Convergence Analysis of the Gaussian Regularized Shannon Sampling Formula

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by  Rongrong Lin

1
Convergence Analysis of the Gaussian Regularized
Shannon Sampling Formula
Rongrong Lin and Haizhang Zhang
Abstract—We consider the reconstruction of a bandlimited where t ξ denotes the standard inner product of t and ξ in
function from its finite localized sample data. Truncating the Rd. The·classical Shannon sampling theorem [2], [3] states
classical Shannon sampling series results in an unsatisfactory that each f (R) can be completely reconstructed from
6 convergence rate due to the slow decayness of the sinc function. its infinite sa∈mpBliπngdata f(j):j Z . Specifically,it holds
1 Toovercomethisdrawback,asimpleandhighlyeffectivemethod, { ∈ }
0 called the Gaussian regularization of the Shannon series, was f(t)= f(j)sinc(t j), t R, f (R), (1)
2 proposed in engineering and has received remarkable attention. − ∈ ∈Bπ
n Isteriwesorwkisthbay rmeguultlaiprliyziantgionthGeausisnscianfufnucntciotinoni.nL.thQeiaSnha(Pnrnoocn. Xj∈Z
a Amer. Math. Soc., 2003) established the convergence rate of where sinc(x) = sin(πx)/(πx). Many generalizations of
7 J Oba(n√dnweidxtph(−anπd−2δnn)i)s tfhoer nthuimsbmeretohfodsa,mwphleerdeatδa.<C. πMicischtehllei Sexhaamnnpolen,’s[4s]a–m[1p2li]nagndthethoerermefehraevnecebsetehnereesitna)b.lished (see, for
et al. (J. Complexity, 2009) proposed a different regularization Inpractice,we can onlysum overfinite sample data“near”
] method and obtained the corresponding convergence rate of
.IT aOll(r√1enguelxapr(iz−atπi−2onδnm))e.thTohdissfloartttehrerSahtaeninsobnysefrairest.hHeobweesvtera,mthoenigr itnina (c1o)nvtoeragpepnrcoexriamteatoeffth(te).orTdreurnOca(ti√n1ng)thdeuesetroietshe(1s)lorwesudlets-
s regularized function involves the solving of a linear system and caynessofthesincfunction,[13]–[16].Besides,thistruncated
c
is implicit and more complicated. The main objective of this series was proved to be the optimal reconstruction method
[
note is to show that the Gaussian regularized Shannon series in the worst case scenario in (R). Dramatic improvement
1 can also achieve the same best convergence rate as that by C. of the convergence rate can bBeπachieved when f (R),
v Micchelli et al. We also show that the Gaussian regularization δ < π. In this case, f(j) : j Z turns out to be∈aBsδet of
3 method can improve theconvergence rate for theuseful average { ∈ }
6 sampling. Numerical experiments are presented to justify the oversamplingdata, where oversamplingmeans to sample at a
3 obtained results. rate strictly larger than the Nyquist rate δ <1.
π
1 IndexTerms—Bandlimitedfunctions,Gaussianregularization, Three explicit methods [1], [14], [17] have been proposed
0 oversampling, Shannon’s sampling theorem, average sampling. in order to reconstruct a univariate bandlimited function
1. f δ(R) (δ < π) from its finite oversampling data
∈ B
0 I. INTRODUCTION f(j) : n+1 j n with an exponentially decaying
6 a{pproxim−ationerr≤or. Th≤ey w}ork by multiplyingthe sinc func-
1 THE main purpose of this paper is to show that the
tionwitharapidly-decayingregularizationfunction.Jagerman
: Gaussian regularized Shannon sampling formula to re-
v [14] used a power of the sinc functionsas the regularizerand
construct a bandlimited function can achieve by far the best
i
X obtained the convergence rate of
convergence rate among all regularization methods for the
r Shannon sampling series in the literature. As a result, we 1 π δ
a O exp( − n) .
improveL. Qian’serrorestimate[1] forthishighlysuccessful n − e
method in engineering. (cid:16) (cid:17)
Micchelliet al. [17] chose a spline functionas the regularizer
We first introduce the Paley-Wiener space (Rd) with the
δ and attained the convergence rate
B
bandwidth δ :=(δ ,δ ,...,δ ) (0,+ )d defined as
1 2 d
∈ ∞ 1 π δ
(Rd):= f L2(Rd) C(Rd): suppfˆ [ δ,δ] , O exp( − n) , (2)
Bδ { ∈ ∩ ⊆ − } √n − 2
where [ δ,δ] := d [ δ ,δ ]. In this paper, the Fourier (cid:16) (cid:17)
transform−of f L1(kR=d1)−takkeskthe form whichisbyfarthebestconvergencerateamongallregulariza-
tionmethodsfortheShannonsamplingseriesinreconstructing
∈ Q
1 a bandlimited function. However, the spline regularization
fˆ(ξ):= f(t)e itξdt, ξ Rd,
(√2π)d ZRd − · ∈ function in [17] is implicit and involves the solving of a
linear system. A third method using a Gaussian function as
This workwas supported inpartbyNatural Science Foundation ofChina the regularizer was first proposed by Wei [18]. Precisely,
undergrants 11222103and11101438.
the Gaussian regularized Shannon sampling series in [18] is
R. Lin is with the School of Mathematics and Computational Sci-
ence, Sun Yat-sen University, Guangzhou 510275, P. R. China (e-mail: defined as
[email protected]).
n
andH.CZomhapnugtat(icoonrarlesSpocinednicneg aanudthoGru)anisgdwonitgh PthroeviSnccehooKleyofLaMboatrhaetomryaticosf (Sn,rf)(t):= f(j)sinc(t−j)e−(t2−rj2)2,t∈(0,1),f ∈Bδ(R).
Computational Science, Sun Yat-sen University, Guangzhou 510275, P. R. j= n+1
China(e-mail:[email protected]). X− (3)
2
A convergenceanalysis of the method was presented by Qian We shall also need a variant of its discrete version:
[1], [8]. Takinga smallcomputationalmistake in (2.33)in [1]
(t j)2 r2 (n 1)2
into consideration and optimizing about the variance r of the e− −r2 < e− −r2 ,r >0,n 2,t (0,1).
n 1 ≥ ∈
Gthaeufsoslilaonwfiunngcctioonnvearsgednocneeriante[1fo7r],(3Q)ian actually established j∈/(X−n,n] − (7)
ThenexttwoareausefulcomputationoftheFouriertransform
π δ
O √nexp( − n) . (4)
(cid:16) − 2 (cid:17) sinc(t j)e−(t2−rj2)2 ˆ(ξ)= e−ijξ ξ+πre−r22η2dη, ξ R
Due to its simplicity and high accuracy (4), the Gaus- (cid:16) − (cid:17) 2π Zξ−π ∈(8)
sian regularized Shannon sampling series (3) has been
and an associated estimate
widely applied to scientific and engineering computations. In
fact, more than a hundred such papers have appeared (see (ξ+π)r (π δ)2r2
http://www.math.msu.edu/ wei/pub-sec.html for the list, and 1 1 √2 e τ2dτ 2e− −2 for all ξ [ δ,δ].
[18]–[23]forcommentsan∼ddiscussion).Wenotethatalthough (cid:12) −√π Z(ξ−√π2)r − (cid:12)≤rπ (π−δ)r ∈ −
the exponential term in (4) is as good as that in (2) for the (cid:12) (cid:12) (9)
(cid:12) (cid:12)
The last one is the upper bound estimate
spline function regularization [17], the first term √n is much
worse than 1 in (2). The first purpose of this paper is to k
√n H (x) (2x)k, x (10)
show that the convergence rate of the Gaussian regularized k
| |≤ | |≥ 2
Shannon sampling series can be improved to (2). Thus, this
for the k-th order Hermite polynomial defined as
method enjoys both simplicity and the best convergence rate
byfar.ThiswillbedoneinSectionII,wherewealsoimprove
k
the convergence rates of the Gaussian regularized Shannon H (x):=k!⌊2⌋( 1)i (2x)k−2i , x R.
k
sampling series for derivatives and multivariate bandlimited − i!(k 2i)! ∈
i=0 −
functions. In Section III, we show that the Gaussian regular- X
ization method can also improve the convergencerate for the We are in a position to present an improved convergence
useful average sampling. In the last section, we demonstrate rate analysis for the Gaussian regularized Shannon sampling
our results via several numerical experiments. series. We follow the methods in [1] and emphasize that the
improvement is achieved by applying the Cauchy-Schwartz
inequality to obtain a better estimate for (2.5) in [1].
II. IMPROVED CONVERGENCE RATES FORTHEGAUSSIAN
Theorem 1: Letδ (0,π),n 2,andchooser := n 1.
REGULARIZATION ∈ ≥ π−δ
TheGaussianregularizedShannonsamplingseries(3)sqatis−fies
In this section, we improve the convergence rate analysis
ficoarlltyh,ewGeausshsaiallnsrheogwulathriazteditSchaannnacohniesvame pblyinfgarsetrhieesb.eSsptercaitfe- f∈Bδ(R),sukfpkL2(R)≤1kf −Sn,rfkL∞((0,1))
(π δ)(n 1) (11)
(2)forunivariatebandlimitedfunctionsanditsderivatives,and √2δ+ 1 e− − 2 − .
for multivariate bandlimited functions. We separate the three ≤ √n π (π δ)(n 1)
(cid:18) (cid:19) − −
cases into different subsections.
Proof: Let f δ(R)pwith f L2(R) 1. Set (f
∈ B k k ≤ −
S f)(t):=E (t)+E (t), t (0,1), where
n,r 1 2
A. Univariate Bandlimited Functions ∈
(t j)2
Let f ∈ Bδ(R) with δ ∈ (0,π) throughout this subsection. E1(t):=f(t)− f(j)sinc(t−j)e− 2−r2 ,
WeshallusetheGaussianregularizedShannonsamplingseries j Z
(lo3c)atloizreedcoonvsetrrusacmt tphleinvgalduaetsaofff(ja)t t:∈(n0,+1)1fromjthe finnite. E2(t):= X∈f(j)sinc(t−j)e−(t2−rj2)2.
We shall need a few technica{l facts in−order to≤impro≤ve th}e j∈/(X−n,n]
convergencerateestablishedin[1].Theywerealso frequently Bound E1(t) by its Fourier transform as follows
used in the estimates in [1], [8].
1
ortFhiornstolyrm, iatlibsawsisellf-okrnown(Rt)h.atAssinfc(·−j()R,)j ∈ Z (foRr)m, wane |E1(t)|≤ √2πkEˆ1kL1(R), t∈(0,1).
π δ π
B ∈ B ⊆ B
have by the Parseval identity ComputingandboundingEˆ by(8),(9)andsimilararguments
1
as those in [1], we have
f 2 = f(j)2, f (R). (5)
k kL2(R) | | ∈Bδ
Xj∈Z E (t) √2δe−(π−δ2)2r2 , t (0,1). (12)
The second result needed is the following upper bound esti- | 1 |≤ π(π δ)r ∈
−
mate of Mills’ ratio [24] of standard normal law:
Observe that
+∞e−t2dt< e−x2, x>0. (6) sinc(t j) 1 , t (0,1), j / ( n,n].
Zx 2x | − |≤ πn ∈ ∈ −
3
Thus, It has been estimated in [1] that
|E2(t)| ≤j∈/1(X−n,n]|f(j)|(cid:12)(cid:12)(cid:12)sinπ(πt((−tt−jj))2j)(cid:12)(cid:12)(cid:12)e−(t2−rj2)2 kE1kL∞((0,1)) ≤r2s2+1δs+π12(eπ−−(πδ−)δ2r)2r2 . (14)
f((cid:12)j)e− 2−r2 ,(cid:12)t (0,1).
≤ πn | | ∈
j∈/(X−n,n] For each t∈(0,1), we calculate
Apply the Cauchy-Schwartz inequality, we get by (5) and (7)
1
E (t) = f(j)
|E2(t)| ≤ π1n |f(j)|2 12 e−(t−r2j)2 12 2 √2πs j∈/(X−n,n]
≤ πren−√(cid:16)(nnj2∈−/r(X12−)21n,,n]t∈(0,1(cid:17)).(cid:16)j∈/(X−n,n] (cid:17) ·"1Xk=0k!(ss−! k)!(cid:16)sinπ(s(πtt−s−!jj)π)(cid:17)(s−k)(cid:16)e−(t2−rj2)2(cid:17)(k)#
− (13) = f(j)
Combining(12) with(13) andoptimallychoosingr = nπ−1δ √2π j∈/(X−n,n] "Xk=0k!
completes the proof. q − s−k πs−k−l−1sin π(t−j+ (s−2k−l)) ( 1)ll!
We remark that the estimate (11) is of the same order as · l!((cid:16)s k l)! (cid:17) · (t−j)l+1!
rthemebareksticsoonnvetrhgeendceegeranteera(2te)dbcyafsaerwinhtehneδlit=eraπt.urIen.tAhisseccaosned, (−Xl=10)kHk(√t−2jr) e−−(t2−rj2)2−. −
theestimatein[1]ortheabove(11)isapparentlymeaningless. · (√2r)k #
To make up for the drawback, a more delicate upper bound
estimate is needed for E1. Specifically, we have Noticing the simple fact
1
E (t) Eˆ (ξ)dξ 1 1
| 1 | ≤ √2π(cid:16)Z[−π,π]\[3−π+n−34,π−n−34]| 1 | (cid:17) t j l+1 ≤ n for j ∈/ (−n,n], t∈(0,1), l ∈Z+, (15)
+ 1 π−n−4 Eˆ (ξ)dξ | − |
√2π 3 | 1 | we obtain by (10)
1 1 Z−π+n2−n434
+ .
≤ √πn38 rπ r E2(t) s! f(j)
| | ≤ π√2π | |
Taking r = n89 above and using (13), we obtain the conver- j∈/(X−n,n]
gence rate 3n−83 for f ∈Bπ(R). s 1 s−k πs−k−l 1 |t−j|k e−(t2−rj2)2
·" k! (s k l)! t j l+1 r2k #
Xks=!0 (cid:16)Xl=0 − − | − | (cid:17)
B. Derivatives of Univariate Bandlimited Functions f(j+1)
≤ nπ√2π | |
infiBnyitetlhyedPifafleerye-nWtiaiebnlee.rItnhetohriesmsu,besaecchtiofunn,cwtieonariencBonδc(Rer)neids s j|jXk|≥n s−k πs−k−l j2
twhiethGtahuessrieacnonrsetgruuclatiroiznedofSthhaenndoenrivsaatmivpeslinogf sfer∈iesB(δ3()R.)Tbhye ·"Xk=0k|!r|2k ·(cid:16)Xl=0 (s−k−l)!(cid:17)#e−2r2.
convergence rate obtained in [1], [8] is also of the order (4).
Since ex > xj for all x>0 and j Z , we have
We shall improve the estimate. j! ∈ +
Theorem 2: Let s N, n max 3, s2 , and r :=
nπ−−2δ. It holds ∈ ≥ { 2(π−δ)} |E2(t)| ≤ nπ√s!2π |f(j+1)| s e|rj2|(s−k+1)eπ e−2jr22
q f∈Bδ(R)s,kufpkL2(R)≤1kf(s)−(Sn,rf)(s)kL∞((0,1)) ≤ eπn((s2π+)232|)jX!|≥n |f(j+1(cid:16))Xk|e=−0j22r+22|j|. (cid:17)
√2δs+12 24(s+2)! e−(π−δ)2(n−2) |jX|≥n
+ .
≤ √2s+1 √n !π (π δ)(n 2)
− − Note that n 1+ s2 and r = n 2 implies n sr.
Proof:Wemaysupposethatf ∈Bpδ(R)withkfkL2(R) ≤ We thus get≥by (5),2((7π)−aδn)d the Caucqhyπ-S−−cδhwartz inequ≥ali√ty2
1. Set f(s)(t) (S f)(s)(t) := E (t)+E (t), t (0,1),
n,r 1 2
− ∈
wheEEre12((tt))::==f√(1s2)π(t)− √12πfXj(∈j)Zfs(ijn)c(cid:16)(sti−ncj()te−−(jt2−)rej2)−2(t2−(rsj2))2.(cid:17)(s), kE2(t)kL∞((0,1)) ≤≤ eeππ((ssn(++(22ππ22))))32!!32ee22rr1122 (cid:16)rne2√−j≥nX(nn2−−r−221)22e.−rj22(cid:17)12(16)
j∈/(X−n,n] (cid:16) (cid:17)
4
Substituting r = n 2 into (14) and (16), we have Note that sinc2(x m)=1, x R. Thus, we have
π−δ m Z − ∈
− ∈
q
k=E(cid:16)2ek−πL√(∞π√−2((sδ20)2(+,n1−))12)+pk(Eπ2−kδLsδ+∞)(12(n(0,−1))2) + eπ(2(πs)+32√2)π!e−2(πnδ−−nδ2)(cid:17) j≤X∈P/JndkY=d1 sinc2(tks−injck2)(et−k(−tk−rj2jkk))e2−(tk−r2jk)2 .
· √2δs+12 24(s+2)! e−(π−δ)2(n−2) Xk=1(cid:16)jk∈/X(−n,n] (cid:17)
< + .
Applying (7) to the last inequality gives the desired result.
√2s+1 √n π (π δ)(n 2)
(cid:16) (cid:17) − − Introduce the important constant
p 3π
In the last inequality, we use r = n 2 1 and e 2
π−δ ≥ √π 2√2π ≤ ∆:=max δk :k =1,2,...,d .
24. The proof is complete. q − { }
Lemma 4: Let δ (0,π)d. It holds for ξ d [ δ ,δ ]
∈ ∈ k=1 − k k
C. Multivariate Bandlimited Functions
In this subsection, we show that the best convergence rate 1 d 1 (ξk√+2π)r e−τ2dτ 2de−Q(π−∆2)2r2 .
(2)canalsobeachievedinthereconstructionofamultivariate −kY=1√π Z(ξk√−2π)r ≤rπ (π−∆)r
bandlimited function by the Gaussian regularized Shannon
sampling series. Techniques to be used for this case is quite Proof: By (6), we have for ξ ∈ dk=1[−δk,δk]
different from those in the univariate case and also much
differs from those in [8]. 1 (ξk√+2π)r e τ2dτ Q
Sjh∈Lanen(t−oδnn:,=sna]m(dδ}p1l,iinnδg2,th.si.es.ri,seδusdb)s[e8∈c]ti(ot0on,.πrTe)dchoeannsGtdrauuJcsntsiaa:=nmr{eujgltui∈lvaarZriizdaetde: =√π1Z(ξk1√−2π)r +∞− e−τ2dτ 1 +∞ e−τ2dτ
dfuenficnteiodnasf ∈ Bδ(Rd) from its finite sample data f(Jn) is − √1π Z(πe−√−ξ2k()πr−ξ2k)2r2 − √e−π(πZ+(ξπ2k+)√2ξ2rk2)r
1 + .
( n,rf)(t):= f(j)sinc(t j)e−kt2−rj2k2,t (0,1)d, ≥ − √π √2(π−ξk)r √2(π+ξk)r
S − ∈  
jX∈Jn (17) 1We tkhen dapwpliyththedelemτen<tar1y,fiatchtotlhdast for constants τk > 0,
where ≤ ≤ k=1 k
d sin(πx ) P d d
sinc(x):= k , x Rd 1 (1 τ ) τ .
πxk ∈ − − k ≤ k
kY=1 kY=1 Xk=1
and x denotes the standard Euclidean norm of x in Rd. As a consequence,
k k
Twolemmasareneededtopresentanimprovedconvergence
analysis for (17). 1 d 1 (ξk√+2π)r e τ2dτ
−
Lemma 3: Let r>0 and n≥2. It holds for all t∈(0,1)d −kY=1√π Z(ξk√−2π)r
sinc2(t−j)e−kt−r2jk2 ≤ πd2rn22e(−n(n−r211))2 . ≤ √1π kX=d1√e−2((ππ−−ξ2k)ξ2kr)2r + √e−2((ππ++ξ2k)ξ2kr)2r
jX∈/Jn(cid:16) (cid:17) − 2 d e−(π−ξ2k)2r2 
.
Proof: Observe that ≤ π (π ξ )r
r k=1 − k
X
j Zd :j / Jn d j Zd :jk / ( n,n] . (18) The proof is completed by noting that x1e−r22x2 is decreasing
{ ∈ ∈ }⊆ { ∈ ∈ − } on (0,+ ).
k[=1 ∞
With the above preparation, we have the following main
By (18), we have for each t (0,1)d theorem.
∈
Theorem 5: Let δ (0,π)d, n 2, and r := n 1. The
sinc2(t−j)e−kt−r2jk2 multivariate Gaussian∈regularized≥Shannon samqpliπn−−g∆series
jX∈/Jn(cid:16) (cid:17) (17) satisfies
d
≤Xk=1(cid:16)jk∈/X(−sni,nnc] s2i(ntlc2(tjkl)−e−j(ktl)−re2j−l)(2tk−r.2jk)2(cid:17) f∈Bδd(R(d2)∆,suk)fpd2kL+2(Rd)d≤1kf −eS−n(π,r−f∆k2)(Ln∞−1()(0,1)d.)
·lY6=k(cid:16)jXl∈Z − (cid:17) ≤ rn!π (π−∆)(n−1)
p
5
Proof: Let f ∈ Bδ(Rd) with kfkL2(Rd) ≤ 1. Set f(t)− where 0 < σ < πδ and ν is a symmetric positive Borel
(S f)(t):= (t)+ (t), t (0,1)d, where probability measure on [ σ/2,σ/2]. It was observed in [25]
n,r 1 2
E E ∈ −
that
t j 2
E21((tt))::==f(t)−f(jjX∈)Zsdinfc((jt)sinjc)e(t−−kt2−jrj2)ke2−.k2−r2k , cos(cid:16)σ2δ(cid:17)kfk2L2(R) ≤Xj∈Z|f˜(j)|2 ≤kfk2L2(R). (21)
E −
jX∈/Jn Thus, {f˜(j):j ∈Z} is in fact induced by a frame in Bδ(R).
By Lemma 4 and similar arguments as those in [1] for the By the standard frame theory, we are able to completely
univariate case, we have reconstruct f from the infinite sample data f˜(j) : j Z
{ ∈ }
througha dualframe.Forstudiesalongthisdirection,see, for
ˆ (ξ) = fˆ(ξ) 1 d 1 (ξk√+2π)r e τ2dτ example, [4], [5], [9], [10], [26], [27].
1 −
|E | | |(cid:12)(cid:12)(cid:12) −kY=1(π√π∆)Z2r(2ξk√−2π)r (cid:12)(cid:12)(cid:12) duMalofrtiavmateedthabtyisthgeenSehraantendonbysatmheplsihnigftsthoefoaresmin,gwleefudnescitrieona.
fˆ(ξ) (cid:12)(cid:12) 2de− −2 , ξ [ δ,δ].(cid:12)(cid:12) In other words, we prefer a complete reconstruction formula
≤| | π (π ∆)r ∈ −
r − of the following form
Bounding by its Fourier transform and then applying the
1
E 1
Cauchy-Schwartz inequality, we get f = f˜(j)φ( j), f (R), (22)
δ
√2π ·− ∈B
1 j Z
ˆ(ξ)dξ X∈
kE1kL∞((0,1)d) ≤ √d2e−π(Zπ−[−∆2δ),2δr]2|E | fˆ(ξ)dξ (19) wφthhe∈enrCe(2(φ2R))ihs∩otlodLs2b(iefRac)nhwdosioethnnl.ysIutipfhpasφˆb⊆eeInδp:=rov[e−d2πin+[2δ5,]2πth−atδif]
≤ π (π ∆)r | |
d(2∆)−d2 e−(π−Z∆2[−)2δr,2δ]. φˆ(ξ)W(ξ)=1 for almost every ξ ∈[−δ,δ], (23)
≤ π (π ∆)r
− where
Recall that σ/2
W(ξ):= eitξdν(t), ξ [ δ,δ]. (24)
kfk2L2(Rd) = |f(j)|2, f ∈Bδ(Rd). Z−σ/2 ∈ −
jX∈Zd Under the above assumptionson ν, W(ξ) is an even function
By this, Lemma 3, and the Cauchy-Schwartz inequality, we on [ δ,δ] and satisfies
−
obtain
σδ σ
kE2kL∞((0,1)d) ≤ |f(j)|2 12 0<γ :=cos(cid:16) 2 (cid:17)≤Wσ2(ξ)≤1, |W′(ξ)|≤ 2, (25)
·(cid:16)jX∈/Jn sinc2((cid:17)t−j)e−kt−r2jk2 21 (20) Inpractice,we|oWnl′y′(hξ)a|ve≤th4efi,nξite∈s[a−mδp,lδe].dataf˜(j), n<
√(cid:16)djX∈/rJen−(n2−r12)2 (cid:17) j ≤n. A natural reconstruction method is by −
. n
≤ π n√n 1 1 f˜(j)φ( j). (26)
−
√2π ·−
The proof is completed by taking r = πn−∆1 in (19) and j=X−n+1
(20). q − The main purpose of this section is to illustrate by an ex-
plicit example that compared to the above direct truncation,
III. GAUSSIAN REGULARIZATIONFOR AVERAGE regularization by a Gaussian function as follows
SAMPLING
In this section, we will apply the method of Gaussian (Sn,rf)(t):= 1 f˜(j)φ(t j)e−(t2−rj2)2,t (0,1),f δ(R)
√2π − ∈ ∈B
regularizationtotheusefulaveragesampling.Amainpurpose j∈(X−n,n] (27)
is again to improve the convergence rate. We first introduce
may lead to a better convergencerate.
some basic facts about the average sampling.
Sampling a function f at j Z can be viewed as applying The function φ satisfying (23) in our example is specified
∈ as
theDeltadistributiontothefunctionf(j+ ).TheDeltadistri-
bution is used theoretically but hard to imp·lement physically. 1 , ξ [ δ,δ],
W(ξ) ∈ −
Hbyenacnea,vaeprargacintigcaflunwcatyioinswtoitahpspmroaxllimsuaptepothretaDroeultnaddtihsteriobruitgiionn. φˆ(ξ):= |ξ|2−π(2π2−δδ) 2P(ξ), δ ≤|ξ|≤2π−δ, (28)
For this sake, we consider the following average sampling  0h, − i elsewhere,
strategy: where 
σ/2
f˜(j):= f(j+ )dν(x), j Z, f δ(R), P(ξ):= 1 W′(δ) ξ + π−2δ + δW′(δ) .
Z−σ/2 · ∈ ∈B (cid:20)(cid:16)(π−δ)W(δ) − W2(δ)(cid:17)| | (π−δ)W(δ) W2(δ)(cid:21)
6
Then suppφˆ [ 2π + δ,2π δ], φˆ C1(R) and φˆ is (28), we have
′
absolutely cont⊆inuo−us on R. By−(25), we∈have Fˆ (ξ)
1
| |
kkφφˆˆk′′LkL∞∞(I(δI)δ)≤≤(2π3σγδπ32σ−−2+δ)δ(γπ+−8πγ3δσ)223,γ + (π−41δσ6)σ2γ2, (29) ≤ +√(cid:12)2|Zfπˆ|(|ηφξˆ|≥)(|ξε)(cid:16)|φ"ˆ((cid:12)(cid:12)(cid:12)ξZ)|η−|<φˆε((cid:16)ξφˆ−(ξη))−(cid:17)rφˆe(−ξr−22η2ηd)η(cid:17)(cid:12)r#e−r22η2dη(cid:12)(cid:12)(cid:12)
kφˆ′′kL1(Iδ) ≤ γ3 + (π δ)2γ + (π δ)γ2. |fˆ(cid:12)(cid:12)(ξ)| φˆ′(ξ)η+ φˆ′′(ξ−θη)η(cid:12)(cid:12)2 re−r22η2dη
− − ≤ √2π "(cid:12)Z|η|<ε(cid:16) 2 (cid:17) (cid:12)
Faninaalyllsyi,s w[2e8]sthhaaltl use an elementary fact from the Fourier +Z|η|≥ε(cid:12)(cid:12)(cid:12)φˆ(ξ)−φˆ(ξ−η)(cid:12)re−r22η2dη# (cid:12)(cid:12)
|φ(t−j)|≤ √12πkφˆt′′kLj1(2Iδ), j ≥2, t∈(0,1). (30) ≤ |√fˆ(2ξπ)|"k(cid:12)(cid:12)φˆ′′k2L∞(Iδ) ·Z|η|<(cid:12)(cid:12)εη2re−r22η2dη
The main result of thi|s−sect|ion is as follows. +2kφˆkL∞(Iδ)·Z|η|≥εre−r22η2dη#
andThφˆeobreemgiv6e:nLaest(δ28∈).(T0h,eπn),thne≥co2n,veγrg=enccoesr(aσt2eδ)o,fr(2=7)nis65 ≤ |√fˆ(2ξπ)|"kφˆ′′k2Lr2∞(Iδ) ·Z|x|<rεx2e−x22dx
+
f∈Bδ(R),sukfpkL2(R)≤1kf −Sn,rfkL∞((0,1)) ≤cn−53, +4√2kφˆkL∞(Iδ)·Z√rε2∞e−x2dx#,
where 0 < θ < 1 and ε := r. Combining the last inequality
where above with Rx2e−x22dx=√2π and (6) yields
c= (√δ+γ32δ)σ2 + (π4σ√δδ)2 + π16σδ +πσ√δ!γ12 |Fˆ1(ξ)| ≤R|√fˆ(2ξπ)| √2πkφˆ2′r′k2L∞(Iδ) + 4e−r24krφˆ2kL∞(Iδ)!
− −
+ 8√δ + 32 +10√δ 1 . |fˆ(ξ)| kφˆ′′kL∞(Iδ) + 4kφˆkL∞(Iδ) .
(π δ)2 π δ !(π δ)γ ≤ r2 2 √2π !
− − −
BoundingF by the L1-normof its Fourier transform,we get
1
Proof: Let f δ(R) with f L2(R) 1. Set f(t) by the above equation
Sn,rf(t):=F1(t)+∈FB2(t), t∈(0,k1),kwhere≤ − F (t) √δ kφˆ′′kL∞(Iδ) + 4kφˆkL∞(Iδ)
F1(t):=f(t)− √12π j Zf˜(j)φ(t−j)e−(t2−rj2)2, | 1 | ≤ rk2φˆ ′′kL∞2(I√δ)π+kφˆkL∞(Iδ√)2√πδ !
F2(t):= 1 X∈f˜(j)φ(t j)e−(t2−rj2)2. ≤ (cid:16) r2 (cid:17) , t∈(0,1)(.31)
√2π −
j∈/(X−n,n] Estimate of F2. By (30) and by the Cauchy-Schwartz
inequality, for each t (0,1),
∈
(ξ√∈E12πsRtiP,maj∈teZf˜(ojf)φˆ(Fξ1).e−iBjξy)χ[−(2δ,2δ)](,ξ),weξ ∈haveR. fTˆ(hξe)n, fo=r |F2(t)| ≤jk∈/φˆ(X′−′knL,n1](|Ifδ˜)(j)||φ(t−fj˜()j|e)−e(−t2−(rtj22−)r2j2)2
Fˆ1(ξ) =·fˆ(Rξ)r−e−√ij(12ξπ−ηXj)∈φˆZ(ξf˜(−j)η√)e12−π(ijη+r22η2)dη ≤≤ kφˆ√√′′22kππLnn1(22Iδ)(cid:16)j∈/j(X∈/−(X−n,nn,]n|]|f˜(j|)|2(cid:17)12(cid:16)j∈/(X−n,n]e−(t−r2j)2(cid:17)12.
=fˆZ(ξ) 1 f˜(j)e ijξφˆ(ξ) 1 It follows from (7) and (21) that
φˆ−(ξ√2ηπ)Xj∈1Z re−r−22η2dη φˆ(ξ) |F2(t)|≤ √r2kπφˆn′′2k√L1n(Iδ)1e−(n2−r12)2 ≤ rkφˆ′′nk52L1(Iδ). (32)
· R − √2π −
= fˆZ(ξ)χφˆ[(−ξδ),δ](ξ)hφˆ(ξ)− √12π ZRφˆ(ξ−η)re−r22η2dηiT.akingfr−=Snn,r65finL(3(1()0,1a)n)d (32), we obtain
(cid:13) (cid:13) ∞ 1
If |ξ| > δ then Fˆ1(ξ) = 0. If |ξ| ≤ δ then by (7), (25) and ≤(cid:13)(cid:13) √δkφˆ′′k(cid:13)(cid:13)L∞(Iδ)+√δkφˆkL∞(Iδ)+kφˆ′′kL1(Iδ) n53,
(cid:16) (cid:17)
7
which, by (29), completes the proof. E(fδ−Tnfδ) Eδ,n E(fδ−Snfδ)
We see that if φˆ C1(R) and φˆ is absolutely continuous n=2 0.0331 0.1663 0.0509
on R then the conv∈ergence rate of′the direcr truncation (26) nn==46 2.207.06019e8-04 9.701.02140e7-04 1.009.03021e7-04
is O(n−32). Therefore, the Gaussian regularized version (27) n=8 0.0024 9.8103e-05 8.5248e-06
indeed results in a better convergencerate. n=10 0.0016 1.0433e-05 7.2866e-07
n=12 3.0405e-05 1.1440e-06 6.8632e-08
n=14 7.9461e-04 1.2799e-07 6.6728e-09
n=16 6.2247e-04 1.4525e-08 6.5985e-10
IV. NUMERICAL EXPERIMENTS n=18 9.1326e-06 1.6661e-09 6.8773e-11
n=20 3.9146e-04 1.9267e-10 7.2196e-12
n=22 3.2878e-04 2.2428e-11 7.5895e-13
We present in this section numerical experiments to illus-
n=24 3.8717e-06 2.6246e-12 8.3156e-14
trate the effectiveness of the Gaussian regularized sampling n=26 2.3225e-04 3.0851e-13 9.2149e-15
formula and to demonstrate the validness of our convergence n=28 2.0277e-04 3.6399e-14 1.4433e-15
n=30 1.9869e-06 4.3080e-15 4.4409e-16
analysis for the formula.
TABLE I: Reconstruction erros for δ = π.
3
A. Univariate Bandlimited Functions E(fδ−Tnfδ) Eδ,n E(fδ−Snfδ)
n=2 0.0051 0.2871 0.0521
Let δ < π. The bandlimited function under investigation n=4 9.6902e-04 0.0316 0.0030
n=6 3.2453e-04 0.0049 3.1393e-04
has the form
n=8 1.4422e-04 8.3589e-04 4.2582e-05
n=10 7.5827e-05 1.5055e-04 6.1037e-06
1 2sinδt sinδ(t 1)
f (t)= + − , t R. n=12 4.4556e-05 2.7936e-05 9.8330e-07
δ π(5δ+4sinδ) t t 1 ∈ n=14 2.8327e-05 5.2865e-06 1.5771e-07
(cid:16) − (cid:17) (33) n=16 1.9097e-05 1.0144e-06 2.7379e-08
We pointpout that fδ δ(R) and fδ L2(R) = 1. We shall nn==1280 19..38457225ee--0056 13..98646481ee--0078 48..64526844ee--0190
∈ B k k
reconstruct the values of f on (0,1) from f(j): n+1 n=22 7.4200e-06 7.5615e-09 1.4863e-10
{ − ≤
j n by the the truncated Gaussian regularized Shannon n=24 5.7257e-06 1.4951e-09 2.7629e-11
≤ } n=26 4.5098e-06 2.9691e-10 4.9942e-12
sampling series
n=28 3.6149e-06 5.9176e-11 9.4547e-13
n=30 2.9418e-06 1.1831e-11 1.7397e-13
n
(Snfδ)(t)= fδ(j)sinc(t−j)e−(t−2j()n2−(π1−)δ), t∈(0,1). TABLE II: Reconstruction errors for δ = π2.
j= n+1
X−
E(fδ−Tnfδ) Eδ,n E(fδ−Snfδ)
The error of reconstruction is measured by n=2 0.0275 0.5074 0.0490
n=4 0.0064 0.0951 0.0049
j n=6 4.6943e-04 0.0249 9.2523e-04
E(fδ Snfδ):= max (fδ Snfδ) n=8 0.0020 0.0072 1.9165e-04
− 1≤j≤99(cid:12) − (cid:16)100(cid:17)(cid:12) n=10 0.0012 0.0022 5.0903e-05
(cid:12) (cid:12) n=12 6.8750e-05 6.9045e-04 1.3828e-05
This error is to be compared wit(cid:12)h the theoretical est(cid:12)imate in n=14 6.5098e-04 2.2083e-04 3.5771e-06
Theorem 1: n=16 4.6868e-04 7.1605e-05 1.0844e-06
n=18 2.1160e-05 2.3456e-05 3.2408e-07
(π δ)(n 1) n=20 3.1800e-04 7.7448e-06 9.1110e-08
E := √2δ+ 1 e− − 2 − ,n 2. n=22 2.5108e-04 2.5733e-06 2.9088e-08
δ,n √n π (π δ)(n 1) ≥ n=24 9.0566e-06 8.5940e-07 9.0859e-09
(cid:16) (cid:17) − − n=26 1.8765e-04 2.8824e-07 2.6710e-09
p n=28 1.5607e-04 9.7020e-08 8.7632e-10
We also compute the reconstruction error resulting from di-
n=30 4.6691e-06 3.2757e-08 2.8080e-10
rectly truncating the Shannon series
TABLE III: Reconstruction errors for δ = 2π.
3
j
E(f T f ):= max (f T f ) .
δ n δ δ n δ
− 1 j 99 − 100
≤ ≤ (cid:12) (cid:16) (cid:17)(cid:12) 0 log(E(fδ−Snfδ))
(cid:12) (cid:12) −20 log(Eδ,n)
for comparison, where (cid:12) (cid:12)
−40
0 5 10 15 20 25 30
0
n
sin(π(t j)) y−20
(T f )(t):= f (j) − , t (0,1).
n δ δ −40
π(t j) ∈ 0 5 10 15 20 25 30
j= n+1 − 0
X−
−20
The above three errors for n=2,4,...,30 and δ = π,π,2π −400 5 10 15 20 25 30
3 2 3 n
are listed in TABLEs I, II, and III, respectively. We also
Fig. 1: Comparisonof logE and logE(f S f ) for δ =
plot logEδ,n and logE(fδ − Snfδ) in Fig. 1. We see that π (Top), δ = π (Middle), aδn,nd δ = 2π (Boδt−tomn). δHere n =
theGaussianregularizedShannonsamplingformulaconverges 3 2 3
2,3,4,...,30.
extremelyfastandourtheoreticalestimatefortheconvergence
rate in Theorem 1 is pretty sharp.
8
B. Derivative of Univariate Bandlimited Functions E(fδ′−(Tnfδ)′) Eδ′,n E(fδ′−(Snfδ)′)
n=3 0.0110 15.7754 0.0419
We now turn to verify the reconstruction error for the first n=5 0.0159 2.4966 0.0071
derivative of a univariate bandlimited function. We use f in n=7 0.0075 0.5774 0.0014
δ
n=9 6.3518e-04 0.1519 3.2892e-04
the above subsection for simulation. Let
n=11 0.0033 0.0427 9.3779e-05
2δ 144 e−(π−δ)2(n−2) n=13 0.0022 0.0125 2.4603e-05
Eδ′,n := r 3 δ+ √n π (π δ)(n 2) nn==1157 1.406.07021e4-04 00..00003182 62..71116536ee--0066
(cid:16) (cid:17) − −
n=19 0.0011 3.6495e-04 6.0578e-07
bethetheoreticalupperboundestabplishedforthederivativein n=21 5.4615e-05 1.1564e-04 1.7733e-07
Theorem 2. This upper bound together with the errors of the n=23 7.5824e-04 3.7009e-05 5.8201e-08
two reconstruction methods are listed in TABLEs IV, V, and n=25 6.1786e-04 1.1941e-05 1.7429e-08
n=27 2.5934e-05 3.8797e-06 5.3009e-09
VI.Inparticular,logE andlogE(f (S f ))are plotted
δ′,n δ′− n δ ′ n=29 4.7534e-04 1.2678e-06 1.7783e-09
in Fig. 2.
TABLEVI:Reconstructionerrorsforthefirstderivativewhen
E(fδ′−(Tnfδ)′) Eδ′,n E(fδ′−(Snfδ)′) δ = 23π.
n=3 0.0064 6.4845 0.0238
n=5 0.0192 0.3582 0.0012
n=7 0.0103 0.0289 8.6147e-05
n=9 2.8936e-04 0.0027 7.2344e-06 C. Bivariate Bandlimited Functions
n=11 0.0041 2.6206e-04 6.5240e-07
n=13 0.0030 2.6989e-05 6.1290e-08 We consider reconstructing bivariate bandlimited functions
n=15 6.3904e-05 2.8410e-06 6.1368e-09
by the regularized Shannon sampling formula in this subsec-
n=17 0.0017 3.0639e-07 6.2431e-10
n=19 0.0014 3.3571e-08 6.4088e-11 tion. The function to be reconstructed is given by
n=21 2.3438e-05 3.7246e-09 6.8596e-12
n=23 9.3372e-04 4.1740e-10 7.3584e-13 f (t ,t )=f (t )f (t ), t ,t R,
n=25 8.0083e-04 4.7166e-11 7.8978e-14 δ 1 2 δ1 1 δ2 2 1 2 ∈
n=27 1.1058e-05 5.3670e-12 9.9087e-15
n=29 5.8806e-04 6.1433e-13 3.9690e-15 where δ = (δ ,δ ) with δ δ < π, and f ,f are
TABLE IV:Reconstructionerrorsforthe firstderivativewhen defined by (33)1. C2learly, fδ1∈≤Bδ(R22) and kfδkL2δ(1R2)δ2= 1.
TheGaussianregularizedanddirecttruncationoftheShannon
δ = π.
3 sampling formula are respectively given by
n n
E(fδ′−(Tnfδ)′) Eδ′,n E(fδ′−(Snfδ)′) (S f )(t ,t ) = f (j ,j )sinc(t j )
n=3 0.0520 9.8133 0.0321 n δ 1 2 δ 1 2 1− 1
nnn===579 000...000100996290 000...910121759345 34..58020.0050482ee8--0045 ·j2s=iXn−cn(+t21j−1=Xj−2)ne+−1((t1−j1)2+2((tn2−−1j)2)2)(π−δ2)
n=11 0.0040 0.0032 7.6107e-06
n=13 0.0029 5.6310e-04 1.1889e-06 and
n=15 0.0022 1.0053e-04 2.0547e-07
n=17 0.0017 1.8324e-05 3.4305e-08 n n
n=19 0.0014 3.3930e-06 6.2129e-09 (Tnfδ)(t) = fδ(j1,j2)sinc(t1 j1)
−
n=21 0.0011 6.3613e-07 1.0797e-09
n=23 9.2764e-04 1.2046e-07 2.0128e-10 j2s=iXn−cn(+t1j1=jX−)n.+1
2 2
n=25 7.8531e-04 2.3001e-08 3.5939e-11 · −
n=27 6.7338e-04 4.4222e-09 6.8315e-12
Similarly, we shall compare the following two reconstruction
n=29 5.8378e-04 8.5523e-10 1.2441e-12
errors:
TABLE V: Reconstruction errors for the first derivative when
δ = π. j j
2 E(f S f ):= max (f S f ) 1, 2 ,
δ n δ δ n δ
− 1≤j1,j2≤49(cid:12) − (cid:16)50 50(cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
0 j j
E(f T f ):= max (f T f ) 1, 2 ,
−−42000 lloogg(( EE(δf,5nδ−) Snfδ)) 10 15 20 25 30 δ− n δ 1≤j1,j2≤49(cid:12) δ− n δ (cid:16)50 50(cid:17)(cid:12)
0 (cid:12) (cid:12)
and the theoretical upper bound(cid:12)for the Gaussian regula(cid:12)rized
y−20
method established in Theorem 5
−400 5 10 15 20 25 30
20
−200 Eδ,n := 4δ2+ 2 e−(π−δ22)(n−1) ,n 2.
−400 5 10 1n5 20 25 30 (cid:16) rn(cid:17)π (π−δ2)(n−1) ≥
Fig. 2: Comparison of logEδ′,n and logE(fδ′ −(Snfδ)′) for The results are shown in TpABLEs VII–IX, and plotted in
δ = π (Top), δ = π (Middle), δ = 2π (Bottom). Fig. 3. Again, we see that our estimate in Theorem 5 is valid
3 2 3
and sharp.
9
E(fδ−Tnfδ) Eδ,n E(fδ−Snfδ)
n=2 0.0316 0.8434 0.0378
n=4 4.6243e-04 0.0971 0.0016
n=6 0.0041 0.0154 1.5469e-04
n=8 1.2792e-04 0.0027 2.1566e-05
n=10 0.0014 4.8514e-04 3.1876e-06
n=12 2.1342e-05 9.0694e-05 4.8749e-07
n=14 7.5353e-04 1.7264e-05 7.7783e-08
n=16 1.6774e-05 3.3288e-06 1.3819e-08
n=18 4.4541e-04 6.4803e-07 2.4148e-09
n=20 4.7213e-06 1.2710e-07 4.1835e-10
n=22 3.0412e-04 2.5075e-08 7.3353e-11
n=24 5.0184e-06 4.9711e-09 1.3924e-11
n=26 2.1432e-04 9.8948e-10 2.5773e-12
n=28 1.7325e-06 1.9762E-10 4.6990e-13
n=30 1.6324e-04 3.9585e-11 8.5987e-14
TABLE VII: Comparison of two reconstruction errors and the theoretical upper bound for δ =(π,π).
4 2
E(fδ−Tnfδ) Eδ,n E(fδ−Snfδ)
n=2 0.0369 0.4005 0.0558
n=4 0.0114 0.0269 0.0020
n=6 2.6001e-04 0.0025 1.2529e-04
n=8 0.0027 2.5544e-04 9.7707e-06
n=10 0.0018 2.7429e-05 8.3512e-07
n=12 3.4742e-05 3.0296e-06 7.8652e-08
n=14 9.1131e-04 3.4093e-07 7.6473e-09
n=16 7.1478e-04 3.8875e-08 7.5614e-10
n=18 1.0435e-05 4.4769e-09 7.8805e-11
n=20 4.4911e-04 5.1951e-10 8.2729e-12
n=22 3.7744e-04 6.0653e-11 8.6942e-13
n=24 4.4241e-06 7.1166e-12 9.4758e-14
n=26 2.6649e-04 8.3846e-13 1.0381e-14
n=28 2.3275e-04 9.9129e-14 1.3323e-15
n=30 2.2703e-06 1.1755e-14 5.5511e-16
TABLE VIII: Comparison of two reconstruction errors and the theoretical upper bound for δ =(π,π).
3 3
E(fδ−Tnfδ) Eδ,n E(fδ−Snfδ)
n=2 0.0232 1.7279 0.0497
n=4 0.0052 0.3392 0.0040
n=6 3.0946e-04 0.0909 6.4739e-04
n=8 0.0014 0.0267 1.4029e-04
n=10 7.9255e-04 0.0082 3.5117e-05
n=12 8.1831e-05 0.0026 9.9855e-06
n=14 4.7669e-04 8.3560e-04 2.4699e-06
n=16 3.4229e-04 2.7222e-04 7.8074e-07
n=18 1.3993e-05 8.9525e-05 2.2438e-07
n=20 2.1941e-04 2.9658e-05 6.5528e-08
n=22 1.7337e-04 9.8829e-06 2.0155e-08
n=24 1.0669e-05 3.3090e-06 6.5146e-09
n=26 1.3496e-04 1.1123e-06 1.8527e-09
n=28 1.1215e-04 3.7516e-07 6.2749e-10
n=30 3.0888e-06 1.2690e-07 1.9506e-10
TABLE IX: Comparison of two reconstruction errors and the theoretical upper bound for δ =(π,2π).
2 3
D. Average Sampling where
Finally, we briefly discuss about the average sampling.
2 δ cos(tξ) 2π−δ 2π δ ξ 2
Consider reconstructing the function f defined by (33) from φ(t) = dξ+ − −
δ
π W(ξ) 2π 2δ
its finite average sampling data r Z0 Zδ (cid:16) − (cid:17)
1 W (δ) π 2δ
′
fδ(j):= 23fδ(j)+fδ(j− 81)+fδ(j+ 81)+12fδ(j− 116)+fδ(j+ 116). ·(cid:20)(cid:16)δW(π−(δ)δ)W(δ) − W2(δ)(cid:17)ξ+ (π−−δ)W(δ)
′
+ cos(tξ)dξ .
By our analysis in Section III, we shall use the following W2(δ)
e (cid:21) (cid:19)
truncated Gaussian regularized reconstruction formula
The reconstruction error is measured by
n
(Snfδ)(t)= √12π f˜δ(j)φ(t−j)e−(2tn−5j/)32, t∈(0,1), Eσ(fδ Snfδ):= max (fδ Snfδ) j .
j= n+1 − 1 j 19 − 20
X− ≤ ≤ (cid:12) (cid:16) (cid:17)(cid:12)
(cid:12) (cid:12)
(cid:12) (cid:12)
10
Results are shown in TABLE X, which are superior than [7] D.K.Hoffman,N.Nayar,O.A.SharafeddinandD.J.Kouri,“Analytic
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REFERENCES R. Lin received the B.S. degree in mathematics and applied mathematics
from Zhangzhou Normal University, Zhangzhou, P.R. China, in 2012. He is
[1] L.Qian,“OntheregularizedWhittaker-Kotel’nikov-Shannnon sampling currentlypursuingthePh.D.degreeincomputationalmathematicsatSunYat-
formula,”Proc.Amer.Math.Soc.,vol.131,no.4,pp.1169–1176,Oct. senUniversity, Guangzhou, P.R.China, throughafive-year Ph.D.program.
2003. Hisresearchinterestsincludetime-frequencyanalysisandkernelmethods
[2] C. E.Shannon, “Communication in the presence of noise,” Proc. IRE, inmachinelearning.
vol.37,pp.10–21,Jan.1949.
[3] E. T. Whittaker, “On the functions which are represented by the H.Zhangreceived theB.S.degreeinmathematics andappliedmathematics
expansion oftheinterpolation theory,”Proc.Roy.Soc.EdinburghSect. from Beijing Normal University, Beijing, P.R. China, in 2003, the M.S.
A,vol.35,pp.181–194,1915. degreeincomputationalmathematicsfromtheChineseAcademyofSciences,
[4] A.Aldroubi,“Non-uniformweightedaveragesamplingandreconstruc- Beijing, P. R. China in 2006, and the Ph.D. degree in mathematics from
tion in shift-invariant and wavelet spaces,” Appl. Comput. Harmon. SyracuseUniversity, NY,in2009.
Anal.,vol.13,no.2,151–161,Sept.2002. From June 2009 to May 2010, he was a postdoctoral research fellow at
[5] K.Gro¨chenig,“Reconstructionalgorithmsinirregularsampling,”Math. UniversityofMichigan,AnnArbor.SinceJune2010,hehasbeenaProfessor
Comput.,vol.59,no.199,pp.181–194,Jul.1992. with the School of Mathematics and Computational Science, Sun Yat-sen
[6] A.J.Jerri,“TheShannonsamplingtheorem–Its variousextensions and University, Guangzhou, P.R.China.
applications: Atutorialreview,”Proc.IEEE,vol.65,no.11,pp.1565– Prof.Zhang’s research interests include computational harmonicanalysis,
1596,Nov.1977. kernelmethods inmachinelearning, andsamplingtheory.