radial function
Another theorem, which is due to Baxter (1992a), Madych (1990), is the re- Another theorem, which is due to Baxter (1992a), Madych (1990), is the re- sult below, which concerns the accuracy of the multiquadric radial function sult below, which concerns the accuracy of the multiquadric radial function
φ
φ((r r ))
==
√ √
r r 22+
+
cc22 interpolants. However, rather than focussing upon how theinterpolants. However, rather than focussing upon how the interpolants converge as the grid spacing decreases, these results examine the interpolants converge as the grid spacing decreases, these results examine the effect of varying the user-defined parametereffect of varying the user-defined parameter cc. As we have seen already in our. As we have seen already in our discussion of condition numbers in Section 4.2, however, the interpolation ma- discussion of condition numbers in Section 4.2, however, the interpolation ma- trix becomes severely ill-conditioned with growing
trix becomes severely ill-conditioned with growing cc (the condition numbers(the condition numbers gr
grooww exexpoponenentintiallally;y; ththeireir exexpoponenentintialal grgroowtwthh isis gegenunuineine andand nonott ononlyly cocontntainaineded in any unrealistic upper bound) and thus virtually impossible to use in the prac- in any unrealistic upper bound) and thus virtually impossible to use in the prac- tical solution of the linear interpolation system. Therefore this result is largely tical solution of the linear interpolation system. Therefore this result is largely of theoretical interest. For the statement of the result we recall once more the of theoretical interest. For the statement of the result we recall once more the notion of a
notion of a band-liband-limited function, that is mited function, that is a a functiofunction whose Fourier transform isn whose Fourier transform is compactly supported. We also recall in this context the Paley–Wiener theorem compactly supported. We also recall in this context the Paley–Wiener theorem (Rudin, 1991) which states that the entire functions of exponential type are (Rudin, 1991) which states that the entire functions of exponential type are those whose Fourier transforms are compactly supported and which we quote those whose Fourier transforms are compactly supported and which we quote in part in a form suitable to our need.
in part in a form suitable to our need.
Paley
Paley–Wien–Wiener er theortheorem.em. LeLet t f f be be an an enentitirre e fufuncnctition on whwhicich h sasatitisfisfies es ththee estimate estimate
||
f f (( z z))| =| =
OO
(1(1++ ||
z z||
))−−
N N exp(exp(r r||
z z||
))
,, zz∈∈
CCnn,, N N∈∈
ZZ+
+
.. TheThenn thetherree exexististss aa comcompacpactlytly supsupporportedted infiinfinitnitelyely difdifferferententiabiablele funfunctctionion whowhosese support is in B
support is in Br r (0)(0) and whose and whose FFourieourier r trantransform is sform is f f . . HerHere,e,
z z denotes denotes thetheimaginary part of the complex quantity vector z. imaginary part of the complex quantity vector z.
With its aid we establish the following theorem. With its aid we establish the following theorem.
The
Theoreoremm 4.14.19.9. Let f Let f
∈∈
C C ((RRnn))∩∩
L L22((RRnn)) bebe anan eentntiirree fufuncnctitionon asas inin tthehe PPalaleeyy– –Wiener theorem of exponential type
Wiener theorem of exponential type ππ , , i.e. i.e. its its FFourier ourier transform transform is is compactlycompactly supported in
supported in TTnn , , so so that that the the apprapproximand oximand is is band-limited. band-limited. Then Then the the cardicardinalnal
interpolants interpolants sscc(( x x ))
==
j j∈∈
ZZnn f f (( j j)) LLcc(( x x−−
jj)),, x x∈∈
RRnn,,with cardinal Lagrange functions for multiquadrics and every positive param- with cardinal Lagrange functions for multiquadrics and every positive param- eter c eter c L Lcc(( x x ))
==
k k∈∈
ZZnn d d k k cc
x x−−
k k
22+
+
cc22,, x x∈∈
RRnn,,96 4. Approximation on infinite grids
that satisfy Lc( j) = δ0 j , j ∈ Zn , enjoy the uniform convergence property
(4.27) sc − f ∞ = o(1), c → ∞.
Proof: Let χ be the characteristic function of [−π, π]n, which is one for an argument inside that cube and zero for all other arguments. Thus, since f
is square-integrable and since the Fourier transform is an isometry on square- integrable functions (cf. the Appendix), the approximation error is the difference (4.28) sc( x ) − f ( x ) = 1 (2π)n
Rn
k ∈Zn ˆ f (t + 2π k )
Lˆc(t ) − χ(t )
ei x t dt .It follows from (4.28) that |sc( x ) − f ( x )| is at most a constant multiple of
Tn
| f ˆ(t )|
k ∈Zn
| Lˆc(t + 2πk ) − χ(t + 2πk )| dt
which is the same as
Tn | f ˆ(t )|
1 − Lˆc(t ) +
k ∈Zn\{0} ˆ Lc(t + 2π k )
dt = 2
Tn | f ˆ(t )|
1 − Lˆc(t )
dt ,because of the form of the Lagrange function stipulated in Theorem 4.3. The same theorem and the lack of sign change of ˆ Lc (coming from the negativity of the multiquadric Fourier transform) gives the two uniform bounds
0 ≤ 1 − Lˆc(t ) ≤ 1.
The following proposition will evidently finish our proof of Theorem 4.19 because the problem now boils down to proving that the Lagrange function’s Fourier transform is, in the limit for c → ∞, pointwise almost everywhere a certain characteristic function.
Proposition 4.20. Let x ∈ Rn. Then lim
c→∞
ˆ
Lc( x ) = χ( x ) , unless x ∞ = π .
Proof: Let x /∈ Tn. There exists a k 0 ∈ Zn\{0} such that x + 2πk 0 < x ,
and the exponential decay of ˆφc (the radial part of the Fourier transform of
· 2 + c2) provides the bounds ˆ φc( x ) ≤ e−c x +c x +2πk 0 φˆc( x + 2πk 0) (4.29) ≤ e−c x +c x +2πk 0
k ∈Zn ˆ φc( x + 2π k ),4.4 Convergence with respect to parameters 97
where we have now assumed without loss of generality that ˆφc > 0 (we may change the sign of ˆφc without altering the cardinal function Lc in any way). Thus, using again Theorem 4.3, we get
0 ≤ Lˆc( x ) ≤ e−c x +c x +2πk 0 → 0 (c → ∞),
as required.
Now let x ∈ (−π, π)n\{0} (the case x = 0 is trivial and our result is true without any further computation). Thus, for allk 0 ∈ Zn\{0}, x < x +2πk 0,
and we have ˆ Lc( x ) =
1 +
k ∈Zn\{0} ˆ φc( x + 2πk ) ˆ φc( x )
−1which means that it is sufficient for us to show lim c→∞
k ∈Zn\{0} ˆ φc( x + 2πk ) ˆ φc( x ) = 0for x < x + 2πk 0. Indeed, according to (4.29), every single entry in the series in the last display satisfies the required limit behaviour. It thus suffices using (4.29) term by term and summing to show that, denoting (1,1, . . . ,1) ∈
Rn by 1, (4.30)
k ≥21 e−c x +2πk +c x → 0 (c → ∞). However, as k ≥ 21 implies x + 2πk − x ≥ 2π
k − 1
≥ πk for x ≤ π1, we get our upper bound of
k ≥21
e−πck
for the left-hand side of (4.30) which goes to zero for c → ∞ and it gives the required result by direct computation.
It follows from the work in Chapter 4 that the Gaussian radial basis function
φ(r ) = e−c2r 2 used for interpolation on a cardinal grid cannot provide any nontrivial approximation order: it satisfies all the conditions (A1), (A2a), (A3a) for µ = 0 and therefore there exist decaying cardinal functions as with all the other radial basis functions we have studied in this book. However, according to Theorem 4.4, there is no polynomial reproduction with either interpolation
98 4. Approximation on infinite grids
or quasi-interpolation using Gaussians. Therefore, as we have seen above, no approximation orders can be obtained. This depends on the parameter c being fixed in the definition of theφ as the spacing of the grid varies. There is, however, a possibility of obtaining convergence orders of Gaussian interpolation on grids if we let c = c(h) vary with h – so we get another result (of Beatson and Light, 1992) on convergence with respect to a parameter: It is the following result on quasi-interpolation as in Section 4.1 that we state without proof.
Theorem 4.21. Let k be a natural number and ψ be a finite linear combi-
nation of multiinteger translates of the Gaussian radial basis function, whose coefficients depend on c but the number of nonzero coefficients is fixed. If c =
2π2/(k |logh|) , then there is a ψ with the above properties such thatquasi-interpolation using (4.14) fulfils the error estimate for h → 0 and f ∈ W ∞k (Rn)
f − sh∞ ≤ Chk |logh|k /2+[(k −1)/2] f k ,∞.
Here, [ ·] denotes the Gauss-bracket and W ∞k (Rn) is the Sobolev space of all
functions with bounded partial derivatives of total order at mostk . Similarly, we recall the definition of thenonhomogeneousSobolev space denoted alternatively by H k (Rn) or by W k 2(Rn) as (4.31)