SHANNON SAMPLING SERIES WITH AVERAGED KERNELS1 1 ...

PROCEEDINGS OF SAMPTA 2007 JUNE 1 - 5, 2007, THESSALONIKI, GREECE
SHANNON SAMPLING SERIES WITH AVERAGED KERNELS 1
SampTA 2007 – June 1 - 5, 2007, Thessaloniki, Greece
ANDI KIVUNUKK (andik@tlu.ee)
Tallinn University, Estonia.
GERT TAMBERG (gert.tamberg@mail.ee)
Tallinn University of Technology, Estonia.
Abstract
The aim of this paper is to study the generalized sampling operators defined by averaged band-limited
kernels. It is shown that averaging leads to sampling operators suitable for functions of bounded variation.
1. Introduction
For the uniformly continuous and bounded functions f ∈ C(R) the generalized sampling series are given by
(t ∈ R; W > 0)
∞
(SW f)(t) := f( k
)s(W t − k), (1)
W
k=−∞
where the condition for the operator SW : C(R) → C(R) to be well-defined is
∞
k=−∞
the absolute convergence being uniform on compact intervals of R.
If the kernel function is the sinc-function
|s(u − k)| < ∞ (u ∈ R), (2)
s(t) = sinc(t) :=
we get the classical (Whittaker-Kotel’nikov-)Shannon operator,
(S sinc
W f)(t) :=
∞
k=−∞
sin πt
,
πt
f( k
) sinc(W t − k).
W
Since (2) is not valid for the sinc-function, the Shannon operator is not defined for all f ∈ C(R). The Whittaker-
Kotel’nikov-Shannon theorem ([7], [9]) states that a set of fixed points of the sampling operator Ssinc W is equal
to the Bernstein class B p
πW (if p = ∞, then Bp σ with σ < πW ) – the class of those bounded functions f ∈ Lp (R)
(1 p ∞) which can be extended to an entire function f(z) (z ∈ C) of exponential type σ ([7] or [20], 4.3.1),
i. e.,
|f(z)| e σ|y| fC (z = x + iy ∈ C).
Explicitly, if f ∈ B p
πW , 1 p < ∞ or f ∈ Bp σ for some σ < πW , then
S sinc
W f = f.
The idea to replace the sinc kernel sinc(·) ∈ L 1 (R) by another kernel function s ∈ L 1 (R) appeared first in
[19], where the case s(t) = (sinc(t)) 2 was considered. A systematic study of sampling operators (1) for arbitrary
kernel functions s with (2) was initiated at RWTH Aachen by P. L. Butzer and his students since 1977 (see [6],
[7], [18] and references cited there).
1 This research was partially supported by the Estonian Science Foundation grants 6943 and 7033. The second author is grateful
to the Johann Radon Institute for Computational and Applied Mathematics (Austrian Academy of Sciences) for support.
AMS Subject Classification 41A25, 41A45, 42A24.
Keywords: sampling operators, band-limited kernels, averaged kernels, functions of bounded variation.

2 A.KIVUNUKK, G.TAMBERG
In this paper we study an even band-limited kernel s, i.e. s ∈ B 1 π, defined by an even window function
λ ∈ C [−1,1], λ(0) = 1, λ(u) = 0 (|u| 1) by the equality
s(t) := s(λ; t) :=
In fact, this kernel is the Fourier transform of λ ∈ L 1 (R),
s(t) =
1
0
λ(u) cos(πtu) du. (3)
π
2 λ∧ (πt). (4)
These types of kernels arise in conjunction with window functions widely used in applications (e.g. [1], [2], [8],
[16]), in Signal Analysis in particular. Many kernels can be defined by (3), e.g.
1) λ(u) = 1 defines the sinc function;
2) λ(u) = 1 − u defines the Fejér kernel sF (t) = 1 2 t
2sinc 2 (cf. [19]);
3) λj(u) := cos π(j + 1/2)u, j = 0, 1, 2, . . . defines the Rogosinski-type kernel (see [11]) in the form
2 πu
4) λH(u) := cos 2
rj(t) := 1
2
sinc(t + j + 1
1
) + sinc(t − j −
2 2 )
= (−1)j
π
1 = 2 (1 + cos πu) defines the Hann kernel (see [13])
(j + 1/2) cos πt
(j + 1/2) 2 ; (5)
− t2 sH(t) := 1 sinc t
; (6)
2 1 − t2 5) λB,a(u) := 1
1
2 + a cos πu + ( 2 − a) cos 3πu defines the Blackman-Harris kernel (a special case of the general
cosine window, where: a0 = 1
2 , a1 = a = 1
2 − a3, a2 = 0, cf. [14], [15])
sB,a(t) := (16a − 9)t2 + 9
2(1 − t 2 )(9 − t 2 )
sinc t = 1
2
3
k=0
ak
sinc(t + k) + sinc(t − k) . (7)
If a = 9/16 the latter has especially rapid decrease at infinity – s B,9/16(t) = O(|t| −5 ) as |t| → ∞.
First we used the band-limited kernel in general form (3) in [10], see also [5].
Now we will study the sampling operators (1) using averaged kernels of the kernel functions (3), i.e.
sm(t) := 1
m
m/2
−m/2
s(t + v) dv (m > 0), (8)
which is suitable for functions of bounded variation. To give concrete examples we restrict ourselves to Blackman-
Harris kernels, because they are rapidly decreasing at infinity and the Blackman-Harris operators have norms,
which are very close to one.
We say that the sampling operator SW : T V (R) → T V (R) has the variation detracting property for functions
f ∈ T V (R) of bounded variation (cf. [4] and references cited therein), if
VR[SW f] SW VR[f]. (9)
As an introductory approach we use averaged kernels to get for these operators the variation detracting property.
2. Preliminary results
The most general kernel for the sampling operators (1) is defined in the following way.

SHANNON SAMPLING SERIES WITH AVERAGED KERNELS 3
Definition 1 ([18, Def. 6.3]) If s : R → C is a bounded function such that
∞
k=−∞
the absolute convergence being uniform on compact subsets of R, and
∞
k=−∞
then s is said to be a kernel for sampling operators (1).
|s(u − k)| < ∞ (u ∈ R), (10)
s(u − k) = 1 (u ∈ R), (11)
The definition formulated above guarantees that operators (1) give approximations for continuous functions
f ∈ C(R).
Theorem 1 ([7, Th. 4.1]) Let s ∈ C(R) be a kernel. Then {SW } W >0 defines a family of bounded linear
operators from C(R) into itself, satisfying
SW = sup
∞
u∈R
k=−∞
|s(u − k)| (W > 0). (12)
In the following we assume that our kernel (3) belongs to L1 (R), which yields s ∈ B1 π, because the Fourier
transform of s,
s ∧ (x) = 1
x
√ λ
2π π
implies s ∧ (x) = 0 for |x| π. (13)
For the band-limited functions s ∈ B p π ⊂ L p (R) the norm (12) is related to the norm sp as shown in following.
Theorem 2 (Nikolskii inequality; [17, p. 124], [9, Th. 6.8]) Let 1 p ∞. Then, for every s ∈ B p σ,
sp sup
u∈R
∞
k=−∞
|s(u − k)| p
1/p
(1 + σ)sp.
From the Nikolskii inequality we see that our assumption s ∈ L 1 (R) is sufficient for (10) and thus s in (3) is
indeed a kernel in the sense of Definition 1.
3. Sampling operators with averaged kernels and the variation detracting property
The variation detracting property of certain sampling operators will be considered for T V (R), the space of
all functions of bounded variation on R endowed with the seminorm
f T V (R) := VR[f] = sup
I⊂R
VI[f] = sup sup
I⊂R xj∈I
j=1
n
|f(xj) − f(xj+1)|,
the first supremum being taken over all intervals I ⊂ R. By AC(R) we denote the subspace of T V (R) consisting
of those functions, which are locally absolutely continuous on R. We say that the sampling operator SW :
T V (R) → T V (R) has the variation detracting property (see [4] and references cited therein), if (9) is valid for
every f ∈ T V (R). This property is important in practice, since very often signals are discontinuous but still
with bounded variation.
Looking for sampling operators with the variation detracting property our first observation was that if we
take λL,m(u) = sinc mu (m ∈ N) as a window function, then by (3) we obtain the averaged sinc-function
sL,m(t) = 1
2m
m
−m
sinc(t + v) dv. (14)

4 A.KIVUNUKK, G.TAMBERG
Here L stands for Lanczos, since sinc as the window is often called the Lanczos window [21].
Although the sinc function does not belong to B 1 π, the kernel sL,m does. Indeed, integrating by parts in (14)
yields for |t| > m
sL,m(t) = (−1)m cos(πt)
π 2 (t 2 − m 2 )
and therefore sL,m ∈ L 1 (R). On the other hand, by (3)
sL,m(t) =
which shows by the Paley-Wiener theorem that sL,m ∈ B 1 π.
Now we will average some other kernels s ∈ B 1 π.
1
0
1
−
2π2m t+m
t−m
sinc(mu) cos(πut) du,
cos(πu)
u 2
Theorem 3 Let the kernel s ∈ B 1 π be defined by (3) using the window function λ. Then the window function
defines by (3) the averaged kernel sm ∈ B 1 π, where
Proof: By (3) we have
Using the representation
it is possible to write (16) in the form
sm(t) = 1
m
= 1
2m
= 1
2m
Since s ∈ L 1 (R), we have by (15)
R
λm(u) := λ(u) sinc mu
2
sm(t) = 1
m
sm(t) :=
m/2
−m/2
m/2
−m/2
m/2
−m/2
1
0
sinc mu
2
0
m/2
−m/2
λ(u) sinc mu
2
1
=
m
m/2
−m/2
(m > 0)
du
s(t + v) dv. (15)
cos(πtu) du. (16)
cos(πvu) dv
1
dv λ(u) cos(πvu) cos(πtu) du
dv
0
1
λ(u) cos π(t − v)u + cos π(t + v)u du
s(t − v) + s(t + v) dv = 1
m
|sm(t)|dt 1
m
m/2
−m/2
R
m/2
−m/2
⎛ ⎞
⎝ |s(t)|dt⎠
dv = s1.
s(t + v) dv.
Hence, by (16), sm ∈ B 1 π.
To obtain the estimates for the variation detracting property we first have to estimate the operator norms.

SHANNON SAMPLING SERIES WITH AVERAGED KERNELS 5
Theorem 4 Let the sampling operator SW be defined by the kernel s ∈ B 1 π from (3) and the averaged sampling
operator SW,m be defined by the kernel sm ∈ B 1 π, m ∈ N, from (15). Then the operator norm of the sampling
operator SW,m has an estimate
SW,m s1 SW .
Proof: For s ∈ B 1 π we have by the definition (15)
∞
k=−∞
|sm(u − k)| =
= 1
m
∞
∞
k=−∞
which by Theorem 1 yields
Using Theorem 2 we get
m−1
1
m
m/2
−m/2
−m/2+j+1
k=−∞ j=0
−m/2+j
s(u − k + v) dv
1
m
|s(k − v − u)| dv = 1
m
∞
SW,m s1.
m/2
k=−∞
−m/2
m−1
SW,m s1 SW .
∞
j=0 k=−∞
k−1
|s(u − k + v)| dv
k
= 1
m−1
m
j=0
−∞
|s(w + m
2
∞
|s(v)| dv =
For some averaged sampling operators we can compute the exact value of the norm.
− j − u)| dw
∞
−∞
|s(v)| dv = s1,
Theorem 5 If there exists a constant a ∈ [0, 1], such that for all k ∈ Z the kernel function s ∈ B1 π does not have
zeroes on (a − k − 1
1
2 , a − k + 2 ), then for the sampling operator SW,1 with the kernel s1 defined by (15) (m = 1)
we get an exact value of the operator norm
SW,1 = s1.
Proof: By the definition of the operator norm
SW,1 = sup
∞
u∈[a−1/2,a+1/2]
k=−∞
|s1(u − k)|
∞
k=−∞
For s ∈ B 1 π we have by the definition (15) and by the assumption on zeroes
By (17) we get
∞
k=−∞
|s1(a − k)| =
∞
1/2
k=−∞
−1/2
=
∞
k
k=−∞
k−1
s(a − k + v) dv
=
|s(w − a + 1
)| dw =
2
SW,1 s1.
Using Theorem 4 completes the proof.
Now we are able to consider the variation detracting property.
∞
1/2
k=−∞
−1/2
∞
−∞
|s1(a − k)|. (17)
|s(a − k + v)| dv
|s(v)| dv = s1.
Theorem 6 Let the sampling operator SW be defined by the kernel s ∈ B 1 π from (3) and the averaged sampling
operator SW,m be defined by the kernel sm ∈ B 1 π , m ∈ N, from (15). Then f ∈ T V (R) implies SW,mf ∈ AC(R)
and
VR[SW,mf] s1 VR[f] SW VR[f], (18)
moreover, under the assumptions of Theorem 5, we have
VR[SW,mf] SW,1 VR[f]. (19)

6 A.KIVUNUKK, G.TAMBERG
Proof: Let f ∈ T V (R), hence f is bounded on R. Since sm ∈ B 1 π, the derivative s ′ m ∈ B 1 π also, and by
Nikolskii inequality
k∈Z |s′ m(u − k)| < ∞. Therefore, there exists the bounded derivative
(SW,mf) ′ (t) = W
∞
k=−∞
f( k
W )s′ m(W t − k),
which implies SW,mf ∈ AC(R). Now we have for the total variation
Since by (15) the derivative
we get
(SW,mf) ′ (t) = W
m
∞
k=−∞
k=−∞
VR[SW,mf] =
s ′ m(t) = 1
m
f( k
W )
∞
−∞
s(t + m
2
s(W t − k + m
2
|(SW,mf) ′ (t)|dt. (20)
m
) − s(t −
2 )
,
m
) − s(W t − k −
2 )
= W
∞
f(
m
k=−∞
k
∞ m
)s(W t − k + ) −
W 2
k=−∞
= W
∞
f(
m
k − m
) − f(k
W W )
s(W t − k + m
2 )
= W
m−1
m
∞
j=0 k=−∞
f(
k − j
W
Now by (20) and Theorem 4 we obtain for the total variation
VR[SW,mf] 1
m−1
m
j=0 k=−∞
∞
− j − j − 1
f(k ) − f(k
W W
)
f( k
m
)s(W t − k −
W 2 )
− j − 1
) − f(k
W
)
s(W t − k + m
2 ).
∞
−∞
s(W t − k + m
2 )
d(W t)
1
m−1
VR[f]s1 = s1VR[f] SW VR[f]. (21)
m
j=0
Under the assumptions of Theorem 5 we have s1 = SW,1, therefore
VR[SW,1f] SW,1VR[f]
4. Examples of operators with variation detracting property
4.1 Hann’s sampling operators
We can provide more examples of operators with variation detracting property (9).
2 πu
Take the Hann window λH(u) := cos . Then by (3)
sH(t) = 1
sinc(t − 1) + 2 sinc t + sinc(t + 1)
4
and its corresponding averaged kernel by (15) is
sH,1(t) = 1
4
2
Sci(t + 1
2
) − Sci(t − 1
2
= 1
r0(t − 1/2) + r0(t + 1/2) =
2
sinc t
3
3
) + Sci(t + ) − Sci(t −
2 2 )
,
2(1 − t 2 )
(22)

where the integral sinc is defined by
SHANNON SAMPLING SERIES WITH AVERAGED KERNELS 7
Sci(x) :=
x
0
sinc(v)dv.
Since sH does not have zeroes on (k, k + 1) for all k ∈ Z, we obtain by Theorems 5 and 6
Corollary 1 Let f ∈ T V (R). Then HW,mf ∈ AC(R) and
VR[HW,mf] HW,1VR[f] = Sci(1) + Sci(2) VR[f] = (1.0409 . . .)VR[f].
Proof: The first inequality follows immediately from (18) of Theorem 6. We need only to compute the norm
sH1. We have by (22)
∞
2
∞
sH1 = 2 |sH(t)| dt = 2 sH(t)dt + 2 (−1) k+1
0
2
= 2 sH(t)dt − 2
0
0
1
0
k=2
k+1
k
sH(t)dt
∞
(−1) k sH(t + k)dt. (23)
The representation of the kernel sH using Rogosinski kernels r0 in (22) allows us to write
2
2n
k=2
(−1) k sH(t + k) =
So, we have by (5)
2
0
1
=
2n
k=2
2n
k=2
k=2
(−1) k
r0(t + k − 1/2) + r0(t + k + 1/2)
(−1) k r0(t + k − 1/2) −
∞
(−1) k sH(t + k)dt = 1
2
k=2
1
0
(−1) k r0(t + k − 1/2) = r0(t + 3/2) + r0(t + 2n + 1/2).
2n+1
k=3
sinc(t + 1) + sinc(t + 2)
The representation of the kernel sH using sinc functions in (22) gives
2
0
2
sH(t)dt = 1
2
2
0
sinc(t − 1) + 2 sinc t + sinc(t + 1)
dt = 1
2
Combining the equalities (25) and (24) we get from (23) by Theorem 5
4.2 Lanczos’ sampling operators
HW,1 = sH1 = Sci(1) + Sci(2).
dt = 1
2
Sci(3) − Sci(1) . (24)
Sci(1) + 2 Sci(2) + Sci(3) . (25)
We can see that the Lanczos’ window function λL,m(u) = sinc(mu) for m = 2j + 1 can be represented via the
Rogosinski’s window function λR,j(u) = cos π(j + 1/2)u as follows,
λL,2j+1(u) = λR,j(u) sinc
(2j + 1)u
.
2
Now, by Theorem 3 (m = 2j + 1), the Lanczos’ sampling operator LW,2j+1 can be considered as the averaging
of the Rogosinski sampling operator RW,j, i.e.
Therefore we can use Theorem 6 getting
LW,2j+1 = RW,j,2j+1.

8 A.KIVUNUKK, G.TAMBERG
Corollary 2 Let the sampling operator LW,2j+1 be defined by the window function λL,2j+1(u) = sinc(2j + 1)u
using (3). Then f ∈ T V (R) implies LW,2j+1f ∈ AC(R) and
In particular,
VR[LW,2j+1f] rj1VR[f] RW,jVR[f].
VR[LW,1f] LW,1VR[f] = 2 Sci(1)VR[f] = (1.178979 . . .)VR[f].
Proof: The first inequalities follow immediately from (18) of Theorem 6. For the second inequality we have
consider the Rogosinski kernel (5) (j = 0)
r0(t) =
cos πt
2π(1/4 − t 2 ) ,
which does not have zeroes on (−k − 1/2, −k + 1/2). Therefore, by (19) we get VR[LW,1f] LW,1VR[f], where
the numerical value of the norm LW,1 was calculated in [12], Theorem 1.
Let us state an elementary proposition using the Rogosinski-type kernels.
Proposition 1 For the kernel (14) we have
thus d
dt sL,m ∈ B 1 π.
d
dt sL,m(t) = 1
m
m
j=1
Proof: From the definition (14) we get
r0(t − m + 2j − 1
2 ) − r0(t − m + 2j − 3
2 )
,
d
dt sL,m(t) = 1
(sinc(t + m) − sinc(t − m))
2m
= 1
m
(sinc(t − m + 2j) − sinc(t − m + 2(j − 1))) ,
2m
j=1
which by (5) gives the assertion.
Although sinc ∈ L 1 (R) we get in rather similar way as in Theorem 6 the following more general result for
the Lanczos operators LW,m defined by the kernel (14).
Theorem 7 Let f ∈ T V (R). Then for the Lanczos sampling operators LW,m defined by the kernel (14) we have
LW,mf ∈ AC(R) and
VR[LW,mf] LW,1VR[f],
where LW,1 = r01 = 1.178979 . . .
Proof: As in Theorem 6 we have
where the derivative is
By Proposition 1 we get
(LW,mf) ′ (t) = W
m
= W
m
∞
k=−∞
m
VR[LW,mf] =
(LW,mf) ′ (t) = W
f( k
W )
∞
j=1 k=−∞
f(
m
j=1
∞
−∞
∞
k=−∞
|(LW,mf) ′ (t)|dt,
f( k
W )s′ L,m(W t − k).
r0(W t − k − m + 2j − 1
2 ) − r0(W t − k − m + 2j − 3
2 )
k + 2j
W
+ 2j − 1
) − f(k ) r0(W t − m − k −
W
1
2 ).

SHANNON SAMPLING SERIES WITH AVERAGED KERNELS 9
For the total variation we obtain
VR[LW,mf] = 1
∞
m ∞
k + 2j + 2j − 1
m f( ) − f(k ) r0(W t − m − k −
W W
−∞
j=1 k=−∞
1
2 )
d(W t)
1
m r01
m ∞
+ 2j + 2j − 1
f(k ) − f(k )
W W r01VR[f]. (26)
j=1 k=−∞
For the numerical value of LW,1 see Proof of Corollary 2.
4.3 Blackman-Harris’ sampling operators
The Blackman-Harris window λB,9/16 := 1
16 (8 + 9 cos πu − cos 3πu) defines by (7) (a = 9/16) the kernel
and by (15) the averaged kernel
sB,9/16;1(t) = 1
16
7 Sci(t + 1
2
s B,9/16(t) :=
) − 7 Sci(t − 1
2
9
2(1 − t 2 )(9 − t 2 )
3
3
) + 9 Sci(t + ) − 9 Sci(t −
2 2 )
+ Sci(t + 5
2
sinc t (27)
) − Sci(t − 5
2
7
7
) − Sci(t + ) + Sci(t −
2 2 )
.
Since sB,9/16 does not have zeroes on (k, k + 1) (k ∈ Z), we can compute by Theorem 5, using the scheme of the
proof of Corollary 1, for the corresponding operator the norm
BW,9/16;1 = sB,9/161 = 1
11 Sci 1 + 15 Sci 2 + Sci 3 − 8 Sci 4 − 2 Sci 5 + Sci 6 = 1.178986 . . .
8
Theorem 6 applies also in this case.
Corollary 3 Let f ∈ T V (R). Then B W,9/16;mf ∈ AC(R) and
VR[B W,9/16;mf] B W,9/16;1VR[f] = (1.178986 . . .)VR[f].
The two-parameter Blackman-Harris window, which is defined by (a0, a1 ∈ R)
defines the kernel function
where
λ B,(a0,a1)(u) := a0 + a1 cos πu + ( 1
2 − a0) cos 2πu + ( 1
2 − a1) cos 3πu, (28)
s B,(a0,a1)(t) =
p (a0,a1)(t)
2(9 − t 2 )(4 − t 2 )(1 − t 2 )
sinc t, (29)
p (a0,a1)(t) := (5 + 8a0 − 16a1)t 4 − (5 + 80a0 − 64a1)t 2 + 72a0. (30)
If the discriminant of the polynomial (30) is negative, i.e. the inequality
25 − 640 a0 + 4096a 2 0 − 640a1 − 5632a0a1 + 4096a 2 1 < 0 (31)
is satisfied, then the polynomial (30) has no zeroes (see [15], Lemma 2). Note that on the a0a1-plane the
inequality (31) determines the interior of an ellipse. The corresponding averaged operator by (15) (m = 1), with
the parameters in that ellipse, has by Theorem 5 the norm
BW,(a0,a1),1 = Sci 1 + Sci 6 + 2a0 Sci 3 − Sci 2 + Sci 4 − Sci 5 + 2a1 Sci 2 − Sci 1 + Sci 5 − Sci 6 (32)
= 1.072 . . . + a0 (0.073 . . .) − a1 (0.202 . . .) (33)
because the kernel (29) in this case do not have zeroes on (k, k + 1) for all k ∈ Z and, therefore, we have by
Theorem 6 the following result.

10 A.KIVUNUKK, G.TAMBERG
Corollary 4 Let f ∈ T V (R). If for a0, a1 ∈ R holds the inequality (31), then for the corresponding averaged
sampling operators we have B W,(a0,a1),mf ∈ AC(R) and
In particular,
VR[B W,(a0,a1),mf] B W,(a0,a1),1VR[f].
VR[B W,(0.352...,0.456...),mf] (1.000104 . . .)VR[f].
Remark. The last inequality corresponds to the situation, when we minimized the norm (32) in the ellipse (31).
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