Operators On Radial Functions

Robert Schaback and Z. Wu

Abstract. A general theory is provided that allows to write multivariate Fourier transforms or convolutions of radial functions as very simple univariate operations. As a byproduct, an interesting group of operators {I_{α}}_{α∈IR} with I_{α+β} = I_{α} ∘I_{β} = I_{β} ∘I_{α} is defined. It contains the classical derivatives as I_{−1} = \frac{d}{dr} and is intimately connected to the Fourier transform. Applications to the construction of new positive definite radial functions and to new identities for special functions are included.

§1. Radial functions

Among all functions g : IR^{d} \ {0} → IR we consider those which can be written as a univariate function of the Euclidean norm ∥x∥_{2} on IR^{d}.

Definition 1.1. A function g : IR^{d} \{0} → IR is radial if there is a function f : IR_{>0} → IR such that

g(x) = f(∥x∥_{2}^{2}/2),  x ∈ IR^{d} \ {0}.  (1.1)

Remark 1.2. Functions defined on all of IR^{d} are treated similarly. Note that we do not use g(x) = f(∥x∥_{2}) for reasons that will soon be apparent. Any univariate function f : IR_{>0} → IR will induce a radial function on each IR^{d} \ {0} via (1.1), and we shall always assume (1.1) when going over from f to a radial function g on IR^{d} \ {0}.

Our major goal is to provide a toolbox of univariate operators that allows to calculate multivariate transforms of radial functions effectively, including multivariate Fourier transforms and convolutions. The results are somewhat related to the work of S.E. Trione [6] based on Laplace transforms, and of M.A. Pinsky et al. [3] on Fourier series of radial functions. They were applied in several papers that constructed compactly supported positive definite functions [7], [8], [9].

Approximation Theory VIII 1 Charles K. Chui and Larry L. Schumaker (eds.), pp. 1–17. Copyright o^{c} 1995 by World Scientific Publishing Co., Inc. All rights of reproduction in any form reserved. ISBN 0-12-xxxxxx-x

§2. Fourier transforms

Here we define the multivariate Fourier transform symmetrically as

ĝ(ω) := (2π)^{−d/2} ∫_{IR^{d}} g(x)e^{−iω^{T} x}dx

and

ǧ(x) := (2π)^{−d/2} ∫_{IR^{d}} g(ω)e^{+iω^{T} x}dω

for g ∈ L_{1}(IR^{d}), where ^{T} stands for vector transposition. Now if g(x) = f(∥x∥^{2}/2) is a radial function, then the d–variate Fourier transform is (see Stein and Weiss [5], p. 155 with a slightly different normalization of the Fourier transform)

ĝ(ω) = ∥ω∥_{2}^{− \frac{d−2}{2}} ∫_{0}^{∞} f(s^{2}/2)s^{d/2}J_{\frac{d−2}{2}} (s · ∥ω∥_{2})ds

= ∫_{0}^{∞} f ( \frac{s^{2}}{2} ) ( \frac{s^{2}}{2} )^{\frac{d−2}{2}} ( \frac{s · ∥ω∥_{2}}{2} )^{− \frac{d−2}{2}} J_{\frac{d−2}{2}} (s · ∥ω∥_{2})s ds

= ∫_{0}^{∞} f ( \frac{s^{2}}{2} ) ( \frac{s^{2}}{2} )^{\frac{d−2}{2}} H_{\frac{d−2}{2}} ( \frac{s^{2}}{2} · \frac{∥ω∥_{2}^{2}}{2} ) s ds

with the functions J_{ν} and H_{ν} defined by

( \frac{z}{2} )^{−ν} J_{ν}(z) = H_{ν}(z^{2}/4) = ∑_{k=0}^{∞} \frac{(−z^{2}/4)^{k}}{k!Γ(k + ν + 1)} = \frac{F_{1}(ν + 1; −z^{2}/4)}{Γ(ν + 1)}

for ν > −1. If we substitute t = s^{2}/2, we find

\begin{aligned}ĝ(ω) & = ∫_{0}^{∞} f(t)t^{\frac{d−2}{2}} H_{\frac{d−2}{2}} (t · \frac{∥ω∥^{2}}{2} ) dt \\ & =: (F_{\frac{d−2}{2}} f) (∥ω∥^{2}/2) \end{aligned}

with the general operator

(F_{ν}f)(r) := ∫_{0}^{∞} f(t)t^{ν}H_{ν}(tr)dt

for ν > −1 defined on all f : IR_{>0} → IR such that

f(t) · t^{ν+ \frac{1}{2}} ∈ L_{1}(IR_{>0}).  (2.1)

Note that both F_{ν} and H_{ν} generalize to arbitrary ν ∈ IR, provided that restrictions like (2.1) hold. Furthermore, by symmetry of radial functions and our definition of Fourier transforms we have

F −1_{\frac{d−2}{2}} = F_{\frac{d−2}{2}} for d ∈ IN.

We shall see later that this generalizes to F_{ν}^{−1} = F_{ν} for all ν ∈ IR, wherever both operators are defined.

In analogy to the classical space of tempered test functions we shall often use the space S of all functions on IR_{≥0} that are infinitely differentiable such that any derivative vanishes faster than polynomially at infinity. We call such functions tempered radial test functions and remark that f(r) = e^{−r} is in S but not in Schwartz’ space. We can easily consider F_{ν} as an operator on S for all ν ∈ IR, and we shall prove F_{ν}^{2} = Id there. However, we postpone the extension of F_{ν} for ν ≤ −1 somewhat, because there will be a more handy definition. For convenience, we shall mainly work on the function space S, but particular results will be extendable by continuity to much more general radial functions, including some with a singularity at zero.

Our basic tool will be the following lemma which provides a formula for the action of F_{μ} on the special function H_{ν}:

Lemma 2.1. For ν > μ > −1 and all r, s > 0 we have

(F_{μ}H_{ν}(s·))(r) = \frac{s^{−ν}(s − r){ν−μ−1 \atop +}}{Γ(ν − μ)}

with the usual definition of the truncated power function.

Proof: The assertion is a consequence of the Weber–Schafheitlin integral (see Abramowitz and Stegun [1], p. 487, 11.4.41) after substitutions of the type t = s^{2}/2. In detail, we have

\begin{matrix}(F_{μ}H_{ν} ( \frac{u^{2}}{2} ·)) & ( \frac{r^{2}}{2} ) \\ = & ∫_{0}^{∞} t^{μ}H_{μ} ( \frac{r^{2}}{2} t) H_{ν} ( \frac{u^{2}}{2} t) dt \\ = & ∫_{0}^{∞} ( \frac{s^{2}}{2} )^{μ} · s · H_{μ} ( \frac{r^{2}}{2} · \frac{s^{2}}{2} ) H_{ν} ( \frac{u^{2}}{2} · \frac{s^{2}}{2} ) ds \\ = & ∫_{0}^{∞} 2^{−μ}s^{2μ+1} ( \frac{rs}{2} )^{−μ} ( \frac{us}{2} )^{−ν} J_{ν}(us)ds \\ = & 2^{ν}r^{−μ}r^{−ν}∫_{0}^{∞}s^{μ−ν+1}J_{μ}(rs)J_{ν}(us)ds \\ = & \frac{2^{ν}r^{−μ}u^{−ν}2^{μ−ν+1}r^{μ}(u^{2} − r^{2}){ν−μ−1 \atop +}}{u^{ν}Γ(ν − μ)} \\ = & \frac{1}{Γ(ν − μ)} ( \frac{u^{2}}{2} )^{−ν} ( \frac{u^{2}}{2} − \frac{r^{2}}{2} ) {ν−μ−1 \atop +}. \end{matrix}

Theorem 2.2. Let ν > μ > −1. Then for all functions f : IR_{>0} → IR with

f(t) · t^{ν−μ−1/2} ∈ L_{1}(IR_{>0})  (2.2)

it follows that

F_{μ} ∘ F_{ν} = I_{ν−μ}  (2.3)

where the integral operator I_{α} is given by

(I_{α}f)(r) = ∫_{0}^{∞} f(s) \frac{(s − r)_{+}^{α−1}}{Γ(α)} ds, r > 0, α > 0.

Proof of Theorem 2.2: For any tempered radial test function f ∈ S we evaluate (F_{μ} · F_{ν})f(r) by means of Lemma 2.1 to obtain

∫_{0}^{∞} H_{μ}(tr)t^{μ} ∫_{0}^{∞} H_{ν}(st)s^{ν}f(s)ds dt

=∫_{0}^{∞} s^{ν}f(s) ∫_{0}^{∞} t^{μ}H_{μ}(tr)H_{ν}(ts)dt ds
=∫_{0}^{∞} s^{ν}f(s) · F_{μ}(H_{ν}(s·))(r)ds
=∫_{0}^{∞} f(s) \frac{(s − r)_{+}^{ν−μ−1}}{Γ(ν − μ)} ds = (I_{ν−μ}f)(r).

Operators on Radial Functions 5 By standard continuity arguments this proof extends to any f with (2.2).

As a step towards proving F {−1 \atop μ} = F_{μ} in general, we need Lemma 2.3. For ν > μ > −1 and all s, r > 0 we have

\begin{aligned}F_{μ} ( \frac{s^{−ν}(s − ·)_{+}^{ν−μ−1}}{Γ(ν − μ)} ) (r) & = F_{μ}^{2}(H_{ν}(s·))(r) \\ & = H_{ν}(rs). \end{aligned}

Proof: We directly calculate the assertion and use Lemma 4.13 of Stein and Weiss (p. 170). In detail,

F_{μ} ( \frac{s^{−ν}(s − ·)_{+}^{ν−μ−1}}{Γ(ν − μ)} ) (r)

=∫_{0}^{∞} t^{μ} \frac{s^{−ν}(s − t)_{+}^{ν−μ−1}}{Γ(ν − μ)} H_{μ}(tr)dt
= \frac{s^{−ν}}{Γ(ν − μ)} ∫_{0}^{s} t^{μ}(s − t)^{ν−μ−1}H_{μ}(tr)dt
= \frac{s^{−ν}}{Γ(ν − μ)} ∫_{0}^{s} t^{μ}(s − t)^{ν−μ−1}J_{μ}(2\sqrt{rt})(rt)^{−μ/2}dt,

and by substitution t = su^{2}, we get

\begin{matrix}= & \frac{s^{−ν}}{Γ(ν − μ)} ∫_{0}^{1} s^{μ}u^{2μ}s^{ν−μ−1}(1 − u^{2})^{ν−μ−1}J_{μ}(2\sqrt{rs}u)(rsu^{2})^{−μ/2}2su du \\ = & \frac{2(rs)^{−μ/2}}{Γ(ν − μ)} ∫_{0}^{1} u^{μ+1}(1 − u^{2})^{ν−μ−1}J_{μ}(2\sqrt{rs}u)du \\ = & \frac{2(rs)^{−μ/2}}{Γ(ν − μ)} \frac{2^{ν−μ−1}Γ(ν − μ)}{(2\sqrt{rs})^{−ν−μ}} J_{ν}(2\sqrt{rs}) \\ = & (\sqrt{rs})^{−ν}J_{ν}(2\sqrt{rs}) \\ = & H_{ν}(rs). \end{matrix}

Corollary 2.4. For all ν > μ > −1 and all s, r > 0

F_{μ}^{2} ( \frac{s^{−ν}(s − ·)_{+}^{ν−μ−1}}{Γ(ν − μ)} ) (r)

=F_{μ}(H_{ν}(s·))(r)
= \frac{s^{−ν}(s − r)_{+}^{ν−μ−1}}{Γ(ν − μ)} .

Theorem 2.5. F 2 μf = f holds for all μ > −1 and on all tempered or compactly supported C^{∞} test functions, and on all continuous and locally L_{1} functions f for which F 2 μf is well–defined.

Proof: The proof follows by density of truncated power functions and by continuity arguments.

§3. Properties of integral operators

We now generalize (2.2) for all ν, μ ∈ IR_{>−1}. To do this, we define

\begin{aligned}(I_{0}f)(r) & := f(r),  f ∈ C(IR_{>0}) \\ (I_{−1}f)(r) & := −f′(r),  f ∈ C^{1}(IR_{>0}) \\ I_{−n} & := I{n \atop −1},  n > 0 \\ I_{−α} & := I_{n−α} ∘ I_{−n}, 0 < α ≤ n = ⌈α⌉ \end{aligned}

and get

Theorem 3.1. The operators I_{α} for α ∈ IR form an Abelian group isomorphic to (IR, +) via α ↦ I_{α}, if the operators are restricted to functions in S. Furthermore, they preserve compact supports for all α, and they are monotonic in the sense

(I_{α}f)(r) ≥ 0 for all r > 0
if  f(r) ≥ 0 for all r > 0,

provided that α ≥ 0.

Proof: For α, β, γ ≥ 0 we have

I_{α} ∘ I_{β} = F_{γ}F_{γ+α}F_{γ+α}F_{γ+α+β}
= F_{γ}F_{γ+α+β} = I_{α+β}
= I_{β} ∘ I_{α}

because of F {−1 \atop γ} = F_{γ} for γ > −1. For n ∈ IN and 0 ≤ n ≤ α it is easy to see by explicit differentiation that

I_{α} ∘ I_{−n} = I_{α−n} = I_{−n} ∘ I_{α}

holds. The definition I_{−α} := I_{n−α} ∘ I_{−n} for 0 < α ≤ n ∈ IN is independent of n, since for any k ∈ IN_{≥0} we have

(I_{n−α} ∘ I_{−n})(f) = (−1)^{n}I_{n−α}f^{(n)}
= (−1)^{n+k}I_{n−α}I_{k}f^{(n+k)}
= I_{n+k−α}I_{−n−k}f.

Thus the identity

I_{−α} = I_{n−α} ∘ I_{−n} = I_{−n} ∘ I_{n−α}

is valid for all 0 < α < n, and

I_{α} ∘ I_{−α} = I_{α} ∘ I_{n−α} ∘ I_{−n} = I_{n} ∘ I_{−n} = I_{0}.

Likewise,

I_{−α} ∘ I_{α} = I_{n−α} ∘ I_{−n} ∘ I_{α} = I_{n−α} ∘ I_{α} ∘ I_{−n} = I_{0}

for all 0 < α < n. If α, β ∈ IR are general, take k ≥ −α, n ≥ −β to find

\begin{aligned}I_{α+β} & = I_{α+β+n+k} ∘ I_{−n−k} = I_{α+k} ∘ I_{β+n} ∘ I_{−n} ∘ I_{−k} \\ & = I_{α+k} ∘ I_{−k} ∘ I_{β+n} ∘ I_{−n} = I_{α} ∘ I_{β} \end{aligned}

and similarly I_{α+β} = I_{β} ∘ I_{α}. The remaining properties of I_{α} follow from the definition.

Note that the operator group {I_{α}}_{α∈IR} nicely interpolates between all classical derivatives and integrals, which occur as special cases I_{n} for n ∈ ZZ. Writing a radial function in the form (1.1) instead of g(x) = f(∥x∥_{2}) pays off by getting nice forms of I_{1} and I_{0}. The identity f = (I_{n} ∘ I_{−n})f is

f(r) = (−1)^{n} ∫_{r}^{∞} f^{(n)}(s) \frac{(s − r)^{n−1}}{(n − 1)!} ds, r > 0

for functions f ∈ C^{n}(IR_{>0}) with f^{(n)}(t) · t^{n+ \frac{1}{2}} ∈ L_{1}(IR_{>0}). This is Taylor’s formula at infinity, and for this reason the group {I_{α}}_{α∈IR} should be called Taylor’s group.

Theorem 3.2. On test functions in S the identity

I_{ν−μ} = F_{μ} · F_{ν}  (3.1)

holds for all ν, μ > −1.

Proof: We only have to treat the case μ > ν > −1, where

\begin{aligned}I_{ν−μ} & = (I_{μ−ν})^{−1} = (F_{ν} ∘ F_{μ})^{−1} \\ & = F {−1 \atop μ} ∘ F {−1 \atop ν} = F_{μ} ∘ F_{ν}. \end{aligned}

Definition 3.3. The operators F_{μ} may be defined for all μ ∈ IR and on S by

F_{μ} := I_{ν−μ} ∘ F_{ν}  (3.2)

where some ν > −1 is taken.

Theorem 3.4. With the above definition,

I_{ν−μ} = F_{μ} ∘ F_{ν}  (3.3)

holds for all μ, ν ∈ IR. Another useful identity is

F_{ν} ∘ I_{α} = I_{−α} ∘ F_{ν}  (3.4)

for all α, ν ∈ IR.

Proof: We first assert that (3.3) holds for all ν > −1, α ∈ IR. It follows from (3.2) for ν + α > −1 or α > 0. The case α < 0 is then implied by

I_{−α} ∘ F_{ν} ∘ I_{−α} = I_{−α} ∘ I_{α} ∘ F_{ν} = F_{ν}.

The independence of (3.4) from the choice of ν > −1 is easily implied by (3.2). Thus F_{μ} for μ ≤ −1 is well–defined by (3.4) and (3.1) now holds for μ ∈ IR, ν > −1. For μ, ν ∈ IR we define

F_{μ} := I_{ρ−μ} ∘ F_{ρ}, F_{ν} := I_{ω−ν} ∘ F_{ω}

for ρ, ω > −1 and get

F_{μ} ∘ F_{ν} = I_{ρ−μ} ∘ F_{ρ} ∘ I_{ω−ν} ∘ F_{ω}
= I_{ρ−μ} ∘ F_{ρ} ∘ F_{ω} ∘ I_{ν−ω}
= I_{ν−μ}.

Finally, for μ ∈ IR and F_{μ} = I_{ν−μ} ∘ F_{ν} with ν > −1 we find

F_{μ} ∘ I_{α} = I_{ν−μ} ∘ F_{ν} ∘ I_{α}
= I_{ν−μ} ∘ I_{−α} ∘ F_{ν}
= I_{−α} ∘ I_{ν−μ} ∘ F_{ν}
= I_{−α} ∘ F_{μ}.

For α > 0 we can use the definition of Γ(α) to see that

I_{α}(e^{−r}) = e^{−r}

holds, and this easily generalizes to all α ∈ IR, since also I_{−1}e^{−r} = −(e^{−r})′ = e^{−r}. By direct calculation we get

F_{ν}(e^{−r}) = e^{−r}

for all ν > −1 and this also generalizes to all ν ∈ IR. Thus the exponential function e^{−r} acts as a universal nontrivial fixed point for the two operator families {I_{α}}_{α∈IR} and {F_{ν}}_{ν∈IR} which deserve further study as actions on S. For each function f ∈ S the orbits

α ↦ f_{α} := I_{α}f  (“Taylor orbit”)
ν ↦ g_{ν} := F_{ν}f  (“Fourier orbit”)

define curves on S. They degenerate into single points for f(r) = 0 and f(r) = e^{−r}, and these two points are the only cases where the curves {f_{α}}_{α∈IR} and {g_{ν}}_{ν∈IR} can intersect at all:

Theorem 3.5. If for some tempered test function f : IR_{≥0} → IR and some α, ν ∈ IR the identity

I_{α}f = F_{ν}f

is valid, then it holds for all α and ν, while f necessarily coincides with e^{−r} or the zero function.

Proof: We consider

F_{μ}I_{β}f = F_{μ}I_{β−α}I_{α}f
= F_{μ}I_{β−α}F_{ν}f
= F_{μ}F_{ν}I_{α−β}f
= I_{ν−μ+α−β}f

and get F_{μ}I_{β}f = I_{β}f whenever ν−μ+α = 2β. Thus for β = \frac{1}{2} (ν + α + \frac{1}{2} ) and μ = −1/2 we have that I_{β}f is a fixed point of the univariate symmetric “radial” Fourier transform in the sense of Section 2. Since e^{−r^{2}/2} is the only nontrivial fixed point of the univariate classical Fourier transform, the assertion follows.

§4. Recursion of Fourier transforms

We can use (3.3) in the form

F_{μ} = I_{ν−μ}F_{ν} = F_{ν}I_{μ−ν}

to express Fourier transforms via other Fourier transforms. With F_{d} := F_{\frac{d−2}{2}} , the d–variate Fourier transform written as a univariate operator on radial functions, we find

F_{n} = I_{(m−n)/2}F_{m} = F_{m}I_{(n−m)/2}

for all space dimensions m, n ≥ 1. Recursion through dimensions can be done in steps of two via

F_{m+2} = I_{−1}F_{m} = F_{m}I_{1}

and in steps of one by

F_{m+1} = I_{−1/2}F_{m} = F_{m}I_{1/2}.

Note that the operators I_{1}, I_{−1}, and I_{1/2} are much easier to handle than the Hankel transforms F_{μ} and F_{m}. This allows simplified computations of Fourier transforms of multivariate radial functions, if the univariate Fourier transforms are known. Furthermore, these operators map compactly supported functions to compactly supported functions, and the I_{α} operators for α ≥ 0 are monotone, i.e. they map nonnegative functions to nonnegative functions. This was successfully used by Wu [8] to construct piecewise polynomial compactly supported radial functions with nonnegative Fourier transforms and to characterize all radial functions with nonnegative multivariate Fourier transforms [9].

To be somewhat more explicit, consider odd-dimensional Fourier transforms

F_{2m+1} = I_{−1}F_{2m−1} = I_{−1}^{m}F_{1}.

Thus, if g is a univariate function that describes the univariate Fourier transform of a radial function f after substitution t ↦ t^{2}/2, then

I{m \atop −1}g = (−1)^{m}g^{(m)}

is the radial Fourier transform of f in 2m + 1 dimensions. This very simple fact should be mentioned in all standard tables of Fourier transforms. Note that Fourier transforms in even dimensions also nicely boil down to bivariate transforms, but the transition between bivariate and univariate transforms does not use a plain derivative, but instead the semi–derivative

(I_{−1/2}f)(r) = (I_{1/2}I_{−1}f)(r) = − ∫_{r}^{∞} f′(s) \frac{(s − r)^{−1/2}}{Γ(1/2)} ds  (4.1)

§5. Convolutions

Two functions g, h ∈ L_{1}(IR^{d}) can be convolved in IR^{d} via

(g ∗_{d} h)(x) = ∫_{y∈IR^{d}} g(y)h(x − y)dy.

If g and h are radial, the result is again a radial function, whose d–variate Fourier transform is the product of the d–variate Fourier transforms of g and h.

Definition 5.1. The operator

C_{ν} : S × S → S

defined by

C_{ν}(f, g) = F_{ν}((F_{ν}f) · (F_{ν}g))

is a generalized convolution operator on radial functions. For ν = \frac{d − 2}{2} it coincides with the operator that takes d–variate convolutions of radial functions and rewrites the result in radial form.

We can now use Theorem 3.2 to describe transforms between two convolutions:

Theorem 5.2. For ν, μ ∈ IR we have

C_{ν}(f, g) = I_{μ−ν}C_{μ}(I_{ν−μ}f, I_{ν−μ}g)  (5.1)

for all f, g ∈ S.

Proof: By definition and (3.1) we find

F_{ν}C_{ν}(f, g) = F_{ν}f · F_{ν}g
= F_{μ}I_{ν−μ}C_{ν}(f, g) = (F_{μ}I_{ν−μ}f) · (F_{μ}I_{ν−μ}g)
= F_{μ}C_{μ}(I_{ν−μ}f, I_{ν−μ}g)

and get

I_{ν−μ}C_{ν}(f, g) = C_{μ}(I_{ν−μ}f, I_{ν−μ}g)

and

C_{ν}(f, g) = I_{μ−ν}C_{μ}(I_{ν−μ}f, I_{ν−μ}g).

The main application of Theorem 5.2 consists in the reduction of multivariate convolutions to univariate convolutions. In fact, for dimensions d ≥ 1 we have

C_{\frac{d−2}{2}} (f, g) = I_{\frac{1−d}{2}} C_{− \frac{1}{2}} (I_{\frac{d−1}{2}} f, I_{\frac{d−1}{2}} g) .

If d is odd, the d–variate convolution of radial functions finally just boils down to a derivative of a univariate convolution of integrals of f and g. For instance,

\begin{aligned}f ∗_{3} g & = I_{−1}((I_{1}f) ∗_{1} (I_{1}g)) \\ & = − \frac{d}{dr} ((∫_{r}^{∞} f) ∗_{1} (∫_{r}^{∞} g)) . \end{aligned}

Since those transformations preserve compactly supported piecewise polynomials, one can very easily calculate multivariate convolutions of compactly supported piecewise polynomial radial functions on odd–dimensional spaces. The same properties hold for transitions between even–dimensional spaces.

To reduce a bivariate convolution to a univariate convolution, one needs the operations

(I_{1/2}f)(r) = ∫_{r}^{∞} f(s) \frac{(s − r)^{−1/2}}{Γ(1/2)} ds

and (4.1) corresponding to integration and differentiation of order 1/2. If applied to compactly supported piecewise polynomials, these operations will generate unpleasant inverse trigonometric functions.

§6. Special functions

We now study the action of the operators I_{α} and F_{ν} on certain fixed functions f. Starting with some test function f_{0}, we can define

f_{α} := I_{α}f_{0} (α ∈ IR)

and get a variety of integral or differential equations from application of the I_{α} operators via the identities

f_{α+β} = I_{β}f_{α} = I_{α}f_{β}.

Furthermore, we can set g_{ν} := F_{ν}f_{0} and get another series of equations

\begin{aligned}I_{α}g_{ν} & = I_{α}F_{ν}f_{0} = F_{ν−α}f_{0} = g_{ν−α} \\ F_{μ}g_{ν} & = F_{μ}F_{ν}f_{0} = I_{ν−μ}f_{0} = f_{ν−μ} \\ F_{μ}f_{α} & = F_{μ}I_{α}f_{0} = F_{μ+α}f_{0} = g_{μ+α} \end{aligned}  (6.1)

that describe the action of F_{μ} or I_{α} operators on the f_{α} or g_{ν} functions. For m ∈ IN_{≥0} fixed we can start with f_{0}(r) = r^{m}e^{−r} and get for α > 0

f_{α}(r) = e^{−r} · ∑_{j=0}^{m} ( \begin{matrix}m \\ j \end{matrix} ) \frac{Γ(α + j)}{Γ(α)} r^{m−j} =: e^{−r}P_{α}(r)

and for ν > −1 by 11.4.28 of [1], p. 486

g_{ν}(r) := (F_{ν}f_{0})(r) = \frac{Γ(m + ν + 1)}{Γ(ν + 1)} M(m + ν + 1, ν + 1, −r)

with the confluent hypergeometric function. The other cases can be constructed by taking derivatives or integrals, e.g.

f_{α}(r) = (−1)^{n} \frac{d^{n}}{dr^{n}} (e^{−r}P_{α+n}(r)) = (I_{−n}f_{α+n})(r)

for n > 0. Or, for μ ∈ IR and n > 0 with n + μ > −1, we have

g_{μ}(r) = F_{μ}f_{0}(r) = (I_{n} ∘ F_{n+μ})f_{0}(r) = (I_{n}g_{n+μ})(r)

taking integrals of the M function. For n ∈ ZZ_{<0} this describes the differentiation rules for M.

Now the equations (6.1) yield other relations between the f_{α} and g_{ν}, for instance

\begin{aligned}(F_{μ}g_{ν})(r) & = \frac{Γ(m + ν + 1)}{Γ(ν + 1)} ∫_{0}^{∞} M(m + ν + 1, ν + 1, −s)s^{μ}J_{μ}(rs)ds \\ & = f_{ν−μ}(r) = e^{−r}P_{ν−μ}(r), \end{aligned}

which is not in [1]. Since the P_{ν−μ} polynomials are positive for r ≥ 0, this application establishes the confluent hypergeometric functions g_{ν}(r) as new examples of positive definite functions, i.e. as functions with a positive Fourier transform when radialized in each IR^{d}.

Another case that is relevant for applications stems from inverse multiquadrics or Bessel potentials

(c^{2} + ∥x∥_{2}^{2})^{β/2}, c > 0, x ∈ IR^{d}

for β < 0. Here, we leave the restricted domain of test functions, but we shall only treat cases that can be handled via approximation by test functions. The function f_{0} now will be

f_{0}(r) = (γ + r)^{β/2}

with γ = c^{2}/2, and we can apply I_{α} for 0 < α < −β/2 to get another multiquadric

f_{α}(r) = ∫_{r}^{∞}(γ + s)^{β/2} \frac{(s − r)^{d−1}}{Γ(α)} ds

= \frac{1}{γ(α)} ∫_{0}^{∞}(γ + r + t)^{β/2+α−1} ( \frac{t}{t + r + γ} )^{α−1}dt

= \frac{(γ + r)^{β/2+α}}{Γ(α)} ∫_{0}^{1} u^{α−1}(1 − u)^{−α−β/2−1}du

= (γ + r)^{β/2+α} \frac{Γ(−α − β/2)}{Γ(−β/2)} .

Taking derivatives, this extends to α in the range −∞ < α < −β/2, but I_{α} is not applicable to f_{0} for α ≥ β/2.

Fourier transforms will now need Bessel or MacDonald functions K_{μ} via

\begin{aligned}g_{ν}(r) := (F_{ν}f_{0})(r) & = ∫_{0}^{∞}(γ + t)^{β/2}t^{ν}H_{ν}(tr)dt \\ & = \frac{2^{1+β/2}}{Γ(−β/2)} ( \frac{u}{\sqrt{γ}} )^{−ν−1−β/2}K_{ν+1+β/2}(u\sqrt{γ}) \end{aligned}

for r = u^{2}/2 and −1 < ν < − \frac{1}{2} − β (see Abramowitz and Stegun [1],p. 488, 11.4.44). Other values can be obtained by

(F_{μ}f_{0})(x) = (I_{ν−μ}(F_{ν}f_{0}))(r)

where μ ≤ −β − 1 − ν is possible. Further cases can be handled by (6.1) for various ranges of α, μ and ν. The identities I_{α}g_{ν} = g_{ν−α} generalize the differentiation rules for functions of the form z^{−ν}K_{ν}(z) =: L_{ν}(z^{2}/2), namely Lν^{′} = L_{ν+1} (see Abramowitz and Stegun [1] p. 376, 9.6.28).

§7. Compactly supported functions

We now take the characteristic function f_{0}(r) = χ_{[0,1]}(r) and get the truncated power function

(I_{α}f_{0})(r) = ∫_{0}^{1} \frac{(s − r){α+1 \atop +}}{Γ(α)} ds = \frac{(1 − r)α +}{Γ(α + 1)} = f_{α}(r), α > 0.

Now Lemma 2.1 yields

f_{α} = F_{μ}H_{ν}

for ν − μ = α + 1, ν > μ > −1 and

F_{μ}f_{α} = H_{μ+α+1}

for α > 0, μ > −1. If h_{α} = f_{α} ∗ f_{α} is the univariate radial convolution of two instances of f_{α} in the sense

h_{α}(s^{2}/2) = ∫_{−∞}^{+∞} f_{α}(r^{2}/s) · f_{α}((s − r)^{2}/2)dr,  (6.2)

then

F_{1}(h_{α}) = (F_{1}(f_{α}))^{2} = (F_{−1/2}f_{α})^{2} = H{2 \atop α+1/2}.

We now fix 0 ≤ k ≤ m ≤ α and get

F_{2k+1}I_{−m}h_{α} = I_{m}F_{k−1/2}h_{α}
= I_{m−k}I_{k}F_{k−1/2}h_{α} = I_{m−k}F_{−1/2}h_{α}
= I_{m−k}H{2 \atop α+1/2} ≥ 0,

proving that the function I_{−m}h_{α} is positive definite on all IR^{d} with d ≤ 2m+1, and the corresponding radial Fourier transform is explicitly known. This generalizes the construction in [8] and provides Fourier transforms of Wu’s functions in the form ϕ_{α,m} = I_{−m}h_{α}.

Now we apply the above technique to calculate inner products

\begin{matrix}(I_{−m}h_{α}(∥ · −x∥_{2}^{2}/2), I_{−m}h_{α}(∥ · −y∥_{2}^{2}/2))_{L_{2}(IR^{2k+1})} \\ = & C_{k−1/2}(I_{−m}h_{α}, I_{−m}h_{α})(∥x − y∥_{2}^{2}/2) \end{matrix}

that arise when the functions I_{−m}h_{α} are used for L_{2} approximation on IR^{2k+1}, 0 ≤ k ≤ m ≤ α. Theorem 5.2 turns the above quantities into

I_{−k}C_{−1/2}(I_{k−m}h_{α}, I_{k−m}h_{α})  (6.3)

such that one only has to • calculate the univariate convolution of I_{k−m}h_{α} with itself in the sense of (6.2)

• take the k–th derivative.

For moderate integer values of α the explicit determination of h_{α}, I_{k−m}(h_{α}), and (6.3) can be easily done by any program for symbolic calculation. This has been done to provide the examples in [4].

If we use the convolution recursion again, we find the representation

I_{−m}h_{α} = I_{−m}C_{−1/2}(f_{α}, f_{α})
= I_{−m}C_{−1/2}(I_{α}f_{0}, I_{α}f_{0})
= I_{α−m}C_{α−1/2}(f_{0}, f_{0})

of Wu’s functions as integrals of (2α+1)–variate convolutions C_{α−1/2}(f_{0}, f_{0}) of the characteristic function f_{0} of the unit ball in IR^{2α+1}. These positive definite compactly supported functions can be called “Euclids Hats”. An explicit construction based on results of H. Wendland is in [4], while the above argument is from Wendland [7].

Another nice construction due to H. Wendland starts with Askey’s functions

A_{ν}(x) := (1 − ∥x∥_{2})_{+}^{ν}, x ∈ IR^{d}

which are compactly supported and positive definite on IR^{d} for ν ≥ ⌊d/2⌋+1 (see Askey [2]). If they are rewritten as radial functions

a_{ν}(r) = (1 − \sqrt{2r})ν +

in our sense, then the functions I_{k}a_{ν} for k ∈ IN are positive definite and compactly supported in IR^{d} for ν ≥ ⌊d/2⌋ + k + 1 because of

F_{\frac{d−2}{2}} I_{k}a_{ν} = F_{\frac{d+2k−2}{2}} a_{ν}.

When written as functions ψ_{ν,k} of ∥x∥_{2} again, the functions I_{k}a_{ν} are in C^{2k} and consist of a single polynomial piece of degree ν + 2k on their support.

Fixing d and k to ensure C^{2k} and positive definiteness on IR^{d}, the lowest possible value of ν is ⌊d/2⌋+k+1, yielding a degree of ⌊d/2⌋+3k+1. Under the above restrictions on d and k, H. Wendland shows [7] that this degree is minimal.

Further examples along the lines of (6.1) and (4.1) are left to the reader. It should be clear by now that the operator families {F_{ν}} and {I_{α}} together with their intrinsic relations form a useful toolbox for handling radial functions.

References

1. Abramowitz, M. and Stegun, I.A., Handbook of Mathematical Functions, Dover, New York, 1970.

2. Askey, R., Radial characteristic functions, MRC Technical Sum: Report no. 1262, University of Wisconsin, 1973.

3. Pinsky, M.A. and Stanton, N.K. and Trapa, P.E., Fourier series of radial functions in several variables, J. Funct. Anal.116 (1993), 111– 132.

4. Schaback, R., Creating Surfaces from Scattered Data Using Radial Basis Functions, in: Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche and L.L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, 1995, 477–496.

5. Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1971.

6. Trione, S.E., On the Laplace transform of radial functions, Studies in Appl. Math.85 1991, 303–315.

7. Wendland, H., Positive definite and compactly supported radial functions, Preprint, Göttingen, 1995.

8. Wu, Z., Multivariate Compactly Supported Positive Definite Radial Functions, Preprint, Göttingen, 1994. To appear in AICM.

9. Wu, Z., Characterization of Positive Definite Radial Functions, in: Mathematical Methods for Curves and Surfaces, M. Daehlen, T. Lyche and L.L. Schumaker (eds.), Vanderbilt University Press, Nashville, TN, 1995, 573–578.

Robert Schaback Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Lotzestraße 16-18, D-37083 Göttingen, Germany

schaback@namu01.gwdg.de

Zong–min Wu Department of Mathematics, Fudan University, 200433 Shanghai, China