Fourier analysis of the approximation power of principal shift-invariant spaces Carl de Boor & Amos

35 Pages·2001·0.23 MB·English

Fourier analysis of the approximation power of
principal shift-invariant spaces
Carl de Boor & Amos Ron
Center for the Mathematical Sciences
and
Computer Sciences Department
University of Wisconsin-Madison
Madison, WI 53706
July 1991
ABSTRACT
The approximation order provided by a directed set fshgh>0 of spaces, each spanned by the
d
hZZ -translatesofonefunction,isanalyzed. The\near-optimal"approximantsof[R2]fromeachsh
to the exponentialfunctions are usedto establishupper bounds onthe approximationorder. These
approximants are also used on the Fourier transform domain to yield approximations for other
smooth functions, and thereby provide lower bounds on the approximation order. As a special
case, the classical Strang-Fix conditions are extended to bounded summable generating functions.
The second part of the paper consists of a detailed account of various applications of these
general results to spline and radial function theory. Emphasis is given to the case when the scale
fshg is obtained from s1 by means other than dilation. This includes the derivation of spectral
approximation orders associated with smooth positive de(cid:12)nite generating functions.
AMS (MOS) Subject Classi(cid:12)cations: primary 41A25, 41A63; secondary 41A15, 42A82.
Key Words and phrases: approximation order, Strang-Fix conditions, exponentials, polynomials,
multivariate, splines, radial basis functions, uniform mesh, regular grids, integer translates, shift-
invariant spaces, principal shift-invariant spaces.
Sponsored in part by the National Science Foundation under grants No. DMS-9000053, DMS-
9102857, and by the United States Army under Contract No. DAAL03-G-90-0090.
Fourier analysis of the approximation power of
principal shift-invariant spaces
Carl de Boor & Amos Ron
1. Introduction
Spaces spanned by (cid:12)nitely or countably many translates of one or several functions play an
important role in spline theory, radial basis function theory, sampling theory and wavelet theory.
Spline theory stresses the case when the generating functions are compactly supported, while sam-
pling theory singles out the case when the spectrum (i.e., the support of the Fourier transform) of
thegeneratingfunctionsiscompact. Intheradialbasisfunctiontheory,neitheroftheseisassumed,
and instead, the computational simplicity as well as the positive de(cid:12)niteness (i.e., the positivity of
the Fourier transform) of the generating functions is preferred. Finally, wavelet theory focuses on
the interrelation between the initial space and its dyadic dilates. In all these areas, the underlying
space s is meant for approximation or decomposition of functions, and thus, the determination of
its approximation properties is of basic signi(cid:12)cance.
The present literature is mainly concerned with a space s which is the algebraic or topological
span of the integer translates of one generating function . More precisely, we hold a collection
f hgh2I of complex-valuedmeasurable functions de(cid:12)ned on IRd, where I is either the open interval
(0dh ), or a discrete subset of such an interval (e.g., f2−n : n 2 INg). For each h, we look at
0 P
all linear combinations (cid:11)2hZZd h((cid:1) − (cid:11))a((cid:11)), for which this sum converges in a certain sense,
and denote by sh the space of all limit functions obtained in this way. Roughly speaking, we
d
call sh the span of the hZZ -translates of h, aPnd this is an exact description of sh in case h is
of compact support, a case in which the sum (cid:11)2hZZd h((cid:1)− (cid:11))a((cid:11)) is locally (cid:12)nite, and hence
arbitrary linear combinations are allowed in this sum. Approximation properties are primarily
studied via approximation orders: for the given scale fshgh, one examines the rate of decay (as
h ! 0) of dist(f;sh), where f varies over some space of admissible functions, which must contain
all test functions in D(IRd) (namely, all C1(IRd) compactly supported functions), and the distance
dist(f;sh) between f and sh is measured in some norm, usually a p-norm (1 (cid:20) p (cid:20) 1), or a
weighted p-norm. We say that the approximation order of the scale fshgh is k (or O(hk)), for
some positive (usually integer) k, if, for every admissible f, dist(f;sh) = O(hk), with a constant
that depends on f (and clearly not on h), while, for some admissible f, dist(f;sh) 6= o(hk).
Although a discussion of the above model can already be found in Schoenberg’s work [S]
(for univariate functions), the (cid:12)rst comprehensive analysis of approximation orders was carried out
about twentyyears ago primarilybypeoplefrom the(cid:12)nite elementgroup, thebest knownreference
for which is [SF]. Strang and Fix considered the \compactly supported stationary case", namely,
when 1 is compactly supported and h is its h-dilate (i.e., h = 1((cid:1)=h)), and showed (for the
2-norm) that approximation orders are characterized by thepolynomialsin s . Some modi(cid:12)cations
1
and improvements of these results (known these days as \The Strang-Fix Conditions") can be
found in [DM2], [BJ] and [R2]. However [DR], the polynomials in spline spaces are unrelated to
approximation orders if the f hgh are not the dilates of one function. Discussion of approximation
1
orders for compactlysupported piecewise-exponentialsf hgh can be found in [DR], [BR] and [LJ].
We know of no study of approximation orders for general compactly supported f hgh.
In the study of the above problem, one usually considers separately the questions of lower
bounds and upper bounds on the approximation order (and hopes of course to match them). The
standardapproachtolowerboundsisviathequasi-interpolationargument: (cid:12)rst,aspaceH (cid:26) \hsh
is identi(cid:12)ed, and then the local approximation properties of H are converted to approximation
orders of fshgh with the aid of local linear operators (=quasi-interpolants) whose restriction to H
is the identity. The space H consists of polynomials in the stationary case, and of exponential-
polynomialsinthepiecewise-exponentialcase,butneednottobesoingeneral(cf.[BAR]).Further,
the condition H (cid:26) sh, all h, is convenient, but not essential, as the quasi-interpolation argument
of [R2] shows. For earlier constructions of quasi-interpolants see, e.g., [SF] and [BF]. An updated
discussion, together with a partial bibliography, can be found in [B2] and [BR].
In contrast to lower bounds, there does not seem to exist a standard approach to the upper
bound question. We already mentioned [SF] and [BJ], and we add [LC], [JL] and [HL], where
weaker forms of approximation orders (\local", \controlled-local") are characterized, under the
assumption that the generating functions are either compactly supported or maintain a high order
0 at 1 (where \high" is de(cid:12)ned relative to the desired approximationorder, andseveral generating
functionsareallowedineachh-layer). However, alltheseresultsarecon(cid:12)nedtothestationarycase,
and further, the fast decay at 1 that is required from the generating functions excludes various
functions of interest. Sharp upper bounds on the approximation order of polynomial box spline
spaces and exponential box spline spaces (integer direction case) were derived in [BH] and [LJ]
respectively, based on the local structure of the spline space, which in general is a rare possibility
(seetheboxsplinesectioninthispaper). Optimalschemesforapproximatingboundedexponentials
in the non-scaling (still, compactly supported) case were introduced in [R2]. These results will be
presented in the sequel, since they form the starting point for the upper bound analysis here.
We introduce and analyze in this paper a new approach for the determination of the approx-
imation orders of the scale fshgh. In this approach, only modest decay rates are required of the
generating function h (e.g., some maximal function h# should be integrable), and the questions
of upper bounds and lower bounds are attacked almost simultaneously, so that, for all special cases
studied here, they match each other and the approximation order is determined. Using Fourier
analysis methods, we further need not restrict our attention to integral approximation orders. On
the other hand, for the lower bound part, we place some smoothness conditions on the generating
functions, whichare met inall exampleswe knowfrom theradial basisfunctiontheory, but exclude
splines of low smoothness, so that we have here the usual smoothness-localization trade-o(cid:11). This
approach makes no use of quasi-interpolation arguments; in particular, polynomial or exponential
reproduction is not required. In addition, the approximation scheme is constructive enough for the
determination of realistic estimates for the constant which is hidden in the O(hk) expression.
In spite of the generality of the results here, we are able to apply them directly to obtain
upper and matching lower bounds for the case when the generating function is a(n exponential)
box spline with rational directions. We believe that none of the methods now in the literature
could provide either bounds. We show the important fact that many of the lower bounds known
for radial basis (and related) functions underestimate the correct approximation order, and explain
2
this phenomenon. Finally, we show that the use of basic molli(cid:12)ers for the generating function (e.g.,
the Gaussian kernel) leads, if properly used, to in(cid:12)nite approximation orders.
As mentioned, lower bounds on the approximation order were derived previously with the
aid of quasi-interpolants, and the di(cid:14)culty we observed in the implementation of this method
encouraged us to start the work reported here. While the quasi-interpolation argument is an
extremely useful and powerful tool in the compactly supported stationary case, its application in
other known situations is complicated. For example, for piecewise-exponentials, the space H of
exponential-polynomials in \hsh might be hard to determine, its local approximation properties
might be even harder to analyze (cf. [DR]), and the lower bounds attained in this way might
underestimate the true approximation order (all these three are valid di(cid:14)culties in the exponential
box spline/rational direction case). But the major drawback of the quasi-interpolation argument
appears in the area of radial basis functions (cf. [P] and the references therein). In almost all
examples there, h is the h-dilate of 1, hence one expects to use polynomial reproduction in
the quasi-interpolation argument. Still, if the function does not decay fast enough, standard
1
polynomial reproduction arguments (namely, Poisson’s summationformula) do not apply. Further,
even if all desired polynomials are shown to be reproduced, more subtle information on the rates of
decay of is required, [DJLR], [Bu1-3]. At the outset of our present study, we tried to apply to
1
these cases the quasi-interpolation argument from [R2], which involves only bounded exponentials,
but found that, although the polynomial reproduction argument can be circumvented in this way,
no better approximation orders are obtained.
The crux of all the analysis here is the linkage between the Fourier transform and Fourier
series via the periodization argument, and which is best expressed by Poisson’s summation for-
mula. Starting with [S], this tool has always been the chief Fourier analysis argument for polyno-
mial/exponential reproduction. The results of this work show that the periodization argument is
not only an important technical tool, but is at the center of the approximation order analysis: the
rearrangement of the error into Fourier series allows us to distinguish between terms that can be
reduced by an optimal selection of the approximant, and terms that can be small only because of
the good approximation properties of the spaces fshgh.
We have chosen in this paper to focus on the L1 case, namely, measure the error in the
1-norm, primarily since this substantially simpli(cid:12)es the analysis of upper bounds (by making the
exponentialfunctionsadmissibleforapproximation). On the other hand, this norm is probablyone
of the harder choices in the lower bound analysis (certainly when compared to the 2-norm): Since
the approximation is performed entirely in the Fourier transform domain, we needed to bound the
Sobolev (or potential) norm of the function to be approximated in terms of its Fourier transform,
andthus our notionof \admissibility"fallsshort of the usualSobolevspace. Further, as we already
mentioned before, our lower bound conditions exclude generating functions of low smoothness, and
thisisagainrelatedtothechoiceofthenorm: theerrorintheapproximationschemecanbewritten
and analyzed in terms of certain Fourier multipliers, whose Fourier transform is explicitly known.
However, to obtain sharp results with the aid of these multipliers requires, because of the use of
the 1-norm, information about the behaviour of the multiplier in the original domain, which, as a
rule, is not easily accessible.
Throughout the paper, C stands for the unit cube [−1=2d1=2]n, and B(cid:17) for the L2(IRd)-ball
3
of radius (cid:17) centered at the origin. We use the notation e(cid:18), (cid:18) 2IRd, for the complex exponential
e(cid:18) : x7! ei(cid:18)(cid:1)x;
and denote by (cid:30)(cid:3)0 the semi-discrete convolution
X
(cid:30)(cid:3)0 : f 7! (cid:30)((cid:1)−(cid:11))f((cid:11));
(cid:11)2ZZd
d
where f is any function de(cid:12)ned (at least) on ZZ . The Fourier transform of the summable function
f is de(cid:12)ned by Z
b
f((cid:18)) := e−(cid:18)(t)f(t)dt;
IRd
and is extended by duality to all distributions in D0(IRd). We also make use of the discrete
e
Fourier transform (or symbol) f of the function f (of polynomial growth), de(cid:12)ned as
X
e
f := e−(cid:11)f((cid:11)):
(cid:11)2ZZd
Note that
(1:1) fe(w) = (f(cid:3)0ew)(0) = (e−wf(cid:3)01)(0)
in case fjZZd 2 ‘1(ZZd). We denote by (cid:5) the ring of all polynomials in d variables, and (cid:5)n is the
subspace of polynomials of degree at most n. Also, (cid:5)<n := (cid:5)n−1.
d d
As a rule, (cid:11), (cid:12) are generic points of ZZ , 2(cid:25)ZZ , respectively, and (cid:18), w are generic points of the
Fourier transform domain. Also, the default norm is k(cid:1)k := k(cid:1)k1, while, for x 2 IRd, jxjp is its
p-norm, and jxj :=jxj is its Euclidean norm.
2
2. BOUNDS ON THE APPROXIMATION ORDER
2.1. Principal shift-invariant spaces
We are interested in characterizing the approximation order of the spaces fshgh. This is,
by de(cid:12)nition, the maximal nonnegative k for which
dist1(f;sh) =O(hk); when h ! 0;
for every k-admissible f.
In order for this de(cid:12)nition to make any sense, we need to de(cid:12)ne precisely the spaces fshgh, as
well as explain the notion of \k-admissible". We start with the former.
4
d
We take sh to be an appropriate closure of the linear hull of the hZZ -translates of some
Pfunction, its generating function. Speci(cid:12)cally, we take sh to consist of functions of the form
(cid:11)2hZZd h((cid:1) − (cid:11))a((cid:11)), with, possibly, some restriction imposed on the coe(cid:14)cient sequence a.
Because of the nature of the results in this paper, it is convenient to scale up sh, i.e., to look at the
space
(2:1) Sh :=ff(h(cid:1)): f 2shg:
ThespaceSh isaprincipal shift-invariantspace,whichmeans,byde(cid:12)nition,thatitis\spanned"
by the integer translates of one generating function (cid:30)h (which happens to be h(h(cid:1))). Denoting by
S((cid:30))
the principal shift-invariant space generated by the integer translates of (cid:30), we can then write
Sh = S((cid:30)h). Since the 1-norm is scale-invariant, we have
dist1(f;sh) = dist1(f(h(cid:1));S((cid:30)h));
hence the change from sh to the scaled space Sh = S((cid:30)h) requires nothing more than switching
from f to the correspondingly scaled f(h(cid:1)). As a simple example, note that in the stationary case,
when h is the h-dilate of 1, the scale-up procedure undoes the dilation and hence (cid:30)h = (cid:30)1 = 1
for all h. In other words, Sh does not change with h.
Thus our setting is as follows: we hold in hand a collection fShgh of spaces, each of which is a
principalshift-invariantspacegeneratedbysomeh-dependentfunction(cid:30)h. Thenfora\reasonable"
function f, we consider the quantities dist1(f(h(cid:1));Sh). Whenever these quantities decay to 0 like
hk, we say that fShgh provides approximation order k for f. If dist1(f(h(cid:1));Sh) = O(hk) for all
k-admissible functions, then we say that fShgh provides approximation order k.
We have not yet de(cid:12)ned the topology used in the de(cid:12)nition of the principal shift-invariant
space S((cid:30)). Whilethe derivationof lowerbounds is largelyindependentof the topologyusedinthe
de(cid:12)nition of this \spline" space (since only a small subset of the space is usually employed in the
analysis), upper bounds are intimately related to the way S((cid:30)) is de(cid:12)ned: results on upper bounds
P
become stronger with the weakening of the topology in which the limits (cid:30)((cid:1)−(cid:11))a((cid:11)) are
(cid:11)2ZZd
calculated. In the absence of a standard de(cid:12)nition for the space S((cid:30)), we have chosen here the
following one, which is motivated by the particular way in which we shall derive upper bounds in
the next section.
De(cid:12)nition. The principal shift-invariant space S((cid:30)) is the space of all locally bounded func-
tions (cid:30)(cid:3)0a, for which the double sum
X
(cid:30)(cid:3)0((cid:30)(cid:3)0a) = (cid:30)(x−(cid:12))(cid:30)((cid:12)−(cid:11))a((cid:11))
(cid:11);(cid:12)2ZZd
is absolutely convergent for every x2 IRd.
If (cid:30) has compact support, then S((cid:30)) contains (cid:30)(cid:3)0a for arbitrary a. Furthermore, if (cid:30) has some
decay at 1, then S((cid:30)) contains all (cid:30)(cid:3)0a for which a does not grow too fast at 1. Here is a sample
proposition:
5
Proposition 2.2. Assume that, for every p 2 (cid:5)n, the series (cid:30)(cid:3)0p converges pointwise absolutely
to a locally bounded function, and let An be the space of all sequences a : ZZd ! C of (at
most) polynomial growth n at 1. Then (cid:30)(cid:3)0An (cid:26) S((cid:30)). In particular, (cid:30)(cid:3)0An (cid:26) S((cid:30)) in case
j(cid:30)(x)j= O(jxj−m) for some m >n+d, as x! 1.
Proof: We will show that, for a 2 An, ((cid:30)(cid:3)0a)j 2 An, from which it will follow (because of the
ZZd
assumption on (cid:30)) that (cid:30)(cid:3)0((cid:30)(cid:3)0a) converges absolutely to a locally bounded function, and therefore,
by the de(cid:12)nition of S((cid:30)), (cid:30)(cid:3)0a 2 S((cid:30)). Without loss, we may assume that both (cid:30) and a are
nonnegative (otherwise take absolute values).
Byassumption, we can(cid:12)nda constantconstsuchthatk(cid:30)(cid:3)0pkL1(C) (cid:20) constforallnormalized
monomials p : x 7! x(cid:11)=(cid:11)!, (cid:11) 2 ZZd, j(cid:11)j (cid:20) n. It follows that
+ 1
k(cid:30)(cid:3)0pkL1(C) (cid:20) constjγmja(cid:20)xnjDγp(0)j
1
for all p 2 (cid:5)n. Now, let y 2 IRd, and set y =: ty + (cid:11)y, with ty 2 C and (cid:11)y 2 ZZd. Since
((cid:30)(cid:3)0p)((cid:1)+(cid:11)y) = (cid:30)(cid:3)0(p((cid:1)+(cid:11)y)), we deduce that ((cid:30)(cid:3)0p)(y) is the value at ty of (cid:30)(cid:3)0(p((cid:1)+(cid:11)y)), and
therefore, by the argument above, j(cid:30)(cid:3)0p(y)j (cid:20) constmaxjγj (cid:20)njDγp((cid:11)y)j. Thus (cid:30)(cid:3)0p = O(j(cid:1)jdegp)
1
at 1, and hence ((cid:30)(cid:3)0p)j 2 An for any p 2(cid:5)n.
ZZd
As for (cid:30)(cid:3)0a, by de(cid:12)nition of An, a 2 An can be bounded by some p 2 (cid:5)n (in the sense that
a((cid:11)) (cid:20) p((cid:11)) for all (cid:11)), hence (cid:30)(cid:3)0a is dominated by (cid:30)(cid:3)0p and therefore ((cid:30)(cid:3)0a)j 2 An.
ZZd
Fortheapproachtakeninthispaper, itisimportantthatthesum(cid:30)(cid:3)0e(cid:18) bewell-de(cid:12)nedforany
exponential e(cid:18), (cid:18) 2 IRd. Therefore, we assume that each operator (cid:30)(cid:3)0 is well-de(cid:12)ned and bounded
as a map from ‘1 to L1, and denote the corresponding norm by k(cid:30)(cid:3)0k. Some conditions related
to the boundedness of k(cid:30)(cid:3)0k are recorded in the following proposition whose proof is standard.
P
Proposition 2.3. The norm of the operator (cid:30)(cid:3)0 is k j(cid:30)((cid:1)−(cid:11))jk, hence, this operator is
P (cid:11)2ZZd
bounded if and only if the series j(cid:30)((cid:1)−(cid:11))j is pointwise convergent to a bounded function.
(cid:11)2ZZd
This proposition implies that (cid:30) 2 L (IRd) whenever (cid:30)(cid:3)0 is bounded, and, hence, that the
1
b
Fourier transform (cid:30) of (cid:30) is a well-de(cid:12)ned continuous function. Also, a su(cid:14)cient condition for the
boundedness of (cid:30)(cid:3)0 is the integrability of the maximal function (cid:30)#(x) :=k(cid:30)kL1(x+C).
2.2. Admissibility
Next,weturntothede(cid:12)nitionofthespaceofadmissiblefunctionsassociatedwiththe1-norm:
De(cid:12)nition. A function f of at most polynomial growth at 1 is termed here k-admissible if
(1+j(cid:1)jk)fbis a Radon measure of (cid:12)nite total mass. For such a function f, we denote by
kfk0
k
the total mass of (1+j(cid:1)jk)fb.
It follows that f is k-admissible (for some k (cid:21) 0) only if fbis a measure of (cid:12)nite total mass.
In particular, any admissible f is bounded. It is worthwhile to keep in mind two examples of
admisssible functions. The (cid:12)rst is the exponential f = e(cid:18), (cid:18) 2IRd. In this case, fb= (cid:14)−(cid:18), and since
fbis compactly supported, f is admissible of all orders. However, kfk0 = 1+j(cid:18)jk, and this grows
k
b
with k and/or (cid:18). The other example occurs when f is a function. In this case, f is k-admissible
whenever (1+j(cid:1)jk)fb2 L (IRd).
1
6
As usual, in case k is integral, the admissibility condition can be interpreted in terms of the
kth order derivatives of f:
Proposition 2.4. A function f is k-admissible for some k 2 ZZ if and only if the Fourier trans-
+
forms of f and of all its kth order derivatives are measures of (cid:12)nite total mass.
Proof: Let f(cid:11) bPe the (cid:11)th order (distributional) derivativ(cid:0)e o(cid:1)f f, hence fc(cid:11) : w 7! (iw)(cid:11)fb(w), and
choosePc(cid:11) so that (cid:11)c(cid:11)jx(cid:11)j = jxjk1 for x 2 IRd, i.e., c(cid:11) = (cid:11)k for j(cid:11)j1 = k and c(cid:11) = 0 otherwise.
Then (cid:11)c(cid:11)jfc(cid:11)j = j(cid:1)jk1jfbj, therefore, if fc(cid:11) is a measure of (cid:12)nite total mass for each (cid:11) 2 ZZd+ with
j(cid:11)j = k, then so is j(cid:1)jkjfbj, hence so is j(cid:1)jkjfbj. Thus, if also fbis of (cid:12)nite mass, then we conclude
1 1
that so is (1 + j (cid:1) jk)jfbj. The converse is even simpler: if f is k-admissible, then fb, as well as
w 7! (iw)(cid:11)fb(w) for j(cid:11)j = k, are majorizedby a measure of (cid:12)nite mass (viz. (1+j(cid:1)jk)fb), and hence
1
the Fourier transform of f and of all its derivatives of order k are measures of (cid:12)nite mass.
2.3. Upper bounds
We obtain upper bounds for the approximation order by considering approximation to expo-
nentials e(cid:18), (cid:18) 2IRd. Our starting point is the following result from [R2]:
Result 2.5. Let (cid:18) 2 IRd, and assume that the sequence f(cid:30)hgh satis(cid:12)es the following conditions:
(a) supp(cid:30)h (cid:26) B, for all h, and for some h-independent compact B.
(b) The functions f(cid:30)hgh are uniformly bounded.
Then,
(2:6) k(cid:30)h(cid:3)0eh(cid:18) −(cid:30)eh(h(cid:18))eh(cid:18)k (cid:20) cdist1(eh(cid:18);S((cid:30)h)):
The proof provided here will make use of the following condition, which is a consequence of
(a)+(b), but implies only (b):
(ab) suphk(cid:30)h(cid:3)0k < 1.
Proof: Fix h, and let f 2 S((cid:30)h). Since (cid:30)h(cid:3)0g = g(cid:3)0(cid:30)h for all g 2 S((cid:30)h), by [B1], and also
e#(cid:3)0(cid:30)h =(cid:30)eh(#)e# for any #, we have
(2:7) k(cid:30)h(cid:3)0eh(cid:18) −(cid:30)eh(h(cid:18))eh(cid:18)k (cid:20) k(cid:30)h(cid:3)0eh(cid:18) −(cid:30)h(cid:3)0fk+kf(cid:3)0(cid:30)h−eh(cid:18)(cid:3)0(cid:30)hk (cid:20) 2k(cid:30)h(cid:3)0kkeh(cid:18) −fk:
Since f 2 S((cid:30)h) was arbitrary, the result follows, with c = 2suphk(cid:30)h(cid:3)0k.
Since the key to the above argument is the \flip" property: (cid:30)(cid:3)0f = f(cid:3)0(cid:30);f 2 S((cid:30)), we
can extend the result to any (cid:30), compactly supported or not, with that property. Our particular
de(cid:12)nition of the space S((cid:30)) was chosen primarily to ensure the \flip" property.
Flip Lemma 2.8. For every f 2 S((cid:30)),
(cid:30)(cid:3)0f = f(cid:3)0(cid:30):
Proof: The argument follows the one given in [B1]. We (cid:12)x f 2 S((cid:30)) and x 2 IRd, and wish to
show that both (cid:30)(cid:3)0f(x) and f(cid:3)0(cid:30)(x) converge, and to the same limit. Since f = (cid:30)(cid:3)0a for some
sequence a, we write explicitly
X X
((cid:30)(cid:3)0f)(x)= (cid:30)(x−(cid:11)) (cid:30)((cid:11)−(cid:12))a((cid:12)):
(cid:11)2ZZd (cid:12)2ZZd
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By the de(cid:12)nition of S((cid:30)), this double sum is absolutely convergent, hence we may rearrange terms
(and replace (cid:11) by (cid:11)+(cid:12)) to get:
X X
(cid:30)((cid:11)) (cid:30)(x−((cid:11)+(cid:12)))a((cid:12))= f (cid:3)0(cid:30)(x):
(cid:11)2ZZd (cid:12)2ZZd
Theorem 2.9. Assume that the operator (cid:30)(cid:3)0 is bounded. Then, for any (cid:18) 2IRd,
(2:10) k(cid:30)(cid:3)0eh(cid:18) −(cid:30)e(h(cid:18))eh(cid:18)k (cid:20) 2k(cid:30)(cid:3)0kdist1(eh(cid:18);S((cid:30))):
Proof: Repeat the proof of Result 2.5, but replace the reference to [B1] by a reference to Lemma
2.8.
Theorem 2.9 can be interpreted in two di(cid:11)erent ways. On the one hand, it suggests that a
‘near-optimal’ approximant for the exponential eh(cid:18) from
Sh :=S((cid:30)h)
is provided by (the supposedly well-de(cid:12)ned) (cid:30)eh(h(cid:18))−1(cid:30)h(cid:3)0eh(cid:18). The following corollary records this
fact.
Corollary 2.11. Assuming that (cid:30)h(cid:3)0 is bounded and that (cid:30)eh(h(cid:18)) 6= 0, we get
(2:12) k(cid:30)eh(h(cid:18))−1(cid:30)h(cid:3)0eh(cid:18) −eh(cid:18)k (cid:20) 2j(cid:30)ke(cid:30)h(hh(cid:3)(cid:18)0k)j dist1(eh(cid:18);Sh):
We note that the ratio k(cid:30)h(cid:3)0k=j(cid:30)eh(h(cid:18))j is independent of the way (cid:30)h is normalized, and hence,
the right hand side of (2.12) is independent of the particular normalization we choose for (cid:30)h. But
the estimate (2.12) is useful for the derivation of bounds for the approximation order only in case
the sequence fk(cid:30)h(cid:3)0k=(cid:30)e(h(cid:18))gh is bounded.
Fortunately,Theorem2.9canbeuseddirectlytoderiveupperbounds. Wesimplyobservethat,
in case the operators f(cid:30)h(cid:3)0gh are uniformly bounded, Theorem 2.9 shows that dist1(eh(cid:18);Sh) =
O(hk) only if the same holds for k(cid:30)h(cid:3)0eh(cid:18) −(cid:30)eh(h(cid:18))eh(cid:18)k, i.e., for ke−h(cid:18)((cid:30)h(cid:3)0eh(cid:18))−(cid:30)eh(h(cid:18))k (since
jeh(cid:18)(x)j= 1 for every (cid:18) 2 IRd and every x 2IRd). Since e−h(cid:18)((cid:30)h(cid:3)0eh(cid:18)) = (e−h(cid:18)(cid:30)h)(cid:3)01, we obtain
(2:13) k(e−h(cid:18)(cid:30)h)(cid:3)01−(cid:30)eh(h(cid:18))k (cid:20) cdist1(eh(cid:18);Sh):
Thus, if we assume that we have approximation order k, then we must have
(2:14) k(e−h(cid:18)(cid:30)h)(cid:3)01−(cid:30)eh(h(cid:18))k = O(hk):
Sincethefunction(e−h(cid:18)(cid:30)h)(cid:3)01isZZd-periodic,(2.14)impliesthatitsFouriercoe(cid:14)cients(excluding
the 0’th coe(cid:14)cient) must be of size O(hk). Furthermore, (2.14) implies, in particular, that
(2:15) k(e−h(cid:18)(cid:30)h)(cid:3)01−(cid:30)eh(h(cid:18))kL2(C) = O(hk);
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which means that the 2-norm of the Fourier coe(cid:14)cient sequence for this periodic function is of
order O(hk). Since (cid:30)eh(h(cid:18)) is part of the constant term of this function, these coe(cid:14)cients are
(e−h(cid:18)(cid:30)h)(cid:3)01)b((cid:12)) for (cid:12) 2 2(cid:25)ZZdn0. We compute
Z
X
((e−h(cid:18)(cid:30)h)(cid:3)01)b((cid:12))= e−h(cid:18)(t−(cid:11))(cid:30)h(t−(cid:11))e−(cid:12)(t) dt
C
(cid:11)2ZZZd
X
(2:16) = e−h(cid:18)(t)(cid:30)h(t)e−(cid:12)(t)dt
C−(cid:11)
Z(cid:11)2ZZd
b
= (cid:30)h(t)e−h(cid:18)−(cid:12)(t) dt =(cid:30)h(h(cid:18)+(cid:12));
IRd
where, for the second equality, the fact that (cid:30)h 2L1 was used. Therefore, we conclude that
X
j(cid:30)bh(h(cid:18)+(cid:12))j2 = O(h2k):
(cid:12)22(cid:25)ZZdn0
As a matter of fact, nothing in the above arguments requires the approximation order to behave
like a power of h, and we thus obtain the following.
The Upper Bound Theorem 2.17. Assume that the (cid:30)h(cid:3)0 are bounded, and let (cid:18) 2 IRd. If
dist(eh(cid:18);S((cid:30)h)) = O((cid:26)(cid:18)(h))
for some (univariate) function (cid:26)(cid:18), then, for every h,
X
j(cid:30)bh(h(cid:18)+(cid:12))j2 (cid:20) constk(cid:30)h(cid:3)0k(cid:26)(cid:18)(h)2:
(cid:12)22(cid:25)ZZdn0
Inparticular,ifwenormalizef(cid:30)hgh toobtainauniformlyboundedf(cid:30)h(cid:3)0gh,thenfS((cid:30)h)gh provides
approximation order k to the exponential function e(cid:18) , (cid:18) 2 IRd, only if
X
j(cid:30)bh(h(cid:18)+(cid:12))j2 (cid:20) c(cid:18)h2k:
(cid:12)22(cid:25)ZZdn0
Note that the above implies that, in order to obtain k-approximation order, it is necessary to
have
(2:18) j(cid:30)bh(h(cid:18)+(cid:12))j(cid:20) c(cid:18)hk; (cid:12) 22(cid:25)ZZdn0; (cid:18) 2IRd:
It is this slightly weaker condition that we use throughout the paper in order to obtain upper
bounds on the approximation orders.
It is remarkable that the result avoids an application of Poisson’s summationformula (namely,
the convergence of the Fourier series of (e−h(cid:18)(cid:30)h)(cid:3)01 was not required), and hence no smoothness
conditions were imposed on f(cid:30)hgh. Also, no \regularity" condition was needed in the upper bound
theorem, i.e., neither f(cid:30)bh(0)gh nor f(cid:30)eh(0)gh were required to stay away from 0. (However, it is
plausiblethat,inthesingularcases,thisupperboundoverestimatestheactualapproximationorder
by the order of the zero of h 7! (cid:30)eh(h(cid:18)) at h = 0.)
The upper bounds were derived under the assumption that the exponential function e(cid:18) is
admissible, hence should be approximated well. Under a stronger assumption on the rates of decay
of each (cid:30)h at 1, we can show that the Upper Bound Theorem 2.17 remains valid even if we only
wish to approximate the test functions in D, i.e., in(cid:12)nitely smooth compactly supported functions.
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