The Euler-MacLaurin Summation Formula, the Sampling Theorem, and Approximate Integration over the Real Axis

P. L. Butzer and R. L. Stem Lehrstuhl A j?ir Mathemutik Aachen University of Technology Aachen, Federal Republic of Germany

Dedicated to Professor Alexander O&row&i on the occasion of his 90th Birthday on 25 September 1983, in admiration and friendship.

Submitted by Walter Gautschi

ABSTRACT

The Euler-MacLaurin summation formula is used to deduce the Whittaker- Shannon sampling theorem for not necessarily band-limited functions, as well as to study numerical integration over the real axis. Concerning the latter, error estimates are determined in case the function to be integrated is smooth but not necessarily analytic. Two characteristic examples are given.

1. INTRODUCTION

One of the results that attracted Professor Ostrowski’s particular attention in the many diverse fields in which he worked is the Euler-MacLaurin summation formula. Professor Ostrowski devoted three important papers [24-261 to this formula, dating back to 1969. The purpose of this note is to show that the Euler formula may be used to deduce the well-known Shannon sampling theorem as well as to study the approximation of /Y, f( x) dx by the Riemann sums (l/W )Cp= _ m f( k/W) for W + cc. As a matter of fact, the Euler formula will be applied to establish a basic lemma from which both the Shannon sampling theorem for not necessarily band-limited functions and the approximate integration result will follow.

The Shannon sampling series expansion, perhaps better known among mathematicians as Whittaker’s cardinal series, has been studied by many different methods so far. There are methods based upon Fourier series expansions [3, 71, Parseval’s formula [4, 29, 321, and the Poisson summation

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0 Ekevier Science Publishing Co., Inc., 1983 52 Vanderbilt Ave., New York, NY 10017 00243795/83,‘$3.00

142 P.L.BUTZERANDR.L.STENS

formula [l, 51, as well as upon Cauchy’s residue formula [34, p. 1141. The Euler-formula approach (using Fourier-analytic tools) will add another to this list.

This note will not be concerned with quadrature formulae for /if( x ) dar( x ) with da(x) a finite Stieltjes measure on the finite or infinite interval [a, b]-a field that has been receiving considerable attention (see e.g. [Ill)-but with the case that a(x) = x and [a, b] is the whole real axis. Cur main emphasis will be placed upon this situation, in particular upon rather sharp error estimates in case the function to be integrated is smooth but not necessarily analytic. Truncation error estimates will also be considered.

The paper is divided up as follows. Section 2 contains the basic tools from Fourier analysis which will be needed. Section 3 deals with the Shannon sampling theorem and further results as an application of the Euler- MacLamin formula. The first part of Section 4 treats the quadrature formula mentioned above, and the second handles the associated error estimates. Finally, two characteristic examples are discussed.

2. PRELIMINARIES

Let C(R) denote the space of all complex-valued continuous and bounded functions defined on the real line W, and LP(W), 1~ p < co, the space of alI measurable functions that are integrable to the pth power over IR, both endowed with the usual norms. Let W(W) be the space of functions which are of bounded variation over W.

The Fourier transform fro) of f E L’(W) is defined by fiu): = (I/G)/, f(u)e+” d u, PER, and that of fcLP(W), l<p<2, by f(o)=s--limp+, (l/&)jP_,f(u)e-‘““du, the limit being understood in the LQ(W) norm with l/p + l/g = 1. If the Fourier transform f A of f E Lp(R!), 1 d p d 2, belongs toL’(R), then

,. -$g Rf 0 eioxdu=f(x) / 0 a.e. (2.1)

Definingtheconvolutionf*goffELP(W),gELi(R), lgp<co,by

then f * g E LP(lp), and one has for 1 Q p 6 2 the convolution theorem

a-e. (2.2)

EULER-MACLAURIN SUMMATION FORMULA 143

The truncated Fourier inversion integral can be written as

--&II fA(u)eioxdu = & jRf(u)q,(x - u)du, (2.3) P

where 0, denotes the Dirichlet kernel on W, namely

(2.4)

with D,(O) = pm. One has by (2.1) for f E V(W), 1~ p f 2, with f-E L’(W)

h-n (f*D,)(x)=f(x) a.e. P-m

(2.5)

Moreover (2.5) can, according to Riemann’s localization principle, be re- placed by

~ia&/bf(u)D,(r-u)du=f(*) a.e., (2.6) Q (1

where [a, b] is any finite interval having x as interior point. Note that (2. l), (2.5), and (2.6) hold in particular at each point of continuity of f.

3. THE EULER-MACIAURIN FORMULA AND THE SAMPLING THEOREM

Let us start off with the Euler-MacLaurin summation formula as it can be found in Ostrowski [24-261, namely

E f(k)=~m~u)~u+h[A~)+f(m)l k=n

+i k=1&[~2k-++ f@k_‘)(n)]+R,, (3.1)

R,: = &#?2,(u)m5’pyk + u)du, k-n

(3.2)

144 P. L. BUTZER AND R. L. STENS

where n<m (m,n~{O,f1,*2 ,... }, r~{1,2,3 ,... }), B, are the Bernoulli numbers, and B,(u) the Bernoulli polynomials.

Let B,*(u) denote the periodic extension to R of B,(u) defined on [0, 11, Using the representation (cf. [16, $$+I])’

B,*,(u) = ( - 1)“-1’ (24 (2n)2’,c +eei2kru (u ER), 00

the remainder (3.2) can be rewritten as

R,=( -1)‘1 fl (27r)2’ k---M

~~m~(2r)(u),-i2~~~du. (3.3) n

As an application of the Euler formula let us first establish the basic lemma mentioned in Section 1.

LEMMA 1. Let f E C(R)n LP(R), some 1~ p d 2, be such that fAE L’(R). For D&t) defined us in (2.4) and W > 0,

= & k=~me-~2k”wt~2k-l)rW (zk+l)rwf A(v)e’“‘&, (t=Q

(3.4)

the series being conditionully convergent.

Proof. One may assume W = 1 without loss of genera&y. Otherwise replace f( .) by f( -/ W) and t by tW. Let us first consider the particular case that f and the derivativef’ both belong to C(R)n L’(W). Then apply (3.1) for T = 1 with remainder term (3.3) to the function g,(u) : = f( u)D,(t - u), t E W fixed. Since g,, gj’ E C(R)n L’(R) with limj, *,g,(j)= limj, *:,g;(j) = 0, one obtains for n + - CO, m + 00 that

‘The prime indicates that the term k = 0 is to be omitted.

EULER-MACLAURIN SUMMATION FORMULA 145

Integrating by parts twice yields

k= -co

Using (2.3) with p = a and the fact that [f( .)e-‘h’]-(u) = f-(u + h), the integrals can be rewritten as

/” gt(u)e-i2ksu -m

du=J” fA(o+2kr)ei"tdo -77

=e -i2knt (2kt1)nf^(u)efut&,, / (2k - 1)~

Inserting this result into (3.5) proves the assertion (3.4) in the particular case. In the general case consider the function

xP being Fejbr’s kernel on R, defined by

sinipu 2 x,(u)=- -

&, +u ’ [ I having the Fourier transform2

x+))= ~A!?! P i 1 P +

(?lE.R, p>O).

One easily verifies that F, satisfies the assumptions of the particular case. Hence (3.4) and (2.2) yield

2 F,(k)D,(t - k) = f

k= -cm k= -co

e-i2knr/(2k+1)nfa( o)( 1 - y) eiot do. (2k - 1)~ +

(3.6)

146 p. L.BU'IZERANDR.L.STENS

The series on the right side, to be denoted by S:(t), is uniformly convergent with respect to p > 0, and its integrals are dominatedly convergent, since f-E L’(W). Therefore

lim s;(t) = c P-a k= -co

Concerning the left side St(t) of (3.6), one has by (2.2) and (2.1)

the interchange of summation and integration being justified by

k~nD_.(t-k)eikol<M (nEI$J),

where M depends only on t. Letting now p + co, interchanging sums and integrals once more, and applying the inversion formula, (2.1) finally yields

(3.8)

So (3.4) follows from (3.6) for p --, co in view of (3.7) and (3.8). ??

A generalization of the Shannon sampling theorem now follows as a simple application.

THEOREM 1. Let f EC(R)~L~(W), pollee 1 Q P G 2, b such that f-E L’(R). Then for t e BP

EULER-MACLAURIN SUMMATION FORMULA 147

in particuZar, unifmly in t E R,

lim -L- E f(+)qw(t-$)=f(t). w+cQ &w kc-m

(3.10)

The proof follows immediately from (3.4) if one writes f via (2.1) in the form

The inequality (3.9) may be used to deduce associated error estimates by applying (4.10) below. See, for example, [7] and also [28].

If one restricts oneself to the case that f (u) = 0 for ]0]> aW, then one obtains the classical sampling theorem, also known as Whittaker’s cardinal series representation.

COROLLARY 1. Let f E C(R)nLp(R), some 1 Q p Q 2, be such that f~(u)=Ofmalmostall~u~>nW. Then

&,$ f(+%w(t-f)=fct, (tER)> (3.11) 00

the series being unifmly and absolutely conuergent.

The absolute and uniform convergence follows from (cf. [33, p. 2331)

{ + ,im IfH)‘” <(l+n)(/m If(U)lpdU)l’p (l<p-w). --Q)

REMARK 1. The assumption f*(u) = 0 for ]u] > 7rW in Corollary 1 could equivalently be expressed by the assumption that f is an entire function of exponential type Q SW. By using this setting CorolIary 1 could be extended to the situation that 2 < p < 00 by approximating f(t) by f(t)(sin et)/Et for E + 0+ (cf. [27]).

4. APPROXIMATE INTEGRATION OVER THE REAL AXIS

4.1. Theory Lemma 1 can also be used to deduce quadrature formulae for /?, f(x) dx

with error estimates. As far as the authors are aware, such formulae have been

148 P. L. BUTZER AND R. L. STENS

mainly studied in the past for functions that are holomorphic in a strip around the real axis; see e.g. [17, 18, 19, 21, 30, 311. Instead, it will be assumed that f E C(R)n L’(R) either has a given rate of decay at infinity or is differentia- ble of a given order.

THEOREM 2. Let f E C(W)n L’(R) be such that

(4.0

is unifmly convergent in u E W for each fixed W > 0. Then, for W > 0,

’ 5 f(f)-s” f(u)du=n@w@ 2’ (l-~)f^(2k~W). - W k= -co -* k- --n

(4.2)

In particular, if the series Zr_ _m f -(2kaW) is convergent, then

’ f f(+)-/wf(u)du=fi e f*(2knW) (W>O). w k---m -CO k= --oo

(4.3)

Proof. Replacing f in (3.4) by f * xPn, pn: = 2naK yields [de (3.611

where CF_ _ m I( f * xP, )( k / W )I < 00 in view of the uniform convergence of (4.1). Furthermore, the series on the right side of (4.4) is finite. So one can integrate (4.4) term by term over [ - R, R] to deduce

EULER-MACLAURIN SUMMATION FORMULA

since

149

J R

eiXt & = 2 sinxRr (XER). -R

Now / !! R D,,( t - k/W) dt is bounded in absolute value by 6, and its limit for R --) cc equals &, so the limit of the left side of (4.5) for R + 00 equals

(4.6)

Concerning the right side of (4.5), the limit for R + cc equals

in view of (2.6). To establish (4.2) it remains to prove that (4.6) converges to

(G/W)cr= - 00 f(k/W) for rr + co. Indeed, (4.6) equals

in view of the uniform convergence of (4.1), and the latter integral converges to the sum in question because the term in curly braces is uniformly continuous and bounded (cf. [6, p. 1221). ??

COROLLARY 2.

(a) Zff E C(w)nL’(R) is such that f(t)= O(ltl-Y), t * f co, for some y>l,then,forW>O,

whenever the series on the right converges. (b) Zf f E C(R)n Ll(R)n BV(W), then (4.7) holds.

Proof. If f(t)= O(ltlpy) for y > 1, then the series (4.1) is uniformly convergent for W > 0. So part (a) follows by Theorem 2. Concerning (b). one

150 P. L. BUIZER AND R. L. STENS

knows that f E By(R) implies the uniform convergence of (4.1) (cf. [6, p. 1241) as well as f (t)) = O(]o]-‘) for o + f co. Now the right side of (4.7) is (C, I)-summable in view of (4.2). Since f-(2knW) = O(]k]-‘), k + f co, it is also convergent in view of the Hardy Tauberian theorem (cf. [14, p. 1211). So the result follows by Theorem 2. ??

REMARK 2. The reader may note that (3.5) or (4.3) is basically the well-known Poisson summation formula, which was actually introduced into number theory by Lejeune Dirichlet and is regarded by Bochner [2, p. 521 as a “broad-gauged duality formula which lies athwart most of analysis.” In other words, in the course of Sections 3 and 4 we indirectly obtained the Poisson formula from the Euler-Ma&auk formula (via the sampling theo- rem). For the fact that an integration of the sampling theorem in its generalized form, basically (3.4), yields the Poisson formula; see [36]. Con- versely, it would also be possible to study the sampling theorem as well as approximate integration as direct applications of the Poisson formula; see especially [l], and also [5, 351. For a detailed investigation of the remainder term in the n-dimensional Euler formula, see [371. It may also be used to deduce the multidimensional Poisson formula; see [22].

There exists a summation formula for the one-sided improper integral /Ff( u) du. It is the summation formula of Abel-Plana; see [15, p. 2741. In [13, 20,231 numerical integration for improper integrals of the first and the second hind is ‘studied by using quite different methods.

4.2. Enor Estimutes In order to give an estimate for the left-hand side in (4.7) it is useful to

consider the modulus of continuity off E L’(R), defined by

COROLLARY 3. Let f E C(R)n L1(R) be such that f”) E L1(R) for some r E {1,2,3,. . .}. Zfw(6; fl”; L1(W)6 ZdPfbr some 0 < a Q 1, then, for w> 0,

where the constant M is gioen by

(4.8)

M: = (2t;ra+a kgl krl,, ( -1 Im”(r + a) Q (2*)r+a(r + a_ q - (4.9)

EULER-MACLAUFUN SUMMATION FORMULA 151

Concerning the proof, one has only to note that r’) E L’(R) implies f E BV(R), and to apply in Corollary 2(b) the estimate (cf. [6, (5.1.3) Proposition 5.1.141)

Note that the rate of convergence in (4.8) is as good as one pleases provided the function is sufficiently smooth. This stands in contrast to the situation when /,“f U) du, [a, b] being finite, is approximated by the con-e sponding Riemann sums; see [8]. For error estimates in case the function in question is holomorphic, one may consult [19, 21, 30, 311.

In practice one is also interested in the “truncation” error which arises when the infinite series in (4.8) is replaced by the truncated sum Cf= _N. This error is additional to that in (4.8). In this situation one has

COROUARY 4. Let f be given as in CmolZuy 3. Moremer, assume that ~f(t)~~Mf/t~-~fora2Z~t~~uaO~andsomey~l.ThenforN>aWundMas given in (4.9) one has

If N=N(W) is chosen such that WOaN(W) for a=(r-ta+y

- l)/(Y - 1x then

Furthermore, one has for l/N = 0(1/w), W + 00,

lim f 5 W-+CO k= -N

f(+)=j” f(u)du. -CO

The proof follows from Corollary 3 and the estimate

152 P. L. BUTZER AND R. L. STENS

Truncation-error estimates using function-theory methods are to be found in [30, pp. 236-2391.

Corollary 1 also enables one to deduce a quadrature formula for the product of two functions f E C(R)n L1(W) and g E L1(R), where f-(v) = 0 for Iv] 2 aW. Indeed, multiplying (3.11) by g(t) and integrating termwise gives

COROUARY 5. Let f E C(R)n LP(Iw), some 1 G p d 2, be such that fiv)=Of or v 2 nW. Then one has for each g E L’(R) I I

(4.13)

where the “weights” A,?, are given by

For the evaluation of integrals over a finite range of integration using Euler’s formula see e.g. [38]. A number of investigations are concerned with evaluating /?, f(u) da(u) for (Y E BV(BB). These are intimately related to the determinacy of the moment problem for da(u) on W. In this respect one may consult [9], and the survey paper [ll, especially pp. 117-1181 and the literature cited there, as well as [lo]. In [12] methods similar to ours are used to study /‘Z, f( u)e& du.

4.3. Applications Let us first consider a standard example, namely the approximate integra-

tion of the function fi(x)= l/(1+ x2) over W; it is analytic in (2 = x + iy; x E If& ]y] < l} (cf. [21]). Since ff(v)=me+l, v E Iw, an application of Corollary 2(a) yields

Concerning the truncation error, one has for W > 0 and N Z+ W

EULER-MACIAURIN SUMMATION FORMULA 153

Note that the error given in [21] in case of (4.14) is %e-3w/sinb3W. Whereas this error was not aimed to be the best possible one, ours is.

As our second example, let us consider h(x)= ]x]/(l+]~]~). It is easy to show that 6 E L’(R), and ~(6; & L’(R) < 86, 6 > 0. This yields by Corol- lary 4

VW J M I4 k= pN W3+lk13 - -co l+]u13

-du < i W-2 +2Wh-‘.

Choosing N- ’ = 0( We3), W + 00, then the combined error is of order O(W_2), w --) 00.

It should be mentioned that fi does not have an analytic extension into the complex plane as fr does. So it cannot be handled by function-theoretic methods.

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Received 19 February 1982