Approximation by Radial Basis Functions

with Finitely Many Centers

Robert Schaback

Institut für Numerische und Angewandte Mathematik

Universität Göttingen

37083 Göttingen Germany

Revised for Constructive Approximation

October 6, 1994

\underline{Abstract}: Interpolation by translates of “radial” basis functions Φ is optimal in the sense that it minimizes the pointwise error functional among all comparable quasi–interpolants on a certain “native” space of functions F_{Φ}. Since these spaces are rather small for cases where Φ is smooth, we study the behavior of interpolants on larger spaces of the form F_{Φ_{0}} for less smooth functions Φ_{0}. It turns out that interpolation by translates of Φ to mollifications of functions f from F_{Φ_{0}} yields approximations to f that attain the same asymptotic error bounds as (optimal) interpolation of f by translates of Φ_{0} on F_{Φ_{0}}.

\underline{AMS Classification}: 41A15, 41A25, 41A30, 41A63, 65D10

\underline{Keywords}: Radial Basis Functions, Multivariate Approximation, Approximation Order

1 Introduction

Given a continuous real–valued function Φ on IR^{d} and a nonnegative integer m, we consider approximations by finitely many translates Φ(· − x_{j}), 1 ≤ j ≤ N, of Φ together with polynomials from the space IP_{m}^{d} of d–variate polynomials of degree less than m. This defines the approximants, but we delay the definition of the functions f that are to be approximated. To comply with the theory of “radial” basis functions (see e.g. review articles by M.J.D. Powell [7], N. Dyn [2] and M. Buhmann [1]), we write the approximants as

p + g_{α} := p + ∑_{j=1}^{N} α_{j}Φ(· − x_{j}),  p ∈ IP_{m}^{d}, α ∈ IR^{N},  (1.1)

for N pairwise distinct “centers” x_{1},…, x_{N} ∈ IR^{d} and with the requirement

∑_{j=1}^{N} α_{j}p(x_{j}) = 0 for all p ∈ IP_{m}^{d}  (1.2)

for the vector α ∈ IR^{N}. Our main concern here is to keep m and Φ fixed and to study the approximation power of functions (1.1) when the number N of centers is large. Another interesting issue is the dependence of the approximation power on the location of the centers, but we do not pursue this question here.

Each function Φ that we shall treat here will implicitly introduce a “native” function space F with a seminorm | . |. But we shall use one function Φ_{0} to define the space F_{0} of functions to be approximated, while the approximants are formed by (1.1) with another, possibly different function Φ_{1}. The corresponding seminorms will be | |_{0} and | |_{1}. Error bounds are known so far only for interpolants with Φ_{0} = Φ_{1}, and for Φ_{0} ≠ Φ_{1} there are some interesting numerical observations (see [8]):

• For F_{0} ⊇ F_{1} the Φ_{1}–interpolants seem to have more or less the same error on the larger space F_{0} as the optimal Φ_{0}–interpolants (quasi–optimality).

• For F_{0} ⊆ F_{1} the Φ_{1}–interpolants seem to behave better on F_{0} than on F_{1} (superconvergence).

The results of this paper serve to support the first statement in case of approximation instead of interpolation.

Bounds for the interpolation error in case Φ_{0} = Φ_{1} are usually of the form

|f(x) − s_{f,X}(x)| ≤ |f|_{0} · P_{0,X}^{∗}(x)  (1.3)

for all x ∈ IR^{d}, f ∈ F_{0}, and all X = {x_{1},…, x_{N}} with the nondegeneracy property

p(X) = {0}, p ∈ IP_{m}^{d} implies p ≡ 0.  (1.4)

Here s_{f,X} is an interpolant to f on X of the form (1.1), and P {∗ \atop 0,X} (x) is the power function that evaluates the norm of the error functional:

P {∗ \atop 0,X} (x) = sup_{|f|_{0}≠0^{f∈F_{0}}} \frac{|f(x) − s_{f,X}(x)|}{|f|_{0}} .

Of course, the error bound (1.3) is large when x is far away from the centers. Therefore there are results that bound P {∗ \atop 0,X} (x) nicely from above whenever x is surrounded by sufficiently many points from X. This is quantified by the “ρ–density”

h_{ρ,X}(x) := sup_{∥y−x∥_{2}≤ρ} min_{z∈X} ∥y − z∥_{2}  (1.5)

of X around x. If X and x satisfy

h_{ρ,X}(x) ≤ h_{0}  (1.6)

for a constant h_{0} depending only on d, ρ and Φ, then error bounds of the form

\begin{aligned}P {∗ \atop 0,X} (x) & ≤ & c · (h_{ρ,X}(x))^{k} \\ P {∗ \atop 0,X} (x) & ≤ & c · exp (−\frac{c}{(hρ,X (x))^{k}} ) \end{aligned}  (1.7)

(see Madych/Nelson [6], [5] and Wu/Schaback [10]) are provided. Here and in the sequel we shall denote generic constants by c.

For approximation one should take x from a compact set Ω ⊂ IR^{d} and then consider all finite sets X such that (1.4) and

h_{ρ,X}(x) ≤ h ≤ h_{0} for all x ∈ Ω  (1.8)

hold. Thus h serves as a scaling parameter to control the approximation quality in terms of the density of points of X with respect to Ω. Note that this requires X to extend at least by a distance ρ out of Ω. But a closer look at the proof technique of [6], [5], and [10] reveals that this is not necessary, provided that the boundary of Ω satisfies a uniform interior cone condition, i.e. there must be a fixed positive angle α such that from each point of the boundary of Ω there is a cone of angle not less than α extending locally into the interior of Ω. The sup in (1.5) is then restricted to the cone instead of a ball. This has independently been observed by W. Light (private communication). In view of (1.7) we should look for bounds like

\begin{aligned}∥f − a_{f,X}∥_{∞,Ω} & ≤ & c · h^{k}  or \\ ∥f − a_{f,X}∥_{∞,Ω} & ≤ & c · exp (−\frac{c}{h^{k}} ) \end{aligned}  (1.9)

for all X satisfying (1.8), where the approximant a_{f,X} is of the form (1.1). In this sense we can compare error orders for interpolation and approximation.

In all interesting cases we shall get that approximation of functions from F_{0} by functions (1.1) with Φ_{1} attains the (optimal) orders of interpolation by Φ_{0} on F_{0}, provided that F_{0} ⊇ F_{1}. This will be done by showing (1.9) for right–hand sides that are comparable to (1.7).

2 Basic assumptions

We assume Φ to be symmetric in the sense Φ(·) = Φ(−·) and to be of at most polynomial growth at infinity. Then Φ has a generalized Fourier transform in the sense of tempered distributions, and we require this (possibly singular) distribution to coincide on IR^{d} \ {0} with a positive continuous function φ in the sense of Jones [3]. The possible polynomial growth at ∞ then corresponds to a singularity of φ at the origin, and we assume

φ(ω) ≤ c · ∥w∥^{−d−s_{0}}  ω ∈ U_{0}  (2.1)

for a fixed and minimal s_{0} ∈ IR in a neighborhood U_{0} of zero. Then m and s_{0} are related by the crucial requirement

2m > s_{0},  (2.2)

and to make the Fourier transform correspondence between φ and Φ analytically sound, we need φ ∈ L_{1}(U_{∞}) for a neighborhood U_{∞} of infinity. Details of this can be found in [4] and [9].

3 Native function spaces

Each pair Φ, φ as defined above will give rise to a “native” function space F_{Φ}. One way of introducing F_{Φ} proceeds by taking generalized Fourier transforms of functions (1.1), resulting in tempered distributions that coincide with functions S_{α} · φ on IR^{d} \ {0}, where

S_{α}(ω) := ∑_{j=1}^{N} α_{j}e^{iω^{T} x_{j}}

is kind of a symbol function that satisfies

|S_{α}(ω)| ≤ c · ∥ω∥_{2}^{m}, c = c(α, x_{1},…, x_{N}, m, N)  (3.1)

due to (1.2). Then the integral

(2π)^{−d} ∫_{IR^{d}} \frac{|S_{α}(ω) · φ(ω)|^{2}}{φ(ω)} dω = (2π)^{−d} ∫_{IR^{d}} φ(ω)|S_{α}(ω)|^{2}dω = |g_{α} − p|_{Φ}^{2} = |g_{α}|_{Φ}^{2}

will exist due to (2.1) and (3.1) and will define a seminorm on the approximants from (1.1). The “native” function space for Φ will now be the largest space to which this seminorm can be properly extended. This will in general be a space of distributions, but for sake of simplicity we restrict ourselves here to the space F_{Φ} of functions f in C(IR^{d}) with a generalized Fourier transform f̂ in the weighted L_{2} space

\{g : ∫_{IR^{d}} \frac{|g(ω)|^{2}}{φ(ω)} dω < ∞\}  (3.2)

such that the Fourier inversion formula

f(x) = (2π)^{−d} ∫_{IR^{d}} f̂(ω)e^{iω^{T} x}dω  (3.3)

holds for all x ∈ IR^{d}. The seminorm in F_{0} = F_{Φ_{0}} then is

|f|_{0}^{2} := |f|_{Φ_{0}}^{2} := (2π)^{−d} ∫_{IR^{d}} \frac{|f̂(ω)|^{2}}{φ_{0}(ω)} dω,

when φ_{0} is the function that coincides with the generalized Fourier transform of Φ_{0} on IR^{d} \ {0}. One of the most important spaces is Sobolew space W_{2}^{k}(IR^{d}) of all functions f ∈ L_{2}(IR^{d}) having distributional derivatives up to order k that coincide with functions in L_{2}(IR^{d}). For k > d/2 this is the native space corresponding to s_{0} = −d, m = 0 and

\begin{aligned}φ(ω) & = & (1 + ∥ω∥_{2}^{2})^{−k} \\ Φ(x) & = & c · ∥x∥_{2}^{k−d/2} · K_{k−d/2}(2π∥x∥_{2}) \end{aligned}  (3.4)

with the Macdonald or modified spherical Bessel function K_{ν}. Due to this observation we shall restrict ourselves to the approximation of functions f from a space F_{0} corresponding to a pair Φ_{0}, φ_{0}. But our approximants (1.1) will use a different pair Φ_{1}, φ_{1}.

4 Basic results

If f is from a “native” space F_{0} := F_{Φ_{0}} with F_{0} larger than F_{1} := F_{Φ_{1}}, we first approximate f by a regularization f_{M} ∈ F_{1} obtained via truncation of the Fourier transform, i.e.:

f̂_{M} := f̂· χ_{M},

χ_{M} being the characteristic function of the Euclidean ball around zero with radius M > 0. Then f_{M} can be defined via (3.3), and there is an easy uniform error bound:

Lemma 4.1 For each function f ∈ F_{0} we have

|f(x) − f_{M}(x)| ≤ |f|_{0} · c_{0}(M),  (4.1)

uniformly in x ∈ IR^{d}, where |.|_{0} is the seminorm in F_{0} and

c_{0}^{2}(M) := (2π)^{−d} ∫_{∥ω∥_{2}≥M} φ_{0}(ω)dω.  (4.2)

Proof: Use (3.3) to get

\begin{aligned}|f(x) − f_{M}(x)| & ≤ & (2π)^{−d}∫_{∥ω∥_{2}≥M} |f̂(ω)|dω \\ & ≤ & ((2π)^{−d} ∫_{∥ω∥_{2}≥M} \frac{|f̂(ω)|^{2}}{φ_{0}(ω)} )^{1/2} · ((2π)^{−d} ∫_{∥ω∥_{2}≥M} φ_{0}(ω) · dω)^{1/2} \end{aligned}

via Cauchy–Schwarz. □

Note that the above proof could allow for an additional o(1) factor in the bound (4.1) for M → ∞, the precise o(1) behavior being dependent on f.

Lemma 4.2 If φ_{0}/φ_{1} is bounded in a neighborhood of zero, then for all M > 0 and all f ∈ F_{0} the function f_{M} lies in F_{1} with seminorm

|f_{M}|_{1} ≤ |f|_{0} · C_{01}(M),

where

C_{01}^{2}(M) := sup_{∥ω∥_{2}≤M} \frac{φ_{0}(ω)}{φ_{1}(ω)} .  (4.3)

Proof: Just evaluate

\begin{aligned}|f_{M}|_{1}^{2} & = & (2π)^{−d}∫_{∥ω∥_{2}≤M} \frac{|f̂(ω)|^{2}}{φ_{1}(ω)} \frac{φ_{0}(ω)}{φ_{0}(ω)} dω \\ & ≤ & |f|_{0}^{2} · sup_{∥ω∥_{2}≤M} \frac{φ_{0}(ω)}{φ_{1}(ω)} . \end{aligned}

□

Note that for M → ∞ the function c_{0}(M) decreases to zero while C_{01}(M) does not decrease. Thus

Lemma 4.3 There is a positive constant c depending only on d, φ_{0}, and φ_{1}, such that for all 0 < ε ≤ c we have an M(ε) with

C_{01}(M(ε)) · ε ≤ c_{0}(M(ε)),  (4.4)

and M(ε) → ∞ for ε → 0. □

From the literature (see e.g. [8]) we cite

Lemma 4.4 Given Φ_{1} with φ_{1} and m_{1}, there is an error bound of the form

|f(x) − s_{f,X}(x)| ≤ |f|_{1} · P_{1,X}^{∗}(x)

for all functions in the native space F_{1} and interpolants s_{f,X} to f by functions (1.1) on sets X = {x_{1},…, x_{N}} ⊂ IR^{d} with (1.4). The power function P_{1,X}^{∗}(x) is the norm of the error functional, i.e.:

P_{1,X}^{∗}(x) = sup_{|f|_{1}≠0} \frac{|f(x) − s_{f,X}(x)|}{|f|_{1}}

and it is the minimum of all such norms, if quasi–interpolants

q_{f,X}(x) := ∑_{j=1}^{N} u_{j}(x)f(x_{j})

with

p(x) = ∑_{j=1}^{N} u_{j}(x)p(x_{j}) for all p ∈ IP_{m}^{d}

are allowed instead of s_{f,X}.

Theorem 4.5 Given two radial basis functions Φ_{0}, Φ_{1} with associated functions φ_{0}, φ_{1} such that φ_{0}/φ_{1} is bounded around zero, there is a positive constant c, depending only on d, φ_{0}, and φ_{1}, such that for all points x ∈ IR^{d} and all sets X of centers satisfying (1.4) and P_{1,X}^{∗}(x) ≤ c there is for all functions f ∈ F_{0} an approximant a_{f,X} of the form (1.1) with Φ = Φ_{1}, satisfying

|f(x) − a_{f,X}(x)| ≤ 2|f|_{0} · c_{0}(M(P_{1,X}^{∗}(x))),

the function M taken from (4.4).

Proof: We pick ε = P_{1,X}^{∗}(x) and M(ε) with (4.4). With this M and a given f ∈ F_{0} we apply Lemmas 4.1, 4.2, and 4.4 for a_{f,X} = s_{f_{M} ,X}. Then the assertion follows from (4.4) and

|f(x) − a_{f,X}(x)| ≤ |f(x) − f_{M}(x)| + |f_{M}(x) − s_{f_{M} ,X}(x)|
≤ |f|_{0} · c_{0}(M) + |f_{M}|_{1} · P_{1,X}^{∗}(x)
≤ |f|_{0} · (c_{0}(M) + C_{01}(M) · ε).

□

Remarks: Under the hypotheses of the theorem, the approximant a_{f,X} can be chosen independent of x, provided that X and x satisfy a uniform bound

P_{1,X}^{∗}(x) ≤ ε ≤ c.

The error bound then is

|f(x) − a_{f,X}(x)| ≤ 2|f|_{0} · c_{0}(M(ε)).

In all practical cases there is an error bound

P_{1,X}^{∗}(x) ≤ F_{1}(h_{ρ,X}(x))

for all x and X satisfying

h_{ρ,X}(x) ≤ h_{0}, F_{1}(h_{ρ,X}(x)) ≤ c

with a monotonic function F_{1} : IR_{>0} → IR_{>0} satisfying F (0) = 0 (see [9] for details). Then for any h with h ≤ h_{0} and F_{1}(h) ≤ c, for all compact sets Ω ⊂ IR^{d} and all functions f ∈ F_{0} we have for all X with

h_{ρ,X}(x) ≤ h for all x ∈ Ω

an error bound

|f(x) − a_{f,X}(x)| ≤ 2|f|_{0} · c_{0}(M(F_{1}(h)))  (4.5)

uniformly for x ∈ Ω. Applications of Theorem 4.5 and (4.5) proceed as follows: First, fix Φ_{0} and φ_{0} to determine c_{0} of (4.2). Then, for any other Φ_{1} and φ_{1}, calculate C_{01} of (4.3) and compare with c_{0} to find the function M that optimally satisfies (4.4). Then Theorem 4.5 or (4.5) can be applied. The following two sections will proceed along these lines.

5 Approximation in Sobolew spaces

We now fix F_{0} = W_{2}^{k}(IR^{d}) with 2k > d and study approximations of the form (1.1) with different functions Φ. Due to (3.4) we have

c_{0}^{2}(M) = (2π)^{−d}∫_{∥ω∥_{2}≥M}(1 + ∥ω∥_{2}^{2})^{−k}dω

= c(d) ∫_{M}^{∞}(1 + r^{2})^{−k} · r^{d−1}dr
= c(d, k)(M^{d−2k} + o(M^{d−2k})) for M → ∞,

while C_{01}(M) will depend on the “radial” basis function Φ_{1} that controls the approximation. Thin–plate splines Φ_{1}(x) = ∥x∥^{β} or ∥x∥^{β} log ∥x∥ for β > 0, β ∉ 2IN or β ∈ 2IN will have

φ_{1}(ω) = c(β, d)∥ω∥^{−d−β}

and φ_{0}/φ_{1} is bounded near zero. Furthermore, for 2k < d + β we have

C_{01}^{2}(M) = c(β, d, k) · M^{d+β−2k}(1 + o(1)) for M → ∞

and (4.4) is satisfied for

M(ε) = c(β, d, k) · ε^{−2/β}(1 + o(1)) for ε → 0

and we have

c_{0}(M(ε)) = c(β, d, k) · ε^{(2k−d)/β}(1 + o(1)) for ε → 0.

Convergence results for optimal interpolants on native spaces guarantee the existence of constants such that for all X and x with h_{ρ,X}(x) ≤ h_{0} the bound

P_{1,X}^{∗}(x) ≤ c · h_{ρ,X}^{β/2}(x) ≤ c · h^{β/2}  (5.1)

holds whenever X and x satisfy h_{ρ,X}(x) ≤ h. For such X and x, we can set ε = h^{β/2} and apply (4.5) to get

|f(x) − a_{f,X}(x)| ≤ 2 · |f|_{0} · c · h^{k−d/2}.

The exponent of h thus is the same as in the optimal error bound

P_{0,X}^{∗}(x) ≤ c · h_{ρ,X}^{k−d/2}(x) ≤ c · h^{k−d/2}

that is attainable on the native space F_{0} itself (see the technique of Wu/Schaback [10]).

We now turn to multiquadrics

Φ_{1}(x) = (1 + ∥x∥_{2}^{2})^{β/2}

for β ∉ 2ZZ, where

φ_{1}(ω) ≤ c(d, β) · \{ \begin{matrix}∥ω∥^{−d−β} & ω near 0 \\ e^{−∥ω∥}∥ω∥^{− \frac{d+β+1}{2}} & ω near infinity \end{matrix} \}

for γ fixed, and where the exponent β in φ_{1} is best possible. Then, up to constants and for M → ∞, we have

C_{01}(M) = M^{−k}e^{M/2}M^{(d+β+1)/4} (1 + o(1)),

and φ_{0}/φ_{1} is bounded near zero. Furthermore, M(ε) has to satisfy

ε · M^{−k}M^{k−d/2}e^{M/2}M^{(d+β+1)/4} = const

which can be done by solving the equation

\frac{1}{2} M(ε) = const − log ε + (\frac{d}{4} − \frac{β}{4} − \frac{1}{4} ) log M(ε)

for sufficiently small ε and large M(ε). Now the optimal error bound on F_{1} is

P_{1,X}^{∗}(x) ≤ c · e^{− \frac{δ}{h_{ρ,X} (x)}} ≤ c · e^{− \frac{δ}{h}} =: ε  (5.2)

for some δ > 0 due to Madych/Nelson [6], and we get log ε ≈ −\frac{δ}{h} and

\begin{aligned}\frac{1}{2} M(ε)(1 + o(1)) & = & \frac{δ}{h} , \\ c_{0}(M(ε)) & ≤ & c(d, k, β, δ)h^{k−d/2}, \end{aligned}

for h → 0 such that (4.5) again equalizes the rate of optimal interpolation in Sobolew space W_{2}^{k}(IR^{d}).

For Gaussians

Φ_{1}(x) = exp(−β∥x∥_{2}^{2})

we find

φ_{1}(ω) = c · exp(−∥ω∥_{2}^{2}/4β)

and φ_{0}/φ_{1} will always be bounded. Clearly

C_{01}(M) = M^{−k}e^{+M^{2}/8β} (1 + o(1))

for M → ∞ and M(ε) has to satisfy

ε · M^{−k}M^{k−d/2}e^{+M^{2}/8β} = const,

or

\frac{M^{2}(ε)}{8β} = const − log ε + \frac{d}{2} log M(ε)

for ε → 0 and M → ∞. Due to Madych/Nelson [6] the optimal error bound is

P_{1,X}^{∗}(x) ≤ c · e^{−_{\frac{δ}{h{2 \atop ρ,X}}} (x)} ≤ c · e^{− \frac{δ}{h^{2}}} ,  (5.3)

and again log ε = (1 + o(1)) (−\frac{δ}{h^{2}} ) yields the optimal order via

c_{0}(M(ε)) ≤ c · h^{k−d/2}.

6 Approximation by Gaussians in spaces defined by multiquadrics

Finally, let us consider approximations of functions from the native space F_{0} for multiquadrics Φ_{0}(x) = (1^{2} + ∥x∥_{2}^{2})^{β/2} by Gaussians Φ_{1}(x) = e^{−α∥x∥^{2}}. Then

c_{0}^{2}(M) ≈ ∫_{M}^{∞} e^{−r}r^{d−1}dr^{−(d+β+1)/2} = c · e^{−M}M^{(d−β−3)/2} · (1 + o(1))

for M → ∞ and

C_{01}^{2}(M) ≤ c · M^{−d−β}e^{M^{2}/4α}(1 + o(1)),
− log ε = const + \frac{M^{2}(ε)}{8α} + \frac{M(ε)}{2} − (\frac{d − 1}{2} + \frac{d + β}{2} ) log M(ε)
= \frac{δ}{h^{2}} (1 + o(1)) (see (5.3)).

for h, ε → 0. This yields

0 < \frac{δ}{h^{2}} − \frac{ζ}{h} ≤ \frac{M^{2}(ε)}{2α} ≤ \frac{δ}{h^{2}} for a suitable ζ > 0

and

c_{0}(M) ≈ (\frac{\sqrt{2αδ}}{h} )^{d−1} exp (−\sqrt{2α (\frac{δ}{h^{2}} − \frac{ζ}{h} )}) ,  (6.4)

which is still an exponential bound, but not of the form (5.2).

7 Final remarks

Unfortunately, the above cases had to be handled individually as special cases of the results of section 4. There would be a fairly general theorem stating quasi–optimality of “radial” basis function approximants on larger spaces, if some additional things, as suggested by the previous section, would hold true. If the interpolation error for Φ_{i} on its own native space is

P_{i,X}^{∗}(x) ≤ F_{i}(h_{ρ,X}(x))

for i = 0, 1, then we can observe that bounds like

\begin{aligned}c_{0} (\frac{c}{h} ) & ≤ & c · F_{0}(h) \\ C_{01} (\frac{c}{h} ) · F_{1}(h) & ≤ & c · F_{0}(h) \end{aligned}  (7.5)

hold for 0 < h ≤ h_{0} in all of the above cases. The technique in Wu/Schaback [10] ties the interpolation error to the behaviour of the Fourier transform around infinity, as required for such bounds as (7.5), but the bounds do not follow from there. There is a gap by a factor of h^{1−d} between (6.4) and (5.2), but the constant δ of (5.2) is not necessarily optimal. The precise form of optimal exponential error bounds for multiquadrics and Gaussians still is unknown: the results of [9] suggest that there might well be a factor of the form h^{k} occurring in the optimal bounds.

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