An Explicit Bound for Stability of Sinc Bases
Antonio Avantaggiati
1
, Paola Loreti
2
and Pierluigi Vellucci
2
1
Via Bartolomeo Maranta, 73, 00156, Roma, Italy
2
Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Via Antonio Scarpa n. 16, 00161, Roma, Italy
Keywords:
Kadec’s 1/4-theorem, Riesz Basis, Exponential Bases, Sinc Bases, Sampling Theorem.
Abstract:
It is well known that exponential Riesz bases are stable. The celebrated theorem by Kadec shows that 1/4 is a
stability bound for the exponential basis on L
2
(−π,π). In this paper we prove that α/π (where α is the Lamb-
Oseen constant) is a stability bound for the sinc basis on L
2
(−π,π). The difference between the two values
α/π − 1/4, is ≈ 0.15, therefore the stability bound for the sinc basis on L
2
(−π,π) is greater than Kadec’s
stability bound (i.e. 1/4).
1 INTRODUCTION
It is well known that exponential Riesz bases
{e
inx
}
n∈Z
are stable, this means that a small pertur-
bation of a Riesz basis produces a Riesz basis (Paley
and Wiener, 1934). See also the Young’s textbook
(Young, 2001). The proof of the Paley-Wiener theo-
rem does not provide an explicit stability bound. The
celebrated theorem by Kadec shows that 1/4 is a sta-
bility bound for the exponential basis on the space
L
2
(−π,π) of square integrable functions.
In this paper we prove, in the spirit of the Kadec’s
result, but with a different mathematical approach, a
stability result for cardinal sine sequence {sinc(x −
n)}
n∈Z
showing that α/π, where α is the Lamb-Oseen
constant (Oseen, 1912), is a stability bound for the
sinc basis on L
2
(−π,π). The difference between the
two values α/π − 1/4, is ≈ 0.15. By L
2
(−∞,+∞)
we denote the Hilbert space of real functions that are
square integrable in Lebesgue’s sense:
L
2
(R) =

f :
Z
+∞
−∞
| f (x)|
2
dx < +∞

with respect to the inner product and L
2
-norm that, on
[−π,π], are
h f ,gi =
1
2π
Z
π
−π
f (x)g(x)dx || f || =
p
h f , f i
Given f ∈ L
2
(R) we denote by
ˆ
f the Fourier trans-
form of f ,
ˆ
f (ω) = F ( f )(ω) =
1
2π
Z
+∞
−∞
f (x)e
−iωx
dx.
Parseval’s equality states
k f k
2
= k
ˆ
f k
2
(1)
If f represents the signal, assuming that f ∈ L
2
(R)
(the energy of the signal is finite), then f is said band-
limited to [−π,π] if
ˆ
f vanishes outside the set [−π,π].
The space of band-limited to [−π,π] functions is the
Paley-Wiener space, usually denoted by PW
π
. It is
defined by { f ∈ L
2
(R) ∩ C(R), supp
ˆ
f ⊂ [−π,π]}
and it follows, for instance, from its characterization
by using the classical Paley-Wiener theorem (Young,
2001), p. 85, i.e.:
{f entire function : | f (z)| 6 Ae
π|z|
,
z ∈ C, f |
R
∈ L
2
(R)} (2)
The space PW
π
play a significant role in signal
processing applications (Higgins, 1985). As well
known, any function f ∈ PW
π
, can be expanded in
terms of the orthonormal basis {e
inx
}
n∈Z
as
ˆ
f (x) =
∑
n∈Z

ˆ
f , e
inx

L
2
(−π,π)
e
inx
(3)
where hg,hi
L
2
(−π,π)
=
1
2π
R
π
−π
g(x)h(x)dx. Taking
the inverse Fourier transform in (3), we obtain the
Whittaker-Kotelnikov-Shannon (WKS) sampling the-
orem,
f (x) =
∑
n∈Z
f (n) sinc(x − n), x ∈ R (4)
with sinc(x) the normalized sinc function commonly
defined as
sinc(x) =
(
sin(πx)
πx
x 6= 0,
1 x = 0,
(5)
473
Avantaggiati A., Loreti P. and Vellucci P..
An Explicit Bound for Stability of Sinc Bases.
DOI: 10.5220/0005512704730480
In Proceedings of the 12th International Conference on Informatics in Control, Automation and Robotics (ICINCO-2015), pages 473-480
ISBN: 978-989-758-122-9
Copyright
c
 2015 SCITEPRESS (Science and Technology Publications, Lda.)

Figure 1: Graph of sinc ξ.
whose graph is shown in figure (1).
The subject of recovery of band-limited signals
from discrete data has its origins in the WKS sam-
pling theorem, historically the first and simplest such
recovery formula. It expresses the possibility of re-
covering a certain kind of signals from a sequence
of regularly spaced samples. Without loss of gener-
ality, the formula (4) recovers a function with a fre-
quency band of [−π,π] given the functions values at
the integers. But the WKS theorem has drawbacks.
Foremost, the recovery formula does not converge
given certain types of error in the sampled data, as
Daubechies and De Vore mention in (Daubechies and
DeVore, 2003). They use oversampling to derive an
alternative recovery formula which does not have this
defect. Furthermore, as already said, for the WKS
theorem, the data nodes have to be equally spaced,
and nonuniform sampling nodes are not allowed but,
from many practical points of view it is necessary to
develop sampling theorems for a sequence of samples
taken with a nonuniform distribution along the real
line.
As discussed in (Zayed, 2000), nonuniform sam-
pling of band-limited functions has its roots in the
work of Paley, Wiener, and Levinson. In fact, the
first answer for this direction was given by Paley and
Wiener (Paley and Wiener, 1934), and later an ad-
vanced result was presented by Levinson (Levinson,
1940). Their sampling formulae recover a function
from nodes {λ
n
}
n
, where {e
iλ
n
x
}
n
forms a Riesz basis
for L
2
[−π,π]. The result is related with the perturba-
tion of a Hilbert basis {e
inx
}
n∈Z
for the function space
L
2
[−π,π] in such a way that the perturbed sequence
{e
iλ
n
x
}
n
is also a Riesz basis for the same space. The
maximum perturbation of the system {e
inx
}
n
is found
by Kadec, whose result is the celebrated Kadec-1/4
theorem (Kadec, 1964). This result plays a very im-
portant role in signal theory; it suffices to think for ex-
ample, that the formula (3) expresses the fact that the
ˆ
f (x) can be seen as infinite sum of elementary con-
tributions of exponential type complex. Modern digi-
tal data processing of functions (or signals or images)
always uses a discretized version of the original sig-
nal f that is obtained by sampling f on a discrete set.
The question then arises whether and how f can be
recovered from its samples. Therefore, the objective
of research on the sampling problem is twofold. The
first goal is to quantify the conditions under which it
is possible to recover particular classes of functions
from different sets of discrete samples. The second
goal is to use these analytical results to develop ex-
plicit reconstruction schemes for the analysis and pro-
cessing of digital data. In particular, the results by
Paley and Wiener, Kadec and others on the nonhar-
monic Fourier bases {e
iλ
n
x
}
n∈Z
can be translated into
results about nonuniform sampling and reconstruction
of band-limited functions: (Benedetto, 1991), (Hig-
gins, 1994), (Pavlov, 1979b), (Seip, 1995), (Zayed,
2000).
Our article concentrates on perturbation of reg-
ular sampling and are therefore similar in spirit to
Kadec’s result for band-limited functions, though it is
based on a different point of view. Is the Riesz bases
{sinc(x − n)}
n∈Z
for the function space L
2
[−π,π] to
being perturbed, in {sinc(x − λ
n
)}
n∈Z
, not the com-
plex exponentials. The result is also extended to
{sinc(z − n)}
n∈Z
for PW
[−π,π]
, with z ∈ C. The stabil-
ity bound for the sinc basis on L
2
(−π,π) (or PW
[−π,π]
)
is greater than Kadec’s stability bound (i.e. 1/4); in
some sense, the result obtained here for sinc basis can
be seen as an improvement of Kadec’s estimate.
Kadec theorem has been extensively generalized
(see, for example (Avdonin, 1974), (Bailey, 2010),
(Khrushchev, 1979), (Pavlov, 1979a), (Savchuk and
Shkalikov, 2006), (Sun and Zhou, 1999), (Vellucci,
2014)) but to the best of our knowledge, there are no
versions of this theorem for sinc bases. The paper is
divided in two sections. For other contributions to ex-
ponential Riesz basis problem and Kadec’s theorem
see survey papers, as: (Ullrich, 1980), (Sedletskii,
2009). The first section contains small overview on
the Lamb-Oseen constant and thereafter we revised
know properties for sinc functions. The second sec-
tion is devoted to {sinc(x − n)}
n∈Z
.
1.1 Lambert Function W, Lamb-Oseen
Constant
The Lambert function W (R. M. Corless, 1996),
(Stewart, 2005), (Hayes, 2005), is defined by the
equation
W (x)e
W (x)
= x (6)
It is direct to find that the function f (ξ) = ξe
ξ
for
ξ ∈ R has a strict minimum point in ξ = −1. In-
deed ξ = −1 is a strict relative point of minimum with
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
474

Figure 2: Diagram of ξe
ξ
= f (ξ).
f (−1) = −
1
e
, and since
lim
ξ→−∞
ξe
ξ
= 0 lim
ξ→+∞
ξe
ξ
= +∞,
the minimum is global, that is
min
ξ∈R
ξe
ξ
= −
1
e
We may draw the diagram of ξe
ξ
= f (ξ). The figure
(2) clarifies the behaviour of the function f (ξ) = ξe
ξ
.
It is clear that the existence of the Lambert function
W (x) depends on the value of x.
More precisely, we have
i) if x < −1/e the equation (6) has no real solutions;
ii) if −1/e 6 x < 0 the equation (6) admits two solu-
tions;
iii) if x > 0 the equation (6) admits one solution.
The statements i), ii) and iii) are specified with the
following properties of f (ξ) = ξe
ξ
, ξ ∈ R.
i The function f (ξ) = ξe
ξ
is strictly increasing in
the interval (−1,+∞);
ii The function f (ξ) = ξe
ξ
is strictly decreasing in
the interval (−∞,−1).
It follows
Proposition 1.1. The function f (ξ) = ξe
ξ
has an in-
creasing inverse in (−1,+∞), and a decreasing in-
verse in (−∞,−1).
We consider f (ξ) = ξe
ξ
restricted to the interval
(−∞,−1] and we denote by W
−1
its inverse. W
−1
is
defined in the interval [−1/e,0). We have two identi-
ties arising from the definition of W
−1
:
W
−1
(ξe
ξ
) = ξ,
h
⇔ W
−1
[ f (ξ)] = W
−1
h
ξe
ξ
i
= ξ
i
,
∀ξ ∈ (−∞, −1] (7)
and
W
−1
( ¯x)e
W
−1
( ¯x)
= ¯x [⇒ f (W
−1
( ¯x)) = ¯x]
∀ ¯x ∈ [−
1
e
,0) (8)
Also we denote by W
0
the restriction to the interval
[−1/e,0) of the increasing inverse of f (ξ) = ξe
ξ
. The
two identities hold true:
W
0
(ξe
ξ
) = ξ,
h
⇔ W
0
[ f (ξ)] = W
0
h
ξe
ξ
i
= ξ
i
,
∀ξ ∈ [−1, 0) (9)
and
W
0
( ¯x)e
W
0
( ¯x)
= ¯x[⇒ f (W
0
( ¯x)) = ¯x]
∀ ¯x ∈ [−
1
e
,0) (10)
1.1.1 Numerical Values
Let us assume that ¯x is a solution of our equation.
e
¯x
− 2 ¯x = 1 (11)
In order to use the Lambert function W , we observe
that from (11) we get the equivalences
e
¯x
− 2 ¯x = 1 ⇔ e
e
¯x
−2 ¯x
= e
whence −
1
2
e
¯x
e
−
1
2
e
¯x
= −
1
2
e
−
1
2
. Therefore we can
identifies −
1
2
e
¯x
with W

−
1
2
e
−
1
2

. Since −
1
e
<
−
1
2
e
−
1
2
< 0, the equation which defines the function
W of Lambert, has two branches which verifies the
same equation W (x)e
W (x)
= −
1
2
e
−
1
2
and we will have
−
1
2
e
¯x
= W
0

−
1
2
e
−
1
2

(12)
and
−
1
2
e
¯x
= W
−1

−
1
2
e
−
1
2

. (13)
We call ¯x
1
the ¯x solution of (12), and ¯x
2
the solution
of (13).
We state easy that ¯x
1
= 0. In fact from (12) we
have
−
1
2
e
¯x
1
= −
1
2
h
= W
0

−
1
2
e
−
1
2
i
and from e
¯x
1
= 1, easily follows ¯x
1
= 0.
From (13), and the relation (8) we get e
¯x
2
=
−2W
−1

−
1
2
e
−
1
2

, and so ¯x
2
:
ln

−2W
−1

−
1
2
e
−
1
2

= − ln
1
−2W
−1

−
1
2
e
−
1
2

AnExplicitBoundforStabilityofSincBases
475

Now we multiply numerator and denominator by e
1
2
= − ln


−
1
2
e
−
1
2
W
−1

−
1
2
e
−
1
2

e
1
2


= −
1
2
− ln
−
1
2
e
−
1
2
W
−1

−
1
2
e
−
1
2

By (8) we have
¯x
2
= −
1
2
−W
−1

−
1
2
e
−
1
2

.
The value −
1
2
−W
−1

−
1
2
e
−
1
2

is called the parame-
ter of Oseen, or Lamb-Oseen constant. It is often de-
noted by the Greek character α. Numerical estimates
give
α = 1.25643...
1.1.2 Properties of α = −
1
2
− W
−1

−
1
2
e
−
1
2

We have introduced the Lambert function W in order
to give an useful expression to the root of equation
e
α
= 2α + 1. (14)
It is easy to prove:
Proposition 1.2. The real number α is transcenden-
tal.
Proof. Such thesis means that α is not root of an al-
gebraic equation with rational coefficients. In fact, if
α were algebraic, then algebraic should be the right
side of (14). But Lindemann - Weierstrass theorem
(Baker, 1990), state that e
α
should be transcendental.
This is a contradiction and the thesis follows.
1.2 Known Results for Whittakers
Cardinal Series
Hardy who was referring to (4) - in literature also
known as basis functions in Whittakers cardinal se-
ries - wrote: ”It is odd that, although these functions
occur repeatedly in analysis, especially in the theory
of interpolation, it does not seem to have been re-
marked explicitly that they form an orthogonal sys-
tem” (Hardy, 1941). See also: (Benedetto, 1998),
(Butzer, 1983), (Whittaker, 1915).
Orthonormality is a fundamental property of the
sinc-function. There is a well-known property (whose
proof is given for the reader convenience), necessary
for the orthonormality proof of the system.
Since for any λ ∈ R, the function
f
λ
= f
λ
(ξ) =
(
0 |ξ| > π
e
iλξ
otherwise,
has Fourier transform F ( f
λ
)(τ) = sinc

τ − λ

. By
Paley Wiener theorem
Suppg ⊂ [−π,π] =⇒ (F g) ∈ PW
π
, (15)
we have that sinc ∈ PW
π
, as consequence of Parseval
identity (1) we see that
∀λ ∈ R sinc

τ − λ

∈ L
2
(R) ⇒
⇒ ksinc

τ − λ

k
L
2
(R)
= 1 (16)
Moreover:
Proposition 1.3. If f ∈ PW
π
, and
Z
R
f (τ)sinc

τ − n

dτ = 0, ∀n ∈ Z,
then f = 0 a.e..
Proof. Indeed we may select f ∈ PW
π
, then by defi-
nition of PW space ∃ g ∈ L
2
(R), such that
F f = g and supp(g) ⊂ [−π, π]. (17)
We now consider
Z
R
f (τ)sinc

τ − n

dτ =
Applying Parseval equality (1) and changing τ in ξ we
have
1
2π
Z
R
f (τ)F (u
n
)dτ =
1
2π
Z
R
(F f )(ξ)u
n
(ξ)dξ
=
1
2π
Z
π
−π
g(ξ)e
inξ
dξ (18)
Then the assumption becomes
1
2π
Z
π
−π
g(ξ)e
inξ
dξ = 0, ∀n ∈ Z,
and the thesis follows since {e
inξ
}
n∈Z
is a basis in
L
2
(−π,π), therefore g = 0, and hence f = 0.
A useful property of sinc-functions, used in this
paper, is the following well-known property whose
proof is given for readers convenience
Proposition 1.4. For any real numbers λ and ν, we
have
Z
R
sinc

τ − λ

sinc

τ − ν

dτ = sinc(λ − ν). (19)
Proof. Consider the LHS multiplied by 4π
2
4π
2
Z
R
sinc

τ − λ

sinc

τ − ν

dτ =
=
Z
R
2πsinc

τ − λ

2πsinc

τ − ν

dτ =
=
Z
R
F (u
λ
)(ξ)F (u
ν
)(ξ)dξ
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
476

which becomes, solving the Fourier transform
= 2π
Z
R
u
λ
(ξ)(u
ν
)(ξ)dξ = 2π
Z
π
−π
e
i(λ−ν)ξ
dξ
= 4π
2
sinπ

λ − ν)
π(λ − ν)
= 4π
2
sinc(λ − ν) (20)
From the previous results follows that

sinc

x − n


n∈Z
is an orthonormal basis in L
2
(R). By (Higgins, 1985),
we have a generalization of this statement: the system
of sinc functions {sinc

z −n

}
n∈Z
is an orthonormal
basis in PW
π
. Shannon mentioned the orthogonality
without proof (Shannon, 1949); Hardy, on the other
hand, proceeds from the complete orthonormal char-
acter of {e
inx
} in L
2
(−π,π) via the Fourier transform,
(Higgins, 1985) and (Hardy, 1941). Accordingly,
since sinc

z − n

, z ∈ C, is the image of e
−inx
under
the inverse transform, the collection {sinc

z − n

}
n∈Z
is an orthonormal basis of PW
π
(Eoff, 1995). There
are some mathematical connections with signal the-
ory. The modern approach to sampling techniques
and, more generally, to signal theory is undoubt-
edly based on Hilbert-space formulation, which al-
lows to reinterpret the original Shannons sampling
procedure as an orthogonal projection onto the sub-
space of band-limited functions. In this mathematical
representation, the continuous signal is considered a
function of the continuous variable x ∈ R. We recall,
however, that very often we do not talk really func-
tions, but rather of equivalence classes of functions,
where two functions are equivalent if they differ in a
set of measure zero. This is the case when we con-
sider the elements of the spaces L
p
with 1 6 p 6 ∞.
Here, change the value of a function at a point does
not change its equivalence class, since the points have
measure zero.
We now assume that the input function f that we
want to sample is in L
2
(−π,π), a space that is consid-
erably larger than the usual subspace of band-limited
functions, which we have called PW
π
to indicate that
we consider only functions defined in [−π,π].
The orthonormality property previously ex-
pressed, greatly simplifies the implementation of the
approximation process by which a function f ∈ L
2
is
projected onto PW
π
. Specifically, the orthogonal pro-
jection operator P : L
2
→ PW
π
can be rewritten as
P f =
∑
n∈Z
h
f ,sinc(x − n)
i
sinc(x − n)
where the inner product represents the signal contri-
bution along the direction specified by sinc(x −n) be-
cause of the orthogonality of the basis functions. This
Figure 3: Frequency interpretation of the sampling theorem.
(a) the Fourier transform of the signal and its bandwidth
B = ω
max
; the signal is band-limited, i.e.
ˆ
f (x) = 0 with-
out the bandwidth (b) sampling in time means to replicate
the Fourier transform in the frequencies, and (c) the analog
signal is reconstructed by ideal low-pass filtering.
Figure 4: A single branch of modulation and filtering: the
channel output is prefiltered by a filter with impulse re-
sponse p(x), modulated by a sequence q(x), post-filtered
by another filter of impulse response s(x). In the traditional
approach, the pre- and postfilters are both ideal low-pass:
p(x) = s(x) = sinc(x). In the more modern schemes, the
filters can be selected more freely under the constraint that
they remain biorthogonal: hp(x − k),s(x − l)i = δ
k−l
.
inner product computation is equivalent to first filter-
ing the input function with the ideal low-pass filter
and sampling thereafter, (Unser, 2000). See figure (3).
In sampling theory, the main goal is to reconstruct
a continuous function g(x) from its samples g(x
i
). If
the sampling is uniform, i.e., the set of these samples
lies on a uniform Cartesian grid (see figure (5)), then
the function g(x) can be recovered exactly from its
samples as long as g ∈ L
2
(R) is band-limited and the
grid-points density is larger than the Nyquist density
(Shannon, 1949), (Aldroubi and Feichtinger, 1998).
This is the meaning of WKS sampling theorem.
The classical result for non-uniform sampling is
due to Paley and Wiener, as widely discussed in the
introduction. See figure (4): optimal branch of mod-
ulation or sampling is employed, then mild perturba-
tion of post-filtering uniform sampling sets does not
degrade the sampled capacity. One general exam-
ple was proved by Kadec (Kadec, 1964). Suppose
that the sampling rate is f
s
and the sampling set sat-
isfies |t
n
− n/ f
s
| 6 f
s
/4. Then {e
it
n
x
}
n∈Z
also forms
a Riesz basis of L
2
(−π,π), thereby preserving infor-
AnExplicitBoundforStabilityofSincBases
477

Figure 5: Sampling grids. Top: uniform cartesian sampling.
Bottom: A typical nonuniform sampling set as encountered
in various signal and image processing applications.
mation integrity. Note that the original Kadec’s the-
orem has a similar formulation: |t
n
− n| 6 1/4, i.e.
with sampling rate = 1. These nonuniform sampling
and reconstruction schemes, while generally compli-
cated to implement in practice, significantly broaden
the class of sampling mechanisms that allow perfect
reconstruction of band-limited signals, and indicate
stability and robustness of the sampling sets. Kadecs
result immediately implies that the sampled capacity
is invariant under mild perturbation of the sampling
sets, (Y. Chen, 2014).
We conclude this section, writing a different ver-
sion of WSK theorem existing in literature, for com-
plex functions, is the following
Theorem 1.5. Let f ∈ PW
π
. Then:
f (z) =
∑
n∈Z
f (n) sinc(z − n), (21)
uniformly on compact subsets of C.
(see, for example: (A. G. Garc
´
ıa, 1998), (Higgins,
1996)).
2 Kadec-type Estimate
Result is given by
Theorem 2.1. If {λ
n
} is a sequence of complex num-
bers for which
|λ
n
− n| 5 L <
α
π
, n = 0, ±1,±2,... (22)
Figure 6: Top: A function f defined on R has been sampled
on a uniformly spaced set. Bottom: The same function f has
been sampled on a non-uniformly spaced set.
then
{
sinc(λ
n
−t)
}
satisfies the Paley-Wiener crite-
rion and so forms a Riesz basis for L
2
(−π,π).
Proof. We show that (Paley-Wiener criterion,
Young’s book (Young, 2001), theorem 13 p. 35)






+∞
∑
n
c
n
(sinc(λ
n
−t)− sinc(n −t))






6 λ < 1 (23)
whenever
∑
n
|c
n
|
2
5 1. We use the Taylor series of
sinc(λ
n
−t):
sinc(n −t) +
+∞
∑
k=1
(λ
n
− n)
k
k!
d
k
dz
k
(sincz)
z=n−t
Therefore we have, since
∑
n
|c
n
|
2
6 1,






∑
n
c
n
(sinc(λ
n
−t)− sinc(n −t))






=
=






∑
n
c
n
+∞
∑
k=1
(λ
n
− n)
k
k!
d
k
dz
k
(sincz)
z=n−t






6
∞
∑
k=1
L
k
k!



d
k
dz
k
sincz



z=n−t
(24)
On the other hand,
sincz =
sinπz
πz
=
Z
1
0
cos(sπz)ds
and
d
k
dz
k

sinπz
πz

=
=π
k
Z
1
0
s
k
cos

sπz + k
π
2

ds (25)
ICINCO2015-12thInternationalConferenceonInformaticsinControl,AutomationandRobotics
478

Putting z = x + ıy in the last formula, we get:
d
k
dz
k

sinπz
πz

6
6 π
k
Z
1
0
|s
k
|



cos

sπx + k
π
2
+ ısπy




ds
6 π
k
Z
1
0
s
k
cosh(sπy)ds
=
π
k
k + 1
coshπy (26)
Putting in formula (24): z = n − t, i.e. x = n − t and
y = 0, it result






∑
n
c
n
(sinc(λ
n
−t)− sinc(n −t))






6
6
∞
∑
k=1
(πL)
k
(k + 1)!
=: λ (27)
and we find λ =
1
M

e
M
− M − 1

where M = πL. In
order to get λ < 1, we solve, in a first moment, the
equation λ = 1, that is
e
M
= 2M + 1 (28)
We obtain same useful properties on the solutions of
equation (28), using the Lambert Function W. Such
function is defined by the equation W(x)e
W (x)
= x or,
for the true meaning, by ξe
ξ
= x. From previous sec-
tion we obtain the thesis.
The proof of the following theorem is the same as
the previous theorem, and we will omit it.
Theorem 2.2. If {λ
n
} is a sequence of complex num-
bers for which
|λ
n
− n| 5 L <
α
π
, n = 0, ±1,±2,... (29)
then
{
sinc(λ
n
− z)
}
satisfies the Paley-Wiener crite-
rion and so forms a Riesz basis for PW
π
.
3 CONCLUSIONS AND FUTURE
DEVELOPMENTS
It is possible to generalize the approach of the sam-
pling theorem to other classes of functions, thanks to
a greater abstractness earned using Hilbert spaces and
projection operators. This is achieved by simply re-
placing sinc x by a more general φ(x), called generat-
ing function. Consequently, we specify the approxi-
mation space V as
V (φ) =
(
∑
k∈Z
c
k
φ(x − k) : {c
n
}
n∈Z
∈ `
2
)
where
`
2
=
(
{x
n
}
n∈Z





∑
k∈Z
|x
k
|
2
< ∞
)
is the space of square-summable sequences. This
means that any function s(x) ∈ V (φ), continuous and
characterized by a sequence of coefficients c
k
is the
discrete representation of the signal in the signal pro-
cessing (note that not necessarily c
k
are the samples
of the signal, and φ can be significantly different from
the sinc(x)). Indeed, one of our motivations is to dis-
cover functions that are simpler to handle numerically
and have a much faster decay. Though we need some
mathematical safeguards:
• The coefficients {c
n
}
n∈Z
∈ `
2
.
• Second, the representation should be stable1 and
unambiguously defined. In other words, the fam-
ily of functions {φ(x − k) : k ∈ Z} should form a
Riesz basis of V (φ).
In this view, and before listing the developments of
this work, we try to point out a possible practical im-
plication of the theorem 2. Assume the existence of a
formula for reconstruction, like the (4) and that, tak-
ing the Fourier transform in this formula, we could
get
ˆ
f (x) =
∑
n∈Z

ˆ
f , sinc(x − n)

L
2
(−π,π)
sinc(x − n)
Thus, reconstruction by means of this formula is
equivalent to the fact that the set
{
sinc(x − λ
n
),n ∈ Z
}
formed an orthonormal basis for L
2
(−π,π). Accord-
ing to this interpretation, our theorem could state that
if we have sampling set
{λ
n
∈ R : |λ
n
− n| 6 L < α/π} (30)
for all k ∈ Z, then the set
{
sinc(x − λ
n
),n ∈ Z
}
is a
Riesz basis for L
2
(−π,π) and, on the other hand, that
{
sinc(x − λ
n
),n ∈ Z
}
is the image of an orthonormal
basis for L
2
(−π,π) under a bounded and invertible
operator from L
2
(−π,π) in itself. So the problem to
recover band-limited function from its samples merit
a deeper investigation. These questions will be inves-
tigated elsewhere.
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