The Summation Formulae of Euler–Maclaurin, Abel–Plana, Poisson, and their Interconnections ...

42 Pages·2011·0.48 MB·English

Results.Math.59(2011),359–400
(cid:2)c 2010SpringerBaselAG
1422-6383/11/030359-42
published onlineApril2,2011 Results in Mathematics
DOI10.1007/s00025-010-0083-8
The Summation Formulae
of Euler–Maclaurin, Abel–Plana,
Poisson, and their Interconnections
with the Approximate Sampling
Formula of Signal Analysis
P. L. Butzer, P. J. S. G. Ferreira, G. Schmeisser and R. L. Stens
In Memory of Alexander Ostrowski, 1893–1986
Dedicated to Heinrich Wefelscheid to mark his 70th birthday
Abstract. This paper is concerned with the two summation formulae of
Euler–Maclaurin(EMSF)andAbel–Plana(APSF)ofnumericalanalysis,
thatofPoisson(PSF)ofFourieranalysis,andtheapproximatesampling
formula(ASF)ofsignalanalysis.Itisshownthatthesefourfundamental
propositions are all equivalent, in the sense that each is a corollary of
anyoftheothers.Forthispurposetenofthetwelvepossibleimplications
are established. Four of these, namely the implications of the grouping
APSF ⇐ ASF ⇒ EMSF ⇔ PSF are shown here for the first time. The
proofs of the others, which are already known and were established by
three of the above authors, have been adapted to the present setting.
In this unified exposition the use of powerful methods of proof has been
avoided as far as possible, in order that the implications may stand in
a clear light and not be overwhelmed by external factors. Finally, the
four propositions of this paper are brought into connection with four
After this manuscript was accepted for publication the authors learned that special issues
werebeingpreparedtohonourHeinrichWefelscheidontheoccasionofhis70thanniversary.
Sincetheyvaluedhisunstintedservicetomathematicsasawhole,thusthefoundingofthe
present journal and his leading editorship over the years, his publication of the Collected
Works,inseveralvolumes,ofW.BlaschkeandE.Landau,theyexpressedinterestthatthe
presentpaper,alreadydedicatedtoA.Ostrowski,beincludedinthespecialissues.
360 P. L. Butzer et al. Results. Math.
propositions of mathematical analysis for bandlimited functions, includ-
ing the Whittaker–Kotel’nikov–Shannon sampling theorem. In conclu-
sion, all eight propositions are equivalent to another. Finally, the first
three summation formulae are interpreted as quadrature formulae.
MathematicsSubjectClassification(2010). 65B15, 65D32, 94A20.
Keywords.Euler–Maclaurinsummationformula,Abel–Planasummation
formula, Poisson summation formula, Whittaker–Kotel’nikov–Shannon
sampling theorem, Approximate sampling formula, Bandlimited signals,
Quadrature formulae.
1. Introduction
First to the Euler–Maclaurin summation formula of the broad area of numer-
ical analysis, a very important one, which has indeed many applications.
Euler–Maclaurin Summation Formula (EMSF)
For n,r ∈N and f ∈C(2r)[0,n], we have
(cid:3)n
(cid:2)n
f(k)= f(x)dx
k=0 0
(cid:2)r (cid:4) (cid:5)
+1[f(0)+f(n)]+ B2k f(2k−1)(n)−f(2k−1)(0)
2 (2k)!
k=1
(cid:3)n
(cid:2)∞ ei2πkt+e−i2πkt
+(−1)r f(2r)(t)dt, (1.1)
(2πk)2r
k=10
where B2k are the Bernoulli numbers.
Eulerdiscoveredthissummationformulainconnectionwiththeso-called
Basel problem, i.e., with determining ζ(2) in modern terminology. It can be
found without proof in [23] (submitted 1732) and with a complete deduc-
tion in [24] (submitted 1735), where he used it to calculate ζ(2),ζ(3),ζ(4),
and the Euler constant γ; for more details see [26]. The formula was found
independently by Maclaurin in 1738 [38]; cf. [39] for the differences between
Euler’s and Maclaurin’s approaches. See also the review of Euler’s life and
works by Gautschi [27], and the overview by Apostol [3]. Of further inter-
est is that Ramanujan [4, Chaps. 6, 8] introduced a method of summation
based on EMSF. Candelpergher et al. [17,18] presented a rigorous treatment
of the method together with many applications. This formula is also treated
in Ostrowski’s three-volume treatise on differential and integral calculus [42,
vol. II, pp. 287–288], which was very popular in German speaking universities
in the 1960s.
Vol. 59 (2011) Summation Formulae of Euler–Maclaurin, Abel–Plana etc 361
Whereas the Euler–Maclaurin summation formula is a standard one in
mathematics per se, our second formula, that of Abel–Plana1 is not. It reads
Abel–Plana Summation Formula (APSF)
Let f be analytic in {z ∈C:(cid:3)z ≥0}. Suppose that
lim |f(x±iy)|e−2πy =0 (1.2)
y→∞
uniformly in x on every finite interval, and that
(cid:3)∞
|f(x+iy)−f(x−iy)|e−2πydy (1.3)
0
exists for every x≥0 and tends to zero when x→∞. Then
⎡ ⎤
N(cid:3)+1/2 (cid:3)∞
⎢(cid:2)N ⎥ 1 f(iy)−f(−iy)
lim ⎣ f(k)− f(x)dx⎦= f(0)+i dy. (1.4)
N→∞ 2 e2πy−1
k=0 0 0
This formula was obtained by Plana [44] in 1820 and independently by Abel
[1, p. 23] in 1823 in the form
(cid:3)∞ (cid:3)∞
(cid:2)∞ 1 f(iy)−f(−iy)
f(k)− f(x)dx= f(0)+i dy. (1.5)
2 e2πy−1
k=0 0 0
As it was customary at that time, neither author gave hypotheses on f.
They proceeded via an integral representation of the Bernoulli numbers. Abel
[1,p.27]alsoestablishedthefollowinginteresting“alternatingseriesversion”:
(cid:3)∞
(cid:2)∞ 1 f(iy)−f(−iy)
(−1)kf(k)= f(0)+i dy.
2 2sinhπy
k=0 0
Although his proof was different, this formula can be obtained by apply-
ing (1.5) to the function z (cid:7)→ 2f(2z) and subtracting the resulting equation
from (1.5).
1 Giovanni(AntonioAmedea)Plana,born1781inVoghera,Lombardy(Italy),died1864in
Turin,studied1800–1803attheE´colePolyt´echniqueinParisunderFourierandLagrange.
OntherecommendationofFourier,Planain1803wasappointedtothemathematicschair
attheArtilleryschoolofPiedmontinTurin.WithLagrange’srecommendationhewasalso
appointed to the chair of astronomy at the University of Turin in 1811. He taught both
astronomy and mathematics at both institutions for the rest of his life. His teaching was
of the highest quality. Plana, one of the major Italian scientists of his time, stood in con-
tact with Francesco Carlini, Cauchy and Babbage. He was elected to the Turin Academy
of Sciences in 1841 and to the Acad´emie des Sciences in Paris in 1860. Plana worked in
astronomy(onmovementofthemoon),Eulerianintegrals,ellipticfunctions,heat,electro-
staticsandgeodesy.SeetheMacTutorhistoryofmathematicsarchivehttp://www-history.
mcs.st-andrews.ac.uk/.
362 P. L. Butzer et al. Results. Math.
Some authors, [19, I, p. 258, III, p. 53(cid:12)], [45, Sect. 2], consider (1.5) to
require in addition to our assumptions that ∞f(x)dx exists and f(N) →0
0
as N →∞. However, the last assumption is superfluous. Indeed, by (1.4) the
existence of the integral implies the convergence of the series which entails
that f(N)→0 as N →∞.
APSF, as stated by us, is exactly what Henrici proves by contour inte-
gration in his book [30, Sect.4.9]. However, his resulting theorem [30, Theo-
rem 4.9c] is presented in a somewhat different form.
TheAbel–Planaformula,connectedwithseveralgreatnamesofthepast,
includingCauchy(1826),Kronecker(1889),Lindelo¨f(1905)(seehisaccountof
the history until 1904 [37, pp. 68–69]), was almost forgotten until Henrici [30,
pp. 270–275] brought it up again in his text-book and applied it. One of the
mostimportantrecentapplications oftheAPSFistothevacuumexpectation
values for the physical observables in the Casimir2 effect; see [21,40,45,50].
ForaproofofthefunctionalequationfortheRiemannzetafunctionusingthe
Abel–Plana formula see [51], and for the interconnections of Plana’s formula
with Ramanujan’s integral formula see [55].
The third formula, which is fundamental in a variety of mathematical
fields, the Poisson summation formula, will be stated in the following form
(see e.g. [11, p. 202]):
Poisson Summation Formula (PSF)
Let f ∈L1(R)∩AC(R). Then
(cid:2)∞ √ (cid:2)∞
(cid:13)
f(k)= 2π f(2πk), (1.6)
k=−∞ k=−∞
where the series on the left-hand side converges absolutely.
Above, f(cid:13)denotes the Fourier transform of f ∈ L1(R); it, as well as the
class AC(R), is defined in Sect. 2.
Although this formula is named after Poisson, it was given in a general
formwithoutproofbyGaußinanotewrittenontheinteriorofthecoverpage
of a book entitled “Opuscula mathematica 1799–1813”; see [25, p. 88]. Jacobi
in his work on the transformation formula for elliptic functions [36, p. 260]
attributesittoPoisson.FurtherremarksconcerningthehistoryofthePoisson
summation formula can be found in [13, p. 28], [53, pp. 36 f.].
Finally to the approximate sampling formula (ASF) of signal analysis,
actually due already to de la Vall´ee-Poussin (see [20], pp. 65–156 for a repro-
duction of this paper and pp. 421–453 for a commentary by Butzer-Stens),
Weiss [54], Brown [6], and Butzer-Splettst¨oßer [14].
2TheDutchphysicistHendrikCasimir(1909–2000)wasawardedanHonoraryDoctorateby
the RWTH Aachen University in 1966, the Laudation having been held by Professor Josef
Meixner.
Vol. 59 (2011) Summation Formulae of Euler–Maclaurin, Abel–Plana etc 363
Approximate Sampling Formula (ASF)
For f belonging to the class F2 (see Sect. 2 for definition), w >0, we have
(cid:14) (cid:15)
(cid:2)∞
k
f(t)= f sinc(wt−k)+(Rwf)(t) (t∈R), (1.7)
w
k=−∞
where the remainder Rwf is defined by
(2k(cid:3)+1)πw
(cid:2)∞ (cid:16) (cid:17)
(Rwf)(t):= √1 1−e−i2πkwt f(cid:13)(v)eivtdv (t∈R),
2π
k=−∞
(2k−1)πw
(1.8)
the sinc-function being given by
(cid:18)
sinπz if z ∈C\{0},
πz
sincz :=
1 if z =0.
It is the sampling theorem for not necessarily bandlimited functions, general-
izing the classical Whittaker–Kotel’nikov–Shannon sampling theorem (CSF)
of signal analysis
(cid:14) (cid:15)
(cid:2)∞
k
f(t)= f sinc(wt−k) (t∈R) (1.9)
w
k=−∞
forf ∈B2 (for definition seeSect.2). Toproceedfrom(1.9) to (1.7) onehas
πw
to add the “error” term (Rwf)(t).
Remark. A consequence of the definition (1.8) of Rwf is the inequality
(cid:20) (cid:3)
(cid:19) (cid:19)
(cid:19)(Rwf)(t)(cid:19)≤ 2 |f(cid:13)(v)|dv, (1.10)
π(cid:19) (cid:19)
(cid:19) (cid:19)
v ≥πw
yielding
(cid:14) (cid:15)
(cid:2)∞
k
lim f sinc(wt−k)=f(t)
w→∞ w
k=−∞
uniformly for t∈R.
The implications to be established in this paper between the assertions
ofthesummationformulaeofEuler–Maclaurin(EMSF),Abel–Plana(APSF),
Poisson(PSF)andtheapproximatesamplingformula(ASF),areindicatedby
thearrowsofthegraphicinFig.1.Asthereaderobserves,thereare12possible
implications; of these we shall prove 10 implications, namely the grouping
APSF⇐=ASF=⇒EMSF⇐⇒PSF
364 P. L. Butzer et al. Results. Math.
PSF
ASF
EMSF APSF
Figure 1. Diagram of the implications to be proven
as well as the grouping
ASF⇐⇒PSF=⇒APSF⇐⇒EMSF=⇒ASF.
The conclusion: All the formulae are equivalent to each other.
Thefirstgroupingconsistsoffournewimplicationswhichareestablished
inthispaperforthefirsttime.Ofthesecondgrouping,theequivalenceASF⇔
PSF,wasfirstpresentedintheproceedingsoftheEdmontonconferenceof1982
[15], the implication EMSF⇒ASF in the special issues dedicated to the 90th
birthday of A. Ostrowski in 1983 [16], and that of PSF⇒APSF⇔EMSF in
memory of U. N. Singh in 1994 [49].
The proofs of five of these six known implications of the second group-
ing, established by three of the present authors since 1982, are to be found
in conference proceedings and special issues not always readily available. For
this reason they are presented here with complete proofs, especially to make
sure that they do not involve “circular arguments”, so important in establish-
ing equivalences between many assertions. Further, they have been adapted
to those of the new implications of the first grouping in order to give a unified
expositionofthematerial.Sothereaderwillfindfullproofsofalloftheresults
discussed in the present paper.
All the proofs use only a handful of modest side results, namely classi-
cal Fourier analysis, the Weierstraß theorem for algebraic (not trigonometric)
polynomials and Cauchy’s theorem of complex function theory. Therefore, if
we prove an implication X ⇒ Y, then it is fully justified to say that Y is a
direct corollary of X. Here “direct” means that the proof of the implication
proceeds directly from X to Y, and is not of the form X⇒Z⇒Y, where Z is
any other of the formulae under consideration.
The implications APSF ⇒ ASF and APSF ⇒ PSF are left open. The
problem is that APSF requires functions which are analytic in a half-plane,
whereas ASF and PSF hold for larger function classes. Thus it seems to be
Vol. 59 (2011) Summation Formulae of Euler–Maclaurin, Abel–Plana etc 365
appropriatetoproceedviaadensityargumentbasedonanapproximationpro-
cess which could be so cumbersome that one could not speak of “establishing
a direct corollary”.
For those readers who are primarily interested in the equivalence of the
fourformulae,thepresentationalsoallowsseveralclosedloopsofimplications,
onebeingPSF⇒APSF⇒EMSF⇒ASF⇒PSF.Inthiscaseonlytheproofs
of four implications are needed.
Concerning contents, Sect. 2 is devoted to notations and side results
needed.Section3isconcernedwiththeproofoftheequivalenceofEMSFand
APSF,andSect.4withtheequivalenceofPSFandASF.Section5dealswith
thePSF⇔EMSFandSect.6withEMSF⇔ASF.Section7treatstheAPSF
as a corollary of ASF and PSF.
Section 8 presents a proof of EMSF; thus since one of the formulae has
been shown to be actually valid, the four formulae are not only equivalent
among themselves, but are all valid too.
Section 9 is concerned with APSF, EMSF and PSF considered as quad-
rature formulae in case the functions in question are bandlimited. Finally,
Sect.10comparestheresultspresentedherewithasimilarequivalencegroup-
ing for bandlimited functions in [7]. Section 11 contains a short biography of
A. Ostrowski.
2. Notations and Side Results
In what follows, the Fourier transform of f ∈Lp(R), p=1,2, is defined by
(cid:3)
f(cid:13)(v):= √1 f(u)e−ivudu (v ∈R),
2π
R
the integral being understood as the limit in the L2(R)-norm for p = 2. We
shall consider the following function spaces:
(cid:21) (cid:22)
Fp := f: R→C;f ∈Lp(R)∩C(R),f(cid:13)∈L1(R) (p=1,2),
where C(R) denotes the space of all uniformly continuous and bounded func-
tions on R. Since f(cid:13)∈L1(R) implies that f is bounded, there holds F1 ⊂F2.
We also make use of the space AC(R)=AC(1)(R). It comprises all func-
tions f: R → C being absolutely continuous on R, i.e., those f having the
representation
(cid:3)t
f(t)= f(cid:5)(u)du+c (t∈R)
−∞
with f(cid:5) ∈ L1(R). More generally, AC(r+1)(R), r ∈ N, consists of all functions
f with f(cid:5) ∈AC(r)(R)∩L1(R).
366 P. L. Butzer et al. Results. Math.
The Bernstein spaces Bp for p = 1,2 and σ > 0 are defined in terms of
σ
the Fourier transform via
(cid:21) (cid:22)
Bp := f ∈Lp(R);f(cid:13)(v)=0 a.e.outside [−πσ,πσ] .
σ
InviewofthePaley–Wienertheorem(seee.g.[56,vol.II,p.272])thesespaces
can be equivalently characterized as the set of all entire functions of exponen-
tial typeσ,therestriction toRofwhich belongs toLp(R). Forthis reasonthe
spaces B2 are also known as Paley–Wiener spaces.
σ
Let us list some well-known results from Fourier analysis; see any text-
book in the matter, e.g. [11, Sects. 5.1, 5.2].
Proposition 2.1. a) If f ∈Lp(R), p=1,2, then for each h∈R,
[e−ih·f(·)]∧(v)=f(cid:13)(v+h) (2.1)
for all v ∈R in case p=1 and a. e. in case p=2.
b) If f ∈L1(R), g ∈Lp(R), p=1,2, then the convolution
(cid:3)
1
(f ∗g)(t):= √ f(u)g(t−u)du (2.2)
2π
R
belongs to Lp(R), and for the Fourier transform one has the convolution
theorem
(f ∗g)∧(v)=f(cid:13)(v)g(cid:13)(v) (2.3)
for all v ∈R in case p=1 and a. e. in case p=2.
c) If f,g ∈Lp(R), p=1,2, then there holds the exchange formula
(cid:3) (cid:3)
f(v)g(cid:13)(v)dv = f(cid:13)(v)g(v)dv. (2.4)
R R
d) If f ∈ L2(R), then f(cid:13)also belongs to L2(R) and (cid:16)f(cid:16)2 = (cid:16)f(cid:13)(cid:16)2. Further-
more, there holds the inversion formula
(cid:3)
f(t)= √1 f(cid:13)(v)eivtdv =f(cid:13)(cid:13)(−t) a. e., (2.5)
2π
R
where the integral is again understood as the limit in L2(R)-norm. If
f ∈Fp, p=1,2, then the integral in (2.5) exists as an ordinary Lebesgue
integral and both equalities hold for all t∈R.
Lemma 2.2. There hold the formulae
(cid:23) 1 (cid:24) · (cid:25) (cid:26)∧
√ rect ei·t (v)=sincw(v−t)=sincw(t−v) (t∈R)
2πw w
(2.6)
(cid:24) (cid:25)
[sincw(·−t)]∧(v)= √ 1 rect v e−ivt (t∈R) (2.7)
2πw w
Vol. 59 (2011) Summation Formulae of Euler–Maclaurin, Abel–Plana etc 367
with the rectangle function
⎧
⎪⎨1, |t|<π
rect(t):= 1, |t|=π .
⎪⎩2
0, |t|>π
Proof. Equation (2.6) isshownbyasimpleintegration, and(2.7)follows from
(2.6)bytheFourierinversionformula.(Foranelementaryproofof (2.7)with-
out using the inversion formula see [5, p. 13f].) (cid:2)
Proposition 2.1 and Lemma 2.2 enable us to prove alternative represen-
tations of the sampling series
(cid:14) (cid:15)
(cid:2)∞
k
(Swf)(t):= f sincw(t−k) (2.8)
w
k=−∞
and the remainder Rwf of (1.8).
Proposition 2.3. For fixed t ∈ R, w > 0 let et,w(v), v ∈ R, be the function
obtained by restricting v (cid:7)→eitv to the interval [−πw,πw), and then extending
it to R by 2πw-periodic continuation. Then for f ∈F2,
(cid:3)
(Swf)(t)= √1 f(cid:13)(v)et,w(v)dv. (2.9)
2π
R
Proof. By a simple calculation et,w has the Fourier expansion
(cid:2)∞
et,w(v)= sinc(wt−k)eikv/w. (2.10)
k=−∞
Sinceet,w isofboundedvariation,itsFourierseriesconvergesand(2.10)holds
at each point of continuity. Moreover, its partial sums
(cid:2)
sn(v):= sincw(t−k)eikv/w (2.11)
|k|≤n
are uniformly bounded with respect to v ∈R and n∈N, that is
|sn(v)|≤C (v ∈R;n∈N)
for some constant C > 0 depending on t only; see [56, p. 90, Theorem 3.7].
Since f(cid:13)∈L1(R), we see that f(cid:13)·sn has an absolutely integrable majorant for
all n∈N. Therefore, Lebesgue’s dominated convergence theorem allows us to
368 P. L. Butzer et al. Results. Math.
conclude that
(cid:3) (cid:3)
√1 f(cid:13)(v)et,w(v)dv = lim √1 f(cid:13)(v)sn(v)dv
2π n→∞ 2π
R ⎧R ⎫
(cid:2) ⎨ (cid:3) ⎬
= lim √1 f(cid:13)(v)eikv/wdv sinc(wt−k)
n→∞ ⎩ 2π ⎭
|k|≤n (cid:14) (cid:15)R
(cid:2)
k
= lim f sincw(t−k),
n→∞ w
|k|≤n
where the Fourier inversion theorem (Proposition 2.1d) has been used in the
last step. (cid:2)
Proposition 2.4. For f ∈F2 and w >0 we have the following two alternative
representations of the remainder Rwf of (1.8),
(cid:3)
(cid:2)∞ (cid:16) (cid:17)
(Rwf)(t)= ei2πkwt−1 f(u)e−i2πkwu sincw(t−u)du (2.12)
k=−∞
R
(cid:3)
(cid:2)∞
=f(t)− f(u)e−i2πkwu sincw(t−u)du (t∈R).
k=−∞
R
(2.13)
Proof. Using (2.4), (2.6) and (2.1) we obtain
(2k(cid:3)+1)πw (cid:3) (cid:24) (cid:25)
(cid:13) itv i2πkwt (cid:13) v itv
f(v)e dv =e f(v+2πkw)rect e dv
w
(2k−1)πw R
(cid:3)
√
= 2πei2πkwt f(u)e−i2πkwu sinc(t−u)du.
R
(2.14)
Inserting this into (1.8) yields (2.12).
Moreover, (1.8) can be rewritten as
(cid:3)∞
(Rwf)(t) = √1 f(cid:13)(v)eivtdv
2π
−∞
(2k(cid:3)+1)πw
(cid:2)∞
−√1 e−i2πkwt f(cid:13)(v)eivtdv.
2π
k=−∞
(2k−1)πw
Noting that the first integral equals f(t) by the Fourier inversion formula
(2.5),andinserting (2.14)intotheinfiniteseries,weobtaintherepresentation
(2.13). (cid:2)