} and ψ(r) = Dψ_{1}(r).

Corollary 3.4 ψ(r) ∈ CS ∩ PD_{n} iff there is function ψ_{1}(r) ∈ CS ∩ PD_{n−2k} and ψ(r) = D^{k}ψ_{1}(r) for some k ≥ 0. If ψ(r) ∈ CS ∩ PD_{1} and ψ_{1}(r) = D^{k}ψ(r) exists, then ψ_{1}(r) ∈ CS ∩ PD_{2k+1}.

Remark 1: D^{k}ψ(r) exists iff ψ(\sqrt{·}) = G(·) ∈ C^{k}.

Example 2: As is shown in section 4, we have B^{2ℓ−1}(ℓ\sqrt{·}) ∈ C^{ℓ−1}, and with corollary 3.4 we find

D^{k}B^{2ℓ−1}(ℓr) ∈ CS ∩ PD_{2k+1} for k < ℓ.

¿From Theorem 3.3 and Lemma 2.2 we can get some interesting further results:

Proposition 3.1 If f(r^{2}), g(r^{2}), h(r^{2}) ∈ CS ∩ PD_{k} and f, g, h ∈ C^{1}, then there exist functions I_{1}(r^{2}), I_{2}(r^{2}) ∈ CS ∩ PD_{k} and I_{1}, i_{2} ∈ C^{1}, such that I′_{1} = f′g′h′ I′_{2} = −f′g′.

Proposition 3.2 If f(r^{2}), g(r^{2}), ∈ CS ∩ PD_{k} and f, g ∈ C^{2}, then there exists a function h(r^{2}) ∈ CS ∩ PD_{k} and h ∈ C^{2}, such that h′′ = f′′g′′.

These results can be used to find some new functions in CS ∩ PD.

4 Univariate positive definite radial functions with compact support

After the discussion in section 3, we only have to investigate sufficiently smooth univariate positive definite radial functions with compact support, and then we can always apply the operator D to ϕ(r) or take the derivative to F (·) = ϕ(\sqrt{·}) a couple of times to get a multivariate positive definite radial function with compact support. For the latter case it is clear that we can take ℓ-times the derivative of F if the function F is in C^{ℓ}. Now we will show how often the operator D can be applied to ϕ(r) if ϕ is in some C^{k}. With the above remark it is clear that the operator F can be applied to ϕ exactly ℓ-times if ϕ(\sqrt{·}) is in C^{ℓ}. A sufficient condition can be presented as a theorem:

Theorem 4.1 If ϕ(r) = F (r^{2}) ∈ C^{2ℓ}(−∞, ∞), then F (·) ∈ C^{ℓ}[0, ∞) and the operator D can be ℓ-times applied to ϕ(r).

Proof: There is no problem for r > 0. By symmetry

ϕ(|x|) = ∑_{0}^{2ℓ} a_{j}|x|^{j} + O(|x^{2ℓ+1}|) for x ∈ IR,

and ϕ ∈ C^{2ℓ} means a_{2k−1} = 0 for k = 1,…, ℓ. Furthermore

F (r^{2}) = ϕ(r) = ∑_{j=0}^{ℓ} a_{2j}r^{2j} + O(r^{2ℓ})  (4.1)

and

F (ξ) = ∑_{0}^{ℓ} a_{2j}ξ^{j} + O(ξ^{ℓ}) ∈ C^{ℓ}

for ξ → 0. This proves our assertion. □

Remark 2: If we start with ϕ(r) = F (r^{2}) and choose

ψ_{k}(r) = G(r^{2}) = (−1)^{k}F^{(k)}(r^{2}) = D^{k}ϕ(r)

then we can see from equation (4.1) that at least ψ_{k} ∈ C^{2ℓ−2k}.

The simplest way to get a smooth univariate positive definite radial function is by convolution. If f ∈ C^{ℓ} then f ∗ f ∈ C^{2ℓ} at least. We can then apply the operator D k-times to get a radial function in C^{2ℓ−2k} ∩ CS ∩ PD_{2k+1}.

Example 3: Let f_{ℓ}(x) = (1 − x^{2})_{+}^{ℓ} ∈ C^{ℓ−1} be a cut-off polynomial. Then the convolution ϕ_{ℓ} = f_{ℓ} ∗ f_{ℓ} is a cut-off polynomial of degree 4ℓ + 1 and in C^{2ℓ} ∩ CS ∩ PD_{1}, and ϕ_{ℓ,k}(r) := D^{k}ϕ_{ℓ}(2r)(where we rescaled ϕ_{ℓk} to the support [0, 1]) is a cut-off polynomial of degree 4ℓ−2k+1 which is in C^{2ℓ−2k} ∩ CS ∩ PD_{2k+1}. Here we use the name cut-off polynomial for symmetric functions that are polynomials in [0, 1] and zero in [1, ∞). Following is a table of such functions for small ℓ. The case ℓ = 0 is the hat function, which is familiar to everybody. We emphasize the fact that this generates multivariate radial “Euclidean” B -splines with an extremely simple mathematical form, i.e.: consisting of a single polynomial function of the Euclidean distance. These B -splines are even simpler to handle than the classical univariate ones, and for ℓ > 0 they even seem to be new in the univariate case.

ℓ = 0  ϕ_{0,0}(r) = (1 − r)_{+} ∈ C^{0} ∩ PD_{1}
ℓ = 1  ϕ_{1,0}(r) ≐ (1 − r)_{+}^{3}(1 + 3r + r^{2}) ∈ C^{2} ∩ PD_{1}
ϕ_{1,1}(r) = Dϕ_{1,0} ≐ (1 − r)_{+}^{2}(2 + r) ∈ C^{0} ∩ PD_{3}
ℓ = 2  ϕ_{2,0}(r) ≐ (1 − r)_{+}^{5}(1 + 5r + 9r^{2} + 5r^{3} + r^{4}) ∈ C^{4} ∩ PD_{1}
ϕ_{2,1}(r) = Dϕ_{2,0}(r) ≐ (1 − r)_{+}^{4}(4 + 16r + 12r^{2} + 3r^{3}) ∈ C^{2} ∩ PD_{3}
ϕ_{2,2}(r) = D^{2}ϕ_{2,0}(r) ≐ (1 − r)_{+}^{3}(8 + 9r + 3r^{2}) ∈ C^{0} ∩ PD_{5}
ℓ = 3 ϕ_{3,0}(r) ≐ (1 − r)_{+}^{7}(5 + 35r + 101r^{2} + 147r^{3} + 101r^{4} + 35r^{5} + 5r^{6}) ∈ C^{6} ∩ PD_{1}
ϕ_{3,1}(r) = Dϕ_{3,0}(r) ≐ (1 − r)_{+}^{6}(6 + 36r + 82r^{2} + 72r^{3} + 30r^{4} + 5r^{5}) ∈ C^{4} ∩ PD_{3}
ϕ_{3,2}(r) = D^{2}ϕ_{3,0}(r) ≐ (1 − r)_{+}^{5}(8 + 40r + 48r^{2} + 25r^{3} + 5r^{4}) ∈ C^{2} ∩ PD_{5}
ϕ_{3,3}(r) = D^{3}ϕ_{3,0}(r) ≐ (1 − r)_{+}^{4}(16 + 29r + 20r^{2} + 5r^{3}) ∈ C^{0} ∩ PD_{7}

Note that ϕ_{ℓ,k}(r) has a zero of order 2ℓ+1−k at r = 1. This implies that for k > 1 the derivative discontinuity of ϕ_{ℓ,k} at r = 1 is much less inportant than the derivative discontinuity at zero. For multivariate applications the effect of “knot lines” at distance 1 to centers x_{j} ∈ {x_{1},…, x_{N}} will be thus much less than the derivative discontinuity effects at the centers themselves. Altogether ϕ_{ℓ,k} ∈ C^{2ℓ−2k} ∩ CS ∩ PD_{2k+1} yields a C^{∞} polynomial function of Euclidean distance to the origin of IR^{d} for d ≤ 2k + 1, except for a C^{2ℓ+1−k} discontinuity on the boundary of the unit ball (its support) and a C^{2ℓ−2k} discontinuity at the origin.

Open problem: For any space dimension not exceeding n and prescribed C^{ℓ} smoothness, our example provides a cut-off polynomial of degree 4⌈\frac{ℓ+1}{2}⌉+2⌈\frac{n}{2}⌉+1 which is in C^{ℓ}∩CS∩PD_{n}. Eventually a lower-degree cut-off polynomial can possess this property, too. More precisely, we can look for a univariate polynomial P_{N} of degree at most N with

P_{N}^{(2k+1)}(0) = 0, 0 < 2k + 1 ≤ ℓ

P_{N}^{(k)}(1) = 0, 0 ≤ k ≤ ℓ

and

∫_{0}^{1} P_{N}(\sqrt{r})I_{n}(r, t)dr≠^{≥}0

Our example shows that such a polynomial exists if N = 4⌈\frac{ℓ+1}{2}⌉+2⌈\frac{n}{2}⌉+1. Therefore we finish this paper with an interesting problem: What is the lowest degree of a cut-off polynomial in C^{ℓ} ∩ CS ∩ PD_{n}?

Acknowledgements: It is a pleasure to acknowledge that R. Schaback helped greatly in clarifying my thinking on a number of matters and in providing a lot of very good comments.

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[12] Wu, Zongmin & Schaback R. Shape preserving interpolation with radial basis functions. Preprint. 1990.

TEX output date: May 17, 1994.