Post on 27-Jan-2023
ARTICLE IN PRESS
Contents lists available at ScienceDirect
Signal Processing
Signal Processing 90 (2010) 145–156
0165-16
doi:10.1
$ The
Interna$$ T
Founda� Cor
E-m
ullrich.m
journal homepage: www.elsevier.com/locate/sigpro
Convergence behavior of non-equidistant sampling series$,$$
Holger Boche �, Ullrich J. Monich
Technische Universitat Berlin, Heinrich-Hertz-Chair for Mobile Communications, Einsteinufer 25, D-10587 Berlin, Germany
a r t i c l e i n f o
Article history:
Received 6 October 2008
Received in revised form
15 April 2009
Accepted 1 June 2009Available online 6 June 2009
Keywords:
Sampling series
Sine type
Non-equidistant sampling
Reconstruction
Stochastic process
84/$ - see front matter & 2009 Elsevier B.V. A
016/j.sigpro.2009.06.003
material in this paper was presented in pa
tional Conference on Acoustics, Speech, and Si
his work was partly supported by the
tion (DFG) under Grant BO 1734/9-1.
responding author.
ail addresses: holger.boche@mk.tu-berlin.de (
oenich@mk.tu-berlin.de (U.J. Monich).
a b s t r a c t
The convergence of sampling series with non-equidistant sampling points cannot be
guaranteed for the Paley–Wiener space PW1p if the class of sampling patterns is not
restricted. In this paper we consider sampling patterns that are made of the zeros of
sine-type functions and analyze the local and global convergence behavior of the
sampling series. It is shown that oversampling is necessary for global uniform
convergence. If no oversampling is used there exists for every sampling pattern a
signal such that the peak value of the approximation error grows arbitrarily large.
Furthermore, we use the findings to derive results about the mean-square convergence
behavior of the sampling series for bandlimited wide-sense stationary stochastic
processes. Finally, a procedure is given to construct functions of sine type and possible
sampling patterns.
& 2009 Elsevier B.V. All rights reserved.
1. Notation and motivation
The reconstruction of bandlimited signals from theirsamples is important for many applications in signalprocessing, communication, and information theory. TheShannon sampling series
X1k¼�1
f ðkÞsinðpðt � kÞÞ
pðt � kÞ(1)
with sinc-kernel is probably the most prominent exampleof a reconstruction process. However, it is not the onlypossible one. In particular for practical applications, non-equidistant sampling patterns are of interest. In this case,more general sampling series can be used for thereconstruction, like
X1k¼�1
f ðtkÞfkðtÞ, (2)
ll rights reserved.
rt at the 2009 IEEE
gnal Processing [1].
German Research
H. Boche),
where fk, k 2 Z, are certain reconstruction functions, andftkgk2Z is the sequence of real sampling points.
There is a vast amount of literature discussing theproperties of sampling series with non-equidistant sam-pling points for the space of bandlimited signals withfinite energy: The papers [2,3] analyze the stability of suchsampling series, and [4] derives series representations.Sampling series involving derivatives are considered in[5], and the case where only a finite number of samplingpoints differs from the integer grid is considered in [6].Aspects of numerical computation in the reconstruction ofbandlimited signals with finite energy from irregularsamples are treated in [7,8].
Only few papers discuss non-equidistant sampling forlarger signal spaces than the space of bandlimited signalswith finite energy [9,10]. In [9] Seip proves the uniformconvergence of (2) on all compact subsets of the complexplane for bounded bandlimited signals if oversampling isused and the sequence ftkgk2Z of real sampling pointsfulfills
supk2Z
jtk � kj � D (3)
for some Do 14. Hinsen [10] analyzes (2) without over-
sampling for bandlimited signals that are in Lp, 1 � po1,when restricted to the real line. He gives a condition on D
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156146
which is sufficient for (2) to be uniformly convergent onall compact subsets of the complex plane.
In this paper we give a stronger result compared to theresult given in [9]. However, we obtain our result for adifferent sampling point sequence and for a smaller signalspace, namely PW1
p. Nevertheless, PW1p is still larger
than all spaces that were considered in [10], andespecially larger than the space of bandlimited signalswith finite energy. Moreover, the space PW1
p is interest-ing because there is a close connection between theconvergence behavior of sampling series for signals inPW1
p and for bandlimited wide-sense stochastic pro-cesses [11]. We prove the local uniform convergence of (2),even when no oversampling is used, and the globaluniform convergence if oversampling is used. Instead ofconsidering sampling point sequences ftkgk2Z that fulfill(3), we consider sampling point sequences that are madeof the zeros of functions of sine type. This class containsmore flexible sampling patterns, but it does not contain allsampling patterns that fulfill (3).
Further results on non-equidistant sampling are pre-sented in [12]. Moreover, sampling series for stochasticprocesses are analyzed in [13]. For an overview on non-equidistant sampling, see [14].
In order to continue, we need some notations anddefinitions. Let f denote the Fourier transform of afunction f, where f is to be understood in the distribu-tional sense. Lp
ðRÞ, 1 � po1, is the space of all pth-powerLebesgue integrable functions on R, with the usual normk � kp, and L1ðRÞ the space of all functions for which theessential supremum norm k � k1 is finite. For s40 and1 � p � 1 we denote by PWp
s the Paley–Wiener space
of signals f with a representation f ðzÞ ¼ 1=ð2pÞR s�s gðoÞeizo do, z 2 C, for some g 2 Lp
½�s;s�. If f 2PWps
then gðoÞ ¼ f ðoÞ. The norm for PWps, 1 � po1, is given
by kfkPWps¼ ð1=ð2pÞ
R s�s jf ðoÞj
p doÞ1=p. As a consequence
of Holder’s inequality we have kfkPWpp� kfkPWs
pand
PWps � PWs
s for 1 � pos � 1. Moreover, it holds
kfk1 � kfkPW1p.
An important class of sampling patterns are completeinterpolating sequences.
Definition 1. We say that a sequence ftkgk2Z is a completeinterpolating sequence for PW2
p if the interpolationproblem f ðtkÞ ¼ ck, k 2 Z, has exactly one solutionf 2 PW2
p for every sequence fckgk2Z satisfyingP1k¼�1jckj
2o1.
Complete interpolating sequences are meaningful sam-pling patterns, because every signal f 2 PW2
p is comple-
tely determined by its sample values ff ðtkÞgk2Z if ftkgk2Z is
a complete interpolating sequence for PW2p. Throughout
the paper we assume that the sequence of samplingpoints ftkgk2Z is real and a complete interpolating
sequence for PW2p. Moreover, we assume, without loss
of generality, that t0 ¼ 0 and that the sequence ofsampling points is ordered strictly increasingly, i.e.,
� � �ot�Not�Nþ1o � � �ot�1ot0ot1o � � �otN�1otNo � � � .(4)
If the sequence of sampling points ftkgk2Z is a completeinterpolating sequence for PW2
p, it follows by definitionthat, for each k 2 Z, there is exactly one function fk 2
PW2p that solves the interpolation problem
fkðtlÞ ¼1; l ¼ k;
0; lak:
((5)
Moreover, the product
fðzÞ ¼ z limN!1
Yjkj�Nka0
1�z
tk
� �(6)
converges uniformly on jzj � R for all Ro1 and f is anentire function of exponential type p [15]. It can bee seenfrom (6) that f, which is often called generating function,has the zeros ftkgk2Z. Since ftkgk2Z is a complete inter-polating sequence, it follows that
fkðtÞ ¼fðtÞ
f0ðtkÞðt � tkÞ(7)
is the unique function in PW2p that solves the interpola-
tion problem (5). For further details we would like to referthe reader to [14, Chapter 3].
For arbitrary complete interpolating sequences, thefunctions f can have a complicated behavior, whichmakes an analysis difficult. Therefore, we restrict ouranalysis to sampling point sequences that are made of thezeros of functions of sine type. In Lemma 1 we will seethat all these sequences are also complete interpolatingsequences, which means that we restrict our analysis to asubclass of complete interpolations sequences. The use ofsine-type functions makes the analysis easier, becausethey have several nice properties. In order to illustratethem, we discuss equivalent definitions and characteriza-tions, and state some of their key properties. For furtherinformation about sine-type functions see [16,15].
Definition 2. An entire function f of exponential type p issaid to be of sine type if
(i)
the zeros of f are separated and simple, and (ii) there exist positive constants A, B, and H such thatAepjyj � jf ðxþ iyÞj � Bepjyj whenever x and y are realand jyj � H.
We use Definition 2 to define sine-type functions in this
paper. Aside from Definition 2 there are other possibledefinitions that have advantages as well. However,Definition 2 explicitly states the structure of sine-typefunctions that we need to obtain our main results.An equivalent definition [17] of sine-type functions isobtained if (i) and (ii) are replaced by the conditions that
(i0)
the zeros of f are separated and lie in fz 2 C : jImðzÞj �hg for some h40, and(ii0)
there is a y0 2 R and A0;B040 such that A0 � jf ðxþiy0Þj � B0 for all x 2 R.
The definitions presented so far are valid for arbitraryfunctions of sine type with complex zeros. However, inthis paper we assume that all zeros are real. With this
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156 147
restriction, it is easier to characterize functions of sinetype, because we can decide the question whether afunction f is a function of sine type solely based on itsbehavior on the real axis. We assume that ftkgk2Z is a realcomplete interpolating sequence. Then the generatingfunction f is not necessarily bounded on the real axis.However, according to part (ii) of Definition 2 and thePhragmen–Lindelof Theorem [16, p. 68], f has to bebounded on the real axis in order to be a sine-typefunction. This is the first condition on f, which refers onlyto its behavior on the real axis. Additionally, we requirethat for every �40 there is a constant C1ð�Þ40 such that
jfðtÞj � C1ð�Þ (8)
for all t 2 RnS
k2Zðtk � �; tk þ �Þ. Of course, every sine-typefunction fulfills (8) [15, p. 163]. However, the converse isnontrivial and contained in the following interestingresult [18].
Let f be the generating function of a real completeinterpolating sequence ftkgk2Z. Then all zeros are simple,and we can assume without loss of generality that fðtÞ40for t 2 ðt0; t1Þ. Next, consider the sequence fckgk2Z definedby
ck ¼
maxt2ðtk ;tkþ1Þ
fðtÞ for k even,
mint2ðtk ;tkþ1Þ
fðtÞ for k odd.
8><>:
We have ckð�1Þk40 for all k 2 Z. f is a function of sinetype if and only if there exist two constants b;B such that
0ob � jckj � Bo1 (9)
for all k 2 Z [18]. Condition (9) implies that f is boundedon the real axis and that the maximum of jfðtÞj on ½tk; tkþ1�
is bounded from below by a positive constant, which isindependent of k 2 Z. Of course, the requirement (9) isweaker than requirement (8) plus boundedness. Never-theless, both conditions are sufficient to characterize sine-type functions with real zeros that are a completeinterpolating sequence.
Example 1. sinðpzÞ is a function of sine type and its zerosare tk ¼ k, k 2 Z.
Although we restrict the sampling patterns to the zeros offunctions of sine type, there are many possible samplingpatterns, because the class of functions of sine type is verylarge. In Section 6 we will present a possibility toconstruct such functions.
There is an important connection between the set ofzeros ftkgk2Z of a function of sine type, the basis propertiesof the system of exponentials feiotk gk2Z, and completeinterpolating sequences [16, pp. 143–144].
Lemma 1. If ftkgk2Z is the set of zeros of a function of sine
type, then the system feiotk gk2Z is a Riesz basis for L2½�p;p�,
and ftkgk2Z is a complete interpolating sequence for PW2p.
Proof. This lemma is a simple consequence of Theorems 9and 10 on pp. 143 and 144, respectively, in [16]. &
Lemma 1 implies that if f is a function of sine type withzeros ftkgk2Z then ffkgk2Z, where fk is given by (7), is aRiesz basis for PW2
p [15, p. 169, Theorem 1].
Definition 3. A system of vectors ffkgk2Z in a separableHilbert space H is called Riesz basis if ff kgk2Z is completein H, and there exist positive constants A and B such thatfor all M;N 2 N and arbitrary scalars ck we have
AXN
k¼�M
jckj2 �
Z 1�1
XN
k¼�M
ck fkðtÞ
����������2
dt � BXN
k¼�M
jckj2. (10)
Eq. (10) is important for the convergence behavior of theseries (2) for signals f 2 PW2
p. If ffkgk2Z is a Riesz basisfor PW2
p then, by virtue of Eq. (10), one has limN!1kf �PNk¼�Nf ðtkÞfkkPW2
p¼ 0 and consequently
limN!1
f �XN
k¼�N
f ðtkÞfk
����������1
¼ 0 (11)
for all signals f 2 PW2p. We will need (11) in the proof of
Theorem 1.In this paper we analyze the convergence behavior of
the sampling series
XN
k¼�N
f ðtkÞfkðtÞ (12)
for f 2PW1bp, 0ob � 1, where ftkgk2Z are the zeros of a
function of sine type and fk is given by (7).A similar problem is studied in [11]. While the goal
here is to reconstruct signals from their samples, the goalin [11] is to approximate some transformation Tf byusing only the samples of the signal f. In particular, theconvergence behavior of the sampling series
XN
k¼�N
f ðtkÞðTfkÞðtÞ (13)
is analyzed for stable linear time-invariant (LTI) systemsT, signals in PW1
p and sampling patterns that arecomplete interpolating sequences. Compared to thispaper, the considered sampling patterns are more generalin [11]. Here, we restrict the class of permissible samplingpatterns to those, which are the zeros of sine-typefunctions. We will resume this discussion in Section 3,where we compare the convergence behavior of (12)and (13).
Example 2. The Shannon sampling series is a special caseof the general sampling series that are considered in thispaper. Let fðtÞ ¼ sinðptÞ with zeros tk ¼ k, k 2 Z. Thenf0ðtkÞ ¼ p cosðptkÞ ¼ pð�1Þk and
fkðtÞ ¼fðtÞ
f0ðtkÞðt � tkÞ¼ð�1Þk sinðptÞ
pðt � tkÞ¼
sinðpðt � kÞÞ
pðt � kÞ
is the well known sinc-kernel of the Shannon samplingseries.
The outline of this paper is as follows. In Section 3 weinvestigate the local convergence behavior of (12) withoutoversampling, i.e., b ¼ 1, and in Section 4 the globalconvergence behavior with oversampling, i.e., 0obo1.The global convergence behavior of (12) without over-sampling is analyzed in Section 5. In Section 6 we give amethod to construct arbitrarily many sampling patterns.
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156148
Section 7 contains some numerical simulations. Finally, inSection 8 we use the results from Sections 3, 4 and 5 toderive results for the convergence behavior of (12) forbandlimited wide-sense stationary stochastic processes.
2. Basic properties of functions of sine type
Functions of sine type have many interesting proper-ties. One concerns their behavior outside circles centeredaround the zeros of the function, another the distributionof their zeros. Since we will need these properties in ourproofs, we state them in Lemmas 2 and 3.
Lemma 2. Let f be a function of sine type, whose zeros
flkgk2Z are ordered increasingly according to their real parts.
Then we have
infk2Zjlkþ1 � lkj � d40 (14)
and
supk2Z
jlkþ1 � lkj � do1 (15)
for some constants d and d.
Proof. Eq. (14) follows directly from Definition 2 and theproof of (15) can be found in [15, p. 164]. &
Lemma 3. Let f be a function of sine type. For each �40there exists a number C240 such that
jf ðxþ iyÞj � C2epjyj
outside the circles of radius � centered at the zeros of f.
Proof. A proof of Lemma 3 can be found in [16,p. 144]. &
3. Local convergence behavior
A well known fact [19–21] about the convergencebehavior of the Shannon sampling series with equidistantsamples (1) is its uniform convergence on compactsubsets of R for all f 2 PW1
p.
Brown’s Theorem. For all f 2 PW1p and T40 fixed we
have
limN!1
maxt2½�T ;T�
f ðtÞ �XN
k¼�N
f ðkÞsinðpðt � kÞÞ
pðt � kÞ
���������� ¼ 0.
This theorem plays a fundamental role in applications,because it establishes the uniform convergence oncompact subsets of R for a large class of signals, namelyPW1
p, which is the largest space within the scale ofPaley–Wiener spaces. It is not possible to extend the localuniform convergence of the Shannon sampling series touniform convergence on whole of R. However, a modifiedseries, which is symmetric around t, converges uniformlyon the whole real axis [22].
In this section we show that the uniform convergenceon compact subsets of R still holds if non-equidistantsampling is used. In this sense Theorem 1 is an extensionof Brown’s Theorem to non-uniform sampling.
Theorem 1. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4). Furthermore,let fk be defined as in (7). Then we have
limN!1
maxt2½�T ;T�
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� ¼ 0
for all T40 and all f 2 PW1p.
Proof. The proof of Theorem 1 is given in Appendix B.
Note that Theorem 1 makes no statement about theglobal convergence behavior of the sampling series.Although T40 can be arbitrary, it has to be fixed. Theglobal convergence behavior will be analyzed in Sections4 and 5.
It is interesting to compare Theorem 1 with an resultthat was given in [11].
Theorem 2. Let ftkgk2Z be a complete interpolating se-
quence for PW2p, and fk as defined in (7). Then, for all t 2 R
there exist a stable LTI system T1 and a signal f 1 2PW1p
such that
lim supN!1
ðT1f 1ÞðtÞ �XN
k¼�N
f 1ðtkÞðT1fkÞðtÞ
���������� ¼ 1.
Theorem 2 shows that, in general, it is not possible toapproximate the output of a stable LTI system Tf by thesampling series (13), because we can find a signal f 2
PW1p and a stable LTI system T such that (13) diverges
for some arbitrary given t 2 R. Thus, there is a significantdifference between the case without a stable LTI system(12), where we have uniform convergence on all compactsubsets of R, and the case with a stable LTI system, wherewe can have divergence for every point t 2 R.
The good local convergence behavior of the samplingseries (12) for sampling points ftkgk2Z that are the zeros ofa sine-type function is a very nice property. For arbitrarycomplete interpolating sequences ftkgk2Z, we conjecturethat the convergence behavior is problematic.
Conjecture 1. For all T40 there exists a complete inter-
polating sequence ftkgk2Z for PW2p and a signal f 2 2 PW1
psuch that
lim supN!1
maxt2½�T ;T�
f 2ðtÞ �XN
k¼�N
f 2ðtkÞfkðtÞ
���������� ¼ 1,
where fk is defined as in Eq. (7).
In the next two sections we will analyze the globalconvergence behavior of the sampling series (12).
4. Global convergence behavior with oversampling
In [23] it has been shown that the Shannon samplingseries with equidistant samples is uniformly convergenton whole of R for all f 2 PW1
p if oversampling is used. InTheorem 3 we will see that this result can be extended tonon-equidistant sampling.
Theorem 3. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4). Furthermore,
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156 149
let fk be defined as in (7). Then, for all 0obo1 and all
f 2 PW1bp, we have
limN!1
maxt2R
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� ¼ 0.
Proof. The proof of Theorem 3 is given in Appendix C. &
A closer look at the proof of Theorem 3 reveals thatoversampling, i.e., bo1, was essential. The same approachcannot be used for b! 1, because certain upper boundsdiverge. The next section will show that this problem isnot due to our proof technique, but due to inherentproperties of the sampling series (12).
5. Global convergence behavior without oversampling
In Sections 3 and 4 we gave two positive results fornon-equidistant sampling, namely the local uniformconvergence of the reconstruction process
XN
k¼�N
f ðtkÞfkðtÞ ¼XN
k¼�N
f ðtkÞfðtÞ
f0ðtkÞðt � tkÞ(16)
for f 2 PW1p, i.e., the case where no oversampling is used,
and the global uniform convergence of (16) for f 2 PW1bp,
0obo1, i.e., the case where oversampling is used.However, so far we made no statement about the globalconvergence behavior of (16) when no oversampling isused. In this section we analyze this remaining question.
Previous investigations [24] have shown for the spacePW1
p and a large class of reconstruction processes thatneither a globally uniformly convergent nor a locallyuniformly convergent and globally bounded signal recon-struction is possible if the samples are taken equidistantlyat Nyquist rate. In contrast, for non-equidistant sampling,the global convergence behavior of sampling basedreconstruction processes is unknown in general. By usingnon-equidistant sampling, an additional degree of free-dom is created, which may help to improve the conver-gence behavior. However, we suspect that non-equidistantsampling is not capable to improve the global conver-gence behavior. This believe is expressed in a conjecture,which was originally posed in [23].
Conjecture 2. For every complete interpolating sequence
ftkgk2Z for PW2p there exists a signal f 3 2 PW1
p such that
lim supN!1
f 3 �XN
k¼�N
f 3ðtkÞfk
����������1
¼ 1,
where fk is defined as in Eq. (7).
In Theorem 4 we show for a significant subclass ofsampling patterns that the global convergence behavior isnot improved by using non-equidistant sampling. Thisnegative result is a partial confirmation of Conjecture 2.
In this section we restrict our analysis to entirefunctions f with separated real zeros ftkgk2Z that have arepresentation as Fourier–Stieltjes integral in the form
fðtÞ ¼1
2p
Z p
�peiot dmðoÞ, (17)
where mðoÞ is a real function of bounded variation on theinterval ½�p;p� and has a jump discontinuity at eachendpoint. It can be shown that all functions f with therepresentation (17) satisfy part (ii) of Definition 2[16, p. 143], and hence the class of functions f that weconsider here is a subclass of the functions of sine type. InSection 6, where we present a simple method to constructsuch functions, we will see that this subclass is still verylarge.
For this subclass the global convergence behavior is notimproved by using non-equidistant sampling.
Theorem 4. Let f be a function of sine type that has the
representation (17), and whose zeros ftkgk2Z are all real and
ordered according to (4). Furthermore, let fk be defined as in
(7). Then there exists a f 4 2 PW1p such that
lim supN!1
maxt2R
f 4ðtÞ �XN
k¼�N
f 4ðtkÞfkðtÞ
���������� ¼ 1.
Remark 1. Theorem 4 shows that Conjecture 2 is correctfor a certain subclass of non-equidistant sampling pat-terns. Nevertheless, we suppose that it is also true forarbitrary complete interpolating sequences for PW2
p.
Proof of Theorem 4. Let M 2 N be arbitrary but fixed andtM ¼ ðtMþ1 þ tMÞ=2. For
AfM;tM
f ¼XM
k¼�M
f ðtkÞfkðtMÞ
we analyze the operator norm kAfM;tMk ¼ supkfk
PW1p�1jA
fM;tM
f j.
There exists a sequence ff ngn2N of functions in L1½�p;p�
with
kf nkL1½�p;p� �
1
2p
Z p
�pdjmjðoÞ; n 2 N,
where jmj denotes the variation of m, and
limn!1
f nðzÞ ¼ limn!1
1
2p
Z p
�peiozf nðoÞdo
¼1
2p
Z p
�peioz dmðoÞ ¼ fðzÞ; z 2 C.
The existence of such a function can be proven as in[25, p. 247]. Furthermore, for R40 and C ¼ fz 2 C : jzj ¼ Rg,we have
f nðtÞ �fðtÞ ¼ 1
2pi
IC
f nðzÞ � fðzÞz� t dz (18)
by Cauchy’s integral formula, and consequently
jf nðtÞ �fðtÞj � R
2pðR� TÞ
Z p
�pjf nðReioÞ � fðReioÞjdo (19)
for all n 2 N and t 2 ½�T; T�, where 0oToR. Since R wasarbitrary and the right-hand side of (19) is independent oft, it follows from Lebesgue’s dominated convergence
theorem that limn!1maxt2½�T ;T�jf nðtÞ � fðtÞj ¼ 0 for all
T40. Moreover, differentiating (18) with respect to t gives
f 0nðtÞ �f0ðtÞ ¼ �1
2pi
IC
f nðzÞ �fðzÞðz� tÞ2
dz
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156150
and, by the same steps as in (19),
jf 0nðtÞ � f0ðtÞj � R
2pðR� TÞ2
Z p
�pjf nðReioÞ � fðReioÞjdo.
As before, it follows from Lebesgue’s dominated conver-gence theorem that limn!1maxt2½�T ;T�jf
0
nðtÞ �f0ðtÞj ¼ 0for all T40. Next, we choose a constant C3 so large thatkf 0nkPW1
p=C3 � 1 for all n 2 N. It follows that
kAfM;tMk �
1
C3
XMk¼�M
f 0nðtkÞfkðtMÞ
����������
for all n 2 N, and consequently
kAfM;tMk � lim
n!1
1
C3
XMk¼�M
f 0nðtkÞfkðtMÞ
����������
¼ limn!1
1
C3
XMk¼�M
f 0nðtkÞfðtMÞ
f0ðtkÞðtM � tkÞ
����������
¼1
C3fðtMÞ
XMk¼�M
1
tM � tk
����������. (20)
Furthermore, since tM ¼ ðtMþ1 þ tMÞ=2 we have, due to(14), tM � tk � ðM þ 1� kÞd for all jkj � M, and it followsthat
XMk¼�M
1
tM � tk�XM
k¼�M
1
ðM þ 1� kÞd
¼1
d
X2Mþ1
k¼1
1
k�
1
dlogð2M þ 2Þ. (21)
Combining (20) and (21), we obtain
kAfM;tMk �jfðtMÞj
C3dlogð2M þ 2Þ.
Because of (14) and Lemma 3, there exists a constantC440, such that jfðtNÞj � C4 for all N 2 N. Consequently,kAf
M;tMk � C5 logð2M þ 2Þ with some constant C540 that is
independent of M. Since M was arbitrary, we obtain
supkfk
PW1p�1
maxt2R
XN
k¼�N
f ðtkÞfkðtÞ
���������� � kAf
N;tNk � C5 logð2N þ 2Þ
for all N 2 N. Thus, the Banach–Steinhaus theorem [25,p. 98] implies that there exists a signal f 4 2 PW1
p suchthat
lim supN!1
maxt2R
XN
k¼�N
f 4ðtkÞfkðtÞ
���������� ¼ 1: &
6. Construction of sine-type functions and possiblesampling patterns
Next, consider for an arbitrary real-valued functiong 2 PW1
p, with kgkPW1po1, the function
fgðtÞ ¼ gðtÞ � cosðptÞ. (22)
Functions of this kind were analyzed for example in[26–28]. It can be shown that the function fg is a functionof sine type. Moreover, the zeros ftkgk2Z of fg are all realand simple, because we assumed that g is real-valued andkgk1 � kgkPW1
po1 [29].
Thus, by Eq. (22) we have a method to constructarbitrarily many functions of sine type fg and hencearbitrarily many sampling patterns ftkgk2Z for which thetheorems in this paper are valid. The sampling pointsftkgk2Z are nothing else than the crossings of somebandlimited function g 2 PW1
p, kgkPW1po1, with the
cosine function.It follows by Theorem 3 that
XN
k¼�N
f ðtkÞfg;kðtÞ, (23)
where
fg;kðtÞ ¼fgðtÞ
f0gðtkÞðt � tkÞ, (24)
is globally uniformly convergent for all f 2 PW1bp,
0obo1. Furthermore, we know by Theorems 1 and 4that, in the case without oversampling, i.e., f 2 PW1
p, (23)is only locally uniformly convergent and not globallyuniformly convergent in general. However, this does notanswer the question whether
XN
k¼�N
gðtkÞfg;kðtÞ,
i.e., the sampling series with matched reconstruction func-tion, is uniformly convergent on whole of R for all g 2 PW1
p.We conjecture that even in this case the series is not
globally uniformly convergent in general.
7. Numerical examples
In this section we illustrate the theorems of thepreceding sections by numerical examples.
To this end, we consider the functions
gðtÞ ¼4
5sin
p2
t� �
and
fgðtÞ ¼ gðtÞ � cosðptÞ.
fg is a sine-type function, and, since kgk1o1, the zerosftkgk2Z of fg are all real [29]. In this example the first fourpositive zeros of fg (rounded to two digits after thedecimal point) are given by t1 ¼ 0:36, t2 ¼ 1:64, t3 ¼ 2:77,and t4 ¼ 3:23. This pattern is recurrent with period 4.
Next, we numerically analyze
ðGNf ÞðtÞ ¼XN
k¼�N
f ðtkÞfg;kðtÞ, (25)
where fg;k is defined as in (24). For og40 and M 2 N,M41=og , consider the signals s
og
M , defined in thefrequency domain by
sog
M ðoÞ ¼Mp; og �
1
M� joj � og ;
0 otherwise;
8<:
all of which have the norm ksog
M kPW1p¼ 1.
To illustrate the convergence behavior of (25) withoversampling, which was discussed in Section 4, we use
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156 151
the signal f 1 ¼ sp=25 , which is bandlimited to p=2. Fig. 1
shows the good convergence behavior of (25) if over-sampling is used.
To illustrate the convergence behavior of (25) withoutoversampling, which was discussed in Section 5, we usethe signal f 2 ¼ sp5 , which is bandlimited to p. Fig. 2 showsthat the convergence speed is slower than in Fig. 1, whereoversampling was used.
Next, we illustrate the global convergence behavior,i.e., the divergence of the peak approximation error, whichwas described in Section 5. Therefore, we analyze
supkfk
PW1p�1
maxt2RjðGNf ÞðtÞj. (26)
Since
jðGNf ÞðtÞj ¼XN
k¼�N
f ðtkÞfg;kðtÞ
���������� � kfkPW1
p
XN
k¼�N
jfg;kðtÞj,
an upper bound for (26) is given by UN ¼
maxt2R
PNk¼�N jfg;kðtÞj: In Table 1 we compare this upper
bound with the lower bound LN ¼ jðGNf 3ÞðtNÞj, where
f 3ðtÞ ¼5
7pf0ðtÞ ¼
2
7cos
p2
t� �
þ5
7sinðptÞ.
10 5 5 10
1.0
0.5
0.5
1.0
Fig. 1. Plot of G2f 1 (dotted line), G5f 1 (dashed-dotted line), G10f 1 (dashed
line), and f 1 (solid line).
0.5
1.0
10 5
1.0
0.5
Fig. 2. Plot of G2f 2 (dotted line), G10f 2 (dashed-dotte
Note that f 3 is not in PW1p. Nevertheless, f 3 can be
approximated arbitrarily close by an signal in PW1p with
norm 1. Thus, for every natural number N we can find a
signal f � 2 PW1p with norm kf �kPW1
p¼ 1 such that
ðGNf �ÞðtNÞ is arbitrarily close to LN . The divergence of UN
and LN is clearly visible in Table 1. In the proof of Theorem4 we have seen that LN increases logarithmically with N.
8. Application: stochastic processes
In this section we analyze the mean square senseconvergence behavior of (12) for bandlimited stochasticprocesses X. We restrict our analysis to wide-sensestationary processes, i.e., the class of continuous-time,complex valued stochastic processes X ¼ XðtÞ with zeromean EðXðtÞÞ ¼ 0 and finite second moment EðjXðtÞj2Þo1for all t 2 R, whose correlation function GXðt; t
0Þ ¼
EðXðtÞXðt0ÞÞ, where z denotes the complex conjugate ofz 2 C, is only a function of the difference t � t0. Thisenables us to define the correlation function as a functionof one variable RXðtÞ ¼ EðXðtÞXð0ÞÞ, and it can be easilyseen that RX is nonnegative definite. Furthermore, weassume that X is mean square continuous, which impliesthat RX is continuous. Under this assumption, the correla-tion function RX has the representation
RXðtÞ ¼1
2p
Z 1�1
eiot dmðoÞ,
with a positive and finite measure m. For details andfurther facts we refer to the standard literature, forexample [30,31]. We additionally assume that the mea-sure m is absolutely continuous with respect to the
5 10
d line), G20f 2 (dashed line), and f 2 (solid line).
Table 1Comparison of the upper bound UN and the lower bound LN for different
values of N.
N 1 13 25 37 49 61 73 85 97
UN 1.83 5.82 6.81 7.41 7.84 8.17 8.45 8.69 8.89
LN 1.05 1.95 2.21 2.37 2.48 2.57 2.65 2.71 2.76
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156152
Lebesgue measure l, which implies that there exists afunction SX 2 L1
ðRÞ with dm ¼ SX dl. Furthermore, since mis positive, it follows that SXðoÞ � 0 almost everywhere(a.e.). SX is called power spectral density. We say the wide-sense stationary process X is bandlimited with bandwidths40, if RX is an entire function and for all �40 there existsa constant Cð�Þ with jRXðtÞj � Cð�Þ expððsþ �ÞjtjÞ for allt 2 R. It follows that almost all sample functions areentire functions of exponential type at most s [32].
Definition 4. We call a bandlimited wide-sense station-ary process X I-process if its correlation function RX has arepresentation
RXðtÞ ¼1
2p
Z p
�pSXðoÞeiot do, (27)
for some non-negative SX 2 L1½�p;p�.
It is well known [33] that for all I-processes X and T40we have
limN!1
maxt2½�T ;T�
E XðtÞ �XN
k¼�N
XðkÞsinðpðt � kÞÞ
pðt � kÞ
����������2
¼ 0, (28)
i.e., the variance of the reconstruction error is bounded onall compact subsets of R and converges to zero for N!1.
Using the findings from Section 3 we can extend thisresult to non-equidistant sampling.
Corollary 1. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4). Furthermore,let fk be defined as in (7). Then, for all T40, there exists a
constant C6 ¼ C6ðTÞo1 such that
maxt2½�T ;T�
maxo2½�p;p�
XN
k¼�N
eiotkfkðtÞ
���������� � C6 (29)
for all N 2 N.
Proof. Since
XN
k¼�N
f ðtkÞfkðtÞ ¼1
2p
Z p
�pf ðoÞ
XN
k¼�N
eiotkfkðtÞdo
we have, according to Lemma 17 in [34],
supkfk
PW1p�1
XN
k¼�N
f ðtkÞfkðtÞ
���������� ¼ max
o2½�p;p�
XN
k¼�N
eiotkfkðtÞ
����������. (30)
Let T40 be arbitrary but fixed. Due to Theorem 1 wehave
limN!1
maxt2½�T ;T�
XN
k¼�N
f ðtkÞfkðtÞ
����������o1
for all f 2 PW1p. It follows by the Banach–Steinhaus
theorem [25, p. 98] that
supN2N
supkfk
PW1p�1
maxt2½�T ;T�
XN
k¼�N
f ðtkÞfkðtÞ
����������o1,
and, by using (30), that
supN2N
maxt2½�T;T�
maxo2½�p;p�
XN
k¼�N
eiotkfkðtÞ
����������o1.
Thus, there exists a constant C6, which does not dependon N such that (29) is true. &
Theorem 5. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4). Furthermore,let fk be defined as in (7). Then, for all I-processes X and all
T40, we have
limN!1
maxt2½�T ;T�
E XðtÞ �XN
k¼�N
XðtkÞfkðtÞ
����������2
¼ 0. (31)
Proof. According to [11] we have (31) if and only if thereexists a constant C7 such that for all N 2 N
maxt2½�T ;T�
maxo2½�p;p�
XN
k¼�N
eiotkfkðtÞ
���������� � C7.
The statement of Theorem 5 follows by using Corollary1. &
Theorem 5 shows that we have a good local conver-gence behavior. However, for practical applications it isimportant to upper bound the mean-square approxima-tion error over whole of R. This is not possible in general,because there exist I-processes such that the global mean-square approximation error increases unboundedly.
Theorem 6. Let f be a function of sine type that has the
representation (17), and whose zeros ftkgk2Z are all real and
ordered according to (4). Furthermore, let fk be defined as in
(7). Then there exists an I-process X1 such that
lim supN!1
supt2R
EXN
k¼�N
X1ðtkÞfkðtÞ
����������2
¼ 1,
and consequently
lim supN!1
supt2R
E X1ðtÞ �XN
k¼�N
X1ðtkÞfkðtÞ
����������2
¼ 1.
Proof. Theorem 6, is a direct consequence of Theorem 5in [11] and Theorem 4. &
The next theorem shows, similar to the deterministiccase in Section 4, that oversampling improves theconvergence behavior of (12) for I-processes. For the proofwe need Corollary 2.
Corollary 2. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4). Furthermore,let fk be defined as in (7). Then, for all 0obo1, there exists a
constant C8o1 such that
supt2R
maxjoj�bp
XN
k¼�N
eiotkfkðtÞ
���������� � C8 (32)
for all N 2 N.
Proof. Since
XN
k¼�N
f ðtkÞfkðtÞ ¼1
2p
Z bp
�bpf ðoÞ
XN
k¼�N
eiotkfkðtÞdo
ARTICLE IN PRESS
Fig. B.1. Path PNðYÞ in the complex plane.
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156 153
we have, according to Lemma 17 in [34],
supkfk
PW1p�1
XN
k¼�N
f ðtkÞfkðtÞ
���������� ¼ max
joj�bp
XN
k¼�N
eiotkfkðtÞ
����������.
Using the Banach–Steinhaus theorem [25, p. 98] andTheorem 3, we obtain (32). &
Theorem 7. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4). Furthermore,let fk be defined as in (7). Then, for all 0obo1 and all
I-processes X, whose power spectral density SXðoÞ is
supported in ½�bp;bp�, we have
supN2N
supt2R
EXN
k¼�N
XðtkÞfkðtÞ
����������2
o1 (33)
and consequently
supN2N
supt2R
E XðtÞ �XN
k¼�N
XðtkÞfkðtÞ
����������2
o1.
Proof. According to [11] we have (33) if and only if thereexists a constant C9 such that for all N 2 N
supt2R
maxjoj�bp
XN
k¼�N
eiotkfkðtÞ
���������� � C9. (34)
The proof is complete because (34) is true byCorollary 2. &
Appendix A. Two lemmas
In this appendix we prove two lemmas, which areneeded for the proof of Theorems 1 and 3, given inAppendices B and C, respectively.
Lemma 4. Let f be a function of sine type, whose zeros are
all real, and Y040. Then there exists a constant C10 such
that, for all 0ob � 1, jYj � Y0, A;B 2 R, A � B, t 2 R, and
f 2 PW1bp, we have
Z B
A
f ðxþ iYÞ
fðxþ iYÞ
fðtÞxþ iY � t
��������dx �
ðB� AÞkfkPW1bpkfk1
C10jY j.
Proof of Lemma 4. For all x;Y 2 R, we havejf ðxþ iYÞj � ebpjYjkfkPW1
bp. Furthermore, since f is a func-
tion of sine type with all zeros being real, it follows fromLemma 3 that there exists a constant C10 such that jfðxþiYÞj � C10epjYj for all x 2 R and all jYj � Y0. Therefore, weobtain
Z B
A
f ðxþ iYÞ
fðxþ iYÞ
fðtÞxþ iY � t
��������dx �
kfkPW1bpkfk1
C10
Z B
A
e�pð1�bÞjYj
jxþ iY � tjdx
�
ðB� AÞkfkPW1bpkfk1
C10jYj,
which completes the proof. &
For the proof of Theorem 3 we need the followinglemma.
Lemma 5. Let f be a function of sine type, whose zeros
ftkgk2Z are all real and ordered according to (4), 0obo1,
and tn as defined in (B.1). Then there exist a constant C10
such that for all jKj 2N, Y40, t 2 R with jt � tK j � d, and
f 2 PW1bp we have
Z Y
�Y
f ðtK þ iyÞ
fðtK þ iyÞ
fðtÞtK þ iy� t
��������dy �
2kfkPW1bpkfk1
pC10 dð1� bÞ.
Proof of Lemma 5. For all x; y 2 R, we havejf ðxþ iyÞj � ebpjyjkfkPW1
bp. Furthermore, since f is a func-
tion of sine type it follows from Lemma 3 and (14) thatthere exists a constant C10 such that jfðtK þ iyÞj � C10epjyj
for all jKj 2 N and y 2 R. Therefore, we obtainZ Y
�Y
f ðtN þ iyÞ
fðtN þ iyÞ
fðtÞtN þ iy� t
��������dy
�
kfkPW1bpkfk1
C10
Z Y
�Y
e�pð1�bÞjyj
tN þ iy� t
��������dy
�
2kfkPW1bpkfk1
C10 d
Z Y
0e�pð1�bÞy dy �
2kfkPW1bpkfk1
pC10 dð1� bÞ,
which completes the proof. &
Appendix B. Proof of Theorem 1
Let T40 and f 2 PW1p be arbitrary but fixed and
tn ¼ðtnþ1 þ tnÞ=2 for n � 1;
ðtn�1 þ tnÞ=2 for n � �1:
((B.1)
Furthermore, consider, for N 2N and Y40, the path PNðYÞ
in the complex plane that is depicted in Fig. B.1. For allN 2 N and t 2 R we have the equality
XN
k¼�N
f ðtkÞfkðtÞ ¼1
2pi
IPN ðYÞ
fðzÞ �fðtÞz� t
f ðzÞfðzÞ
dz. (B.2)
Eq. (B.2) can be easily seen by using the method ofresidues. Note that by the choice of PNðYÞ we have fðzÞa0for all z 2 PNðYÞ. Furthermore, for all N 2 N and t 2 R witht�NototN , we have
1
2pi
IPN ðYÞ
fðzÞ � fðtÞz� t
f ðzÞfðzÞ
dz
¼1
2pi
IPN ðYÞ
f ðzÞz� t
dz�1
2pi
IPN ðYÞ
fðtÞz� t
f ðzÞfðzÞ
dz
¼ f ðtÞ �1
2pi
IPN ðYÞ
fðtÞz� t
f ðzÞfðzÞ
dz. (B.3)
Combining (B.2) and (B.3), it follows that
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ ¼1
2pi
IPN ðYÞ
fðtÞz� t
f ðzÞfðzÞ
dz (B.4)
for all N 2 N and t 2 R with t�NototN .
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156154
According to Corollary 2 there exist two positiveconstants d and d such that (14) and (15) are fulfilled.Next choose YN ¼ Nd, N 2 N. Since jKjdojtK joðjKj þ 1Þd,jKj 2 N, it follows that there are two positive constantsC11 and C12 such that
Y jKj=C11ojtK joY jKj=C12 (B.5)
for all jKj 2 N.Let N0 be the smallest natural number for which
minðtN0; jt�N0
jÞ4T . By using the identity (B.4), we obtain
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
����������
�1
2p
Z YN
�YN
f ðtN þ iyÞ
fðtN þ iyÞ
�������� jfðtÞjjtN þ iy� tj
dy
þ1
2p
Z YN
�YN
f ðt�N þ iyÞ
fðt�N þ iyÞ
�������� jfðtÞjjt�N þ iy� tj
dy
þ1
2p
Z tN
t�N
f ðxþ iYNÞ
fðxþ iYNÞ
�������� jfðtÞjjxþ iYN � tj
dx
þ1
2p
Z tN
t�N
f ðx� iYNÞ
fðx� iYNÞ
�������� jfðtÞjjx� iYN � tj
dx (B.6)
for all N � N0 and t 2 ½�T; T�. Next, we will upper boundthe right-hand side of (B.6) by analyzing each integralseparately. It is important that this bound is independentof N.
For all x; y 2 R, we have jf ðxþ iyÞj � epjyjkfkPW1p.
Furthermore, since f is a function of sine type it followsfrom (14) and Lemma 3 that there exists a constant C13
such that jfðtK þ iyÞj � C13epjyj for all jKj 2 N and all y 2 R.Consequently, for the first term on the right-hand side of(B.6) we have
1
2p
Z YN
�YN
f ðtN þ iyÞ
fðtN þ iyÞ
�������� jfðtÞjjtN þ iy� tj
dy
�kfkPW1
pkfk1
C13
1
2p
Z YN
�YN
1
jtN þ iy� tjdy
�kfkPW1
pkfk1
C13
YN
pðtN � TÞ�kfkPW1
pkfk1
C13
tNC11
ðtN � TÞ
for all N � N0, where we used (B.5) in the last inequality.Similarly, for the second term we obtain
1
2p
Z YN
�YN
f ðt�N þ iyÞ
fðt�N þ iyÞ
�������� jfðtÞjjt�N þ iy� tj
dy
�kfkPW1
pkfk1
C13
jt�NjC11
ðjt�Nj � TÞ
for all N � N0. The third term on the right-hand side of(B.6) can be upper bounded by using Lemma 4. For allN � N0 we have
1
2p
Z tN
t�N
f ðxþ iYNÞ
fðxþ iYNÞ
�������� jfðtÞjjxþ iYN � tj
dx
�kfkPW1
pkfk1
pC13
ðtN � t�NÞ
2YN
�kfkPW1
pkfk1
pC13C12,
where we used (B.5) again in the last inequality. Similarly,we obtain for the fourth term that
1
2p
Z tN
t�N
f ðx� iYNÞ
fðx� iYNÞ
�������� jfðtÞjjx� iYN � tj
dx �kfkPW1
pkfk1
pC13C12
for all N � N0. Next, we choose N1 � N0 such that
maxtN1
tN1� T
;jt�N1j
jt�N1j � T
� �� 2.
Note that N1 depends only on T and not on f. Consequently,for all N � N1 and t 2 ½�T; T�, we have
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� � C14kfkPW1
p, (B.7)
where C14 is some constant.Let �40 be arbitrary but fixed. There exists a function
f � 2 PW2p such that kf � f �kPW1
po�. As a consequence, we
have
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
����������
� f ðtÞ � f �ðtÞ �XN
k¼�N
ðf ðtkÞ � f �ðtkÞÞfkðtÞ
����������
þ f �ðtÞ �XN
k¼�N
f �ðtkÞfkðtÞ
����������
� C14kf � f �kPW1pþ f �ðtÞ �
XN
k¼�N
f �ðtkÞfkðtÞ
����������
� C14�þ f �ðtÞ �XN
k¼�N
f �ðtkÞfkðtÞ
���������� (B.8)
for all N � N1 and t 2 ½�T; T�. Furthermore, sincef � 2 PW2
p, we can use (11), i.e., the uniform convergenceof the series. It follows that there exists a N2ð�Þ � N1 suchthat
f �ðtÞ �XN
k¼�N
f �ðtkÞfkðtÞ
����������o� (B.9)
for all N � N2ð�Þ and t 2 R. Combining (B.8) and (B.9) gives
maxt2½�T ;T�
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� � C14 þ 1ð Þ�
for all N � N2ð�Þ, which completes the proof, because �was arbitrary.
Appendix C. Proof of Theorem 3
Let 0obo1 and f 2 PW1bp be arbitrary but fixed.
Furthermore, let tn be defined as in (B.1) and consider, forM;N 2 N and Y40, the path PM;NðYÞ in the complex planethat is depicted in Fig. C.1. For all M;N 2 N and t 2 R wehave
XN
k¼�M
f ðtkÞfkðtÞ ¼1
2pi
IPM;N ðYÞ
fðzÞ � fðtÞz� t
f ðzÞfðzÞ
dz. (C.1)
ARTICLE IN PRESS
Fig. C.1. Path PM;NðYÞ in the complex plane.
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156 155
Note that by the choice of PM;NðYÞ we have fðzÞa0 for allz 2 PM;NðYÞ. If t 2 ðt�M ; tNÞ, we have
1
2pi
IPM;N ðYÞ
fðzÞ �fðtÞz� t
f ðzÞfðzÞ
dz
¼ f ðtÞ �1
2pi
IPM;N ðYÞ
fðtÞz� t
f ðzÞfðzÞ
dz, (C.2)
and, by combining (C.1) and (C.2), it follows that
f ðtÞ �XN
k¼�M
f ðtkÞfkðtÞ ¼1
2pi
IPM;N ðYÞ
fðtÞz� t
f ðzÞfðzÞ
dz (C.3)
for all M;N 2 N and t 2 ðt�M ; tNÞ. If t 2 Rn½t�M ; tN� we haveIPM;N ðYÞ
f ðzÞz� t
dz ¼ 0,
and it follows that
XN
k¼�M
f ðtkÞfkðtÞ ¼�1
2pi
IPM;N ðYÞ
fðtÞz� t
f ðzÞfðzÞ
dz (C.4)
for all M;N 2 N and t 2 Rn½t�M ; tN�.Let N 2 N, N42 be arbitrary, but fixed. According to
Corollary 2 there exist two positive constants d and d suchthat (14) and (15) are fulfilled. Next choose YN ¼ Nd,N 2 N. Without loss of generality, we can assume thatt40, because negative t are treated analogously topositive. We have to distinguish two cases, first0 � t � tN , and second t4tN .
For 0 � t � tN we have
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
����������
� kfkPW1bpkfNþ1k1 þ f ðtÞ �
XNþ1
k¼�N
f ðtkÞfkðtÞ
����������. (C.5)
Furthermore, by using the identity (C.3), we obtain
f ðtÞ �XNþ1
k¼�N
f ðtkÞfkðtÞ
����������
�1
2p
Z YN
�YN
f ðtNþ1 þ iyÞ
fðtNþ1 þ iyÞ
�������� jfðtÞjjtNþ1 þ iy� tj
dy
þ1
2p
Z YN
�YN
f ðt�N þ iyÞ
fðt�N þ iyÞ
�������� jfðtÞjjt�N þ iy� tj
dy
þ1
2p
Z tNþ1
t�N
f ðxþ iYNÞ
fðxþ iYNÞ
�������� jfðtÞjjxþ iYN � tj
dx
þ1
2p
Z tNþ1
t�N
f ðx� iYNÞ
fðx� iYNÞ
�������� jfðtÞjjx� iYN � tj
dx. (C.6)
Since tNþ1 � t4d and t � t�N4 d, we can use Lemma 5 toupper bound the first and second term on the right-hand
side of (C.6). In particular we get
1
2p
Z YN
�YN
f ðtNþ1 þ iyÞ
fðtNþ1 þ iyÞ
�������� jfðtÞjjtNþ1 þ iy� tj
dy
�
kfkPW1bpkfk1
p2C10 dð1� bÞ(C.7)
and
1
2p
Z YN
�YN
f ðt�N þ iyÞ
fðt�N þ iyÞ
�������� jfðtÞjjt�N þ iy� tj
dy �kfkPW1
bpkfk1
p2C10 dð1� bÞ,
(C.8)
where C10 is some constant, which is independent of N.The third term on the right-hand side of (C.6) is upperbounded by
1
2p
Z tNþ1
t�N
f ðxþ iYNÞ
fðxþ iYNÞ
�������� jfðtÞjjxþ iYN � tj
dx
�
ðtNþ1 � t�NÞkfkPW1bpkfk1
2pC10YN
� C15kfkPW1bpkfk1, (C.9)
because of Lemma 4 and (15). C15 is a constant, which isindependent of N. Analogously, the fourth term on theright-hand side of (C.6) is upper bounded by
1
2p
Z tNþ1
t�N
f ðx� iYNÞ
fðx� iYNÞ
�������� jfðtÞjxþ iYN � tj
dx � C15kfkPW1bpkfk1.
(C.10)
Combining (C.7)–(C.10) gives
f ðtÞ �XNþ1
k¼�N
f ðtkÞfkðtÞ
���������� � C16kfkPW1
bpkfk1,
which inserted in (C.5) leads to
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� � kfkPW1
bpðkfNþ1k1 þ C16kfk1Þ
(C.11)
if 0ot � tN . The constant C16 depends on b. Since weassumed b to be fixed throughout the proof we suppressthis dependence.
Next, we treat the case t4tN . We have
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� � kf kPW1
bpð1þ kfNk1Þ þ
XN�1
k¼�N
f ðtkÞfkðtÞ
����������.
(C.12)
Using (C.4) and Lemma 5, the sum on the right-hand sideof (C.12) can be upper bounded, similarly to the previouscase, by
XN�1
k¼�N
f ðtkÞfkðtÞ
���������� � C17kfkPW1
bpkfk1,
where the constant C17 is independent of N. Thedependence of C17 on b is suppressed, because weassumed b to be fixed throughout the proof. Moreover,
ARTICLE IN PRESS
H. Boche, U.J. Monich / Signal Processing 90 (2010) 145–156156
by (C.12), we finally obtain
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� � kfkPW1
bpð1þ kfNk1 þ C17kfk1Þ.
(C.13)
if t4tN .
From (C.11) and (C.13) we see that jf ðtÞ �PNk¼�Nf ðtkÞfkðtÞj � kfkPW1
bpC18 for all t40, with some
constant C18 that is independent of t and N. Since N42was arbitrary and the same upper bound holds fornegative t, it follows that
maxt2R
f ðtÞ �XN
k¼�N
f ðtkÞfkðtÞ
���������� � kfkPW1
bpC18 (C.14)
for all N42.Finally, using the same arguments as in the proof of
Theorem 1 we obtain limN!1maxt2Rjf ðtÞ�PN
k¼�Nf ðtkÞ
fkðtÞj ¼ 0, which completes the proof.
For the proof it was important to find (C.14), i.e., anupper bound that is independent of t. A closer look at theproof of Theorem 3 reveals that oversampling, i.e., bo1, isessential for the approach that was used to obtain (C.14).The same approach cannot be used for b! 1, because theright-hand sides of (C.7) and (C.8) diverge.
References
[1] H. Boche, U.J. Monich, Local and global convergence behavior ofnon-equidistant sampling series, in: IEEE International Conferenceon Acoustics, Speech, and Signal Processing (ICASSP ’09), 2009,pp. 2945–2948.
[2] K. Yao, J.B. Thomas, On some stability and interpolatory propertiesof nonuniform sampling expansions, IEEE Transactions on Circuitsand Systems 14 (4) (1967) 404–408.
[3] H.J. Landau, Sampling, data transmission, and the Nyquist rate,Proceedings of the IEEE 55 (10) (1967) 1701–1706.
[4] J.R. Higgins, A sampling theorem for irregularly spaced samplepoints, IEEE Transactions on Information Theory 22 (5) (1976)621–622.
[5] M.M. Milosavljevic, M.R. Dostanic, On generalized stable nonuni-form sampling expansions involving derivatives, IEEE Transactionson Information Theory 43 (5) (1997) 1714–1716.
[6] K.M. Flornes, Y. Lyubarskii, K. Seip, A direct interpolation method forirregular sampling, Applied and Computational Harmonic Analysis7 (3) (1999) 305–314.
[7] H.G. Feichtinger, K. Grochenig, Theory and practice of irregularsampling, in: Wavelets: Mathematics and Applications, Studies inAdvanced Mathematics, CRC, Boca Raton, FL, 1994, pp. 305–363.
[8] K. Grochenig, Irregular sampling, Toeplitz matrices, and theapproximation of entire functions of exponential type, Mathematicsof Computation 68 (226) (1999) 749–765.
[9] K. Seip, An irregular sampling theorem for functions bandlimited ina generalized sense, SIAM Journal on Applied Mathematics 47 (5)(1987) 1112–1116.
[10] G. Hinsen, Irregular sampling of bandlimited Lp-functions, Journalof Approximation Theory 72 (3) (1993) 346–364.
[11] H. Boche, U.J. Monich, Non-uniform sampling—signal and systemrepresentation, in: Proceedings of the 2008 International Sympo-sium on Information Theory and its Applications (ISITA2008), 2008,pp. 1576–1581.
[12] B. Lacaze, Reconstruction formula for irregular sampling, SamplingTheory in Signal and Image Processing 4 (1) (2005) 33–43.
[13] C. Houdre, Reconstruction of band limited processes from irregularsamples, The Annals of Probability 23 (2) (1995) 674–696.
[14] F. Marvasti (Ed.), Nonuniform Sampling: Theory and Practice,Kluwer Academic/Plenum Publishers, 2001.
[15] B.Y. Levin, Lectures on Entire Functions, AMS, 1996.[16] R.M. Young, An Introduction to Nonharmonic Fourier Series,
Academic Press, New York, 1980.[17] B.J. Levin, I.V. Ostrovskii, On small perturbations of the set of zeros
of functions of sine type, Mathematics of the USSR Izvestiya 14 (43)(1980) 79–101.
[18] A.E. Eremenko, M.L. Sodin, Parametrization of entire functions ofsine-type by their critical values, Advances in Soviet Mathematics11 (1992) 238–242.
[19] J.L. Brown Jr., On the error in reconstructing a non-band-limited function by means of the bandpass sampling theorem,Journal of Mathematical Analysis and Applications 18 (1967) 75–84Erratum, Journal of Mathematical Analysis and Applications 21(1968) 699.
[20] P.L. Butzer, W. Splettstoßer, R.L. Stens, The sampling theorem andlinear prediction in signal analysis, Jahresbericht der DeutschenMathematiker-Vereiningung 90 (1) (1988) 1–70.
[21] P.L. Butzer, R.L. Stens, Sampling theory for not necessarily band-limited functions: a historical overview, SIAM Review 34 (1) (1992)40–53.
[22] H. Boche, U.J. Monich, On stable Shannon type reconstructionprocesses, Signal Processing 88 (6) (2008) 1477–1484.
[23] H. Boche, U.J. Monich, Global and local approximation behavior ofreconstruction processes for Paley–Wiener functions, SamplingTheory in Signal and Image Processing 8 (1) (2009) 23–51.
[24] H. Boche, U.J. Monich, There exists no globally uniformly con-vergent reconstruction for the Paley–Wiener space PW1
p ofbandlimited functions sampled at Nyquist rate, IEEE Transactionson Signal Processing 56 (7) (2008) 3170–3179.
[25] W. Rudin, Real and Complex Analysis, third ed., McGraw-Hill, NewYork, 1987.
[26] I. Bar-David, An implicit sampling theorem for boundedbandlimited functions, Information and Control 24 (1) (1974)36–44.
[27] E. Masry, S. Cambanis, Consistent estimation of continuous-timesignals from nonlinear transformations of noisy samples, IEEETransactions on Information Theory 27 (1) (1981) 84–96.
[28] B.F. Logan Jr., Signals designed for recovery after clipping—I.Localization of infinite products, AT&T Bell Laboratories TechnicalJournal 63 (2) (1984) 261–285.
[29] R. Duffin, A.C. Schaeffer, Some properties of functions of exponen-tial type, Bulletin of the American Mathematical Society 44 (4)(1938) 236–240.
[30] M. Loeve, Probability Theory, third ed., Van Nostrand, Princeton, NJ,1963.
[31] H. Cramer, M.R. Leadbetter, Stationary and Related StochasticProcesses, Dover Publications, New York, 2004.
[32] Y.K. Belyaev, Analytic random processes, Theory of Probability andits Applications 4 (4) (1959) 402–409.
[33] J.L. Brown Jr., Truncation error for band-limited random processes,Information Sciences 1 (1969) 171–261.
[34] H. Boche, U.J. Monich, Limits of signal processing performanceunder thresholding, Signal Processing 89 (8) (2009) 1634–1646.