On Boundedness Inequalities of Some
Semi-Discrete Operators in Connection with
Sampling Operators
Andi Kivinukk
Tarmo Metsmägi
Dept. of Mathematics
Tallinn University
Narva mnt 25
Tallinn 10120, Estonia
Email: andik@tlu.ee
Dept. of Mathematics
Tallinn University
Narva mnt 25
Tallinn 10120, Estonia
Email: tmetsmag@tlu.ee
Abstract—The main aim of this paper is to study some boundedness inequalities of certain semi-discrete operators. These
operators allow to unify some inequalities for both the Shannon
sampling operators and Kantorovich-type operators.
I. I NTRODUCTION
It is well-known that the classical Whittaker-Kotel’nikovShannon operator
∞
X
k
sinc
(Sw f )(t) :=
f ( ) sinc(wt − k),
(1)
w
k=−∞
where the kernel function
sin πt
,
πt
do not form a family of uniformly bounded operators from
Lp (R) to Lp (R), 1 ≤ p ≤ ∞. Here we identify L∞ (R)
with C(R), the space of uniformly continuous and bounded
functions on R. To overcome this shortness P. L. Butzer and
his school at Aachen proposed several ideas since 1977.
1. To get bounded operators Sw : C(R) → C(R), the sinckernel has been replaced by a quite general suitable kernel
function, s ∈ L1 (R) ∩ C(R) (see [11] and literature, cited
there). It appears [11] that the generalized Shannon sampling
operators
∞
X
k
(Sw f )(t) :=
f ( )s(wt − k),
(2)
w
sinc(t) :=
k=−∞
have the finite operator norms
kSw kC→C = m0 (s) := sup
u∈R
∞
X
|s(u − k)| < ∞,
k=−∞
k/w
K
It appears that [5] the operators (5), Sw
: Lp (R) → Lp (R),
are uniformly bounded in the case 1 ≤ p ≤ ∞, but the
operators (2) don’t in the case 1 ≤ p < ∞.
The aim of this note is, firstly, to consider jointly all these
three ideas above in a context of some semi-discrete operators
∞
X
Sw y (t) :=
yk s(wt − k)
(w > 0, t ∈ R),
(6)
k=−∞
where the sequence y = (yk )k∈Z belongs to a suitable class.
A similar abstract class of semi-discrete sampling operators
was introduced also in [16], on the basis of two papers by
Butzer and Lei [7], [8]. The basic classes are the sequence
spaces lp (Z), 1 ≤ p < ∞, consisting of all complex-valued
sequences c = (cj )j∈Z with
(3)
kcklp (Z) :=
k=−∞
∞
X
|cj |p
1/p
< ∞.
(7)
j=−∞
and
lim kSw f − f kC = 0,
w→∞
provided
Note that m0 (s) < ∞ yields s ∈ L1 (R).
2. Considering Lp -spaces, 1 ≤ p < ∞, it is proved in
[2] (also [6], [10], [3]) that for boundedness of (1) and (2) we
have to restrict the operators to a certain subspace Λp of Lp . It
sinc
: Λp → Lp , 1 < p < ∞,
appears that [2] operators (1), Sw
are uniformly bounded and [10], [3] operators (2), Sw : Λp →
Lp are uniformly bounded in the case 1 ≤ p < ∞. Note that
the case p = ∞ was considered in the previous paragraph.
3. Another idea for Lp -spaces is to consider the
Kantorovich-type sampling operators [5] (w > 0, t ∈ R)
∞ Z (k+1)/w
X
K
(Sw f )(t) :=
w
f (u)du s(wt − k). (5)
∞
X
s(u − k) = 1
(u ∈ R).
k=−∞
c
978-1-4673-7353-1/15/ $31.00 2015
European Union
(4)
Here, in (6), the kernel s ∈ L1 (R) ∩ C(R) can sometimes be
the sinc-kernel, although sinc does not belong to L1 (R).
Secondly, we want to consider some other function spaces
like AC r (R), the space of all r-times absolutely continuous
functions on R, or BV (R), the space of functions of bounded
variation on R.
The semi-discrete operators (6) are connected with sampling
k
), where the function
operators (1) or (2) taking yk = f ( w
f : R → R (or C) belongs to classes as mentioned before.
Another possible choice of (yk ) would be
Z
yk = w q(wu − k)f (u)du,
(8)
P
Denote m(wt) := k |s(wt − k)|. Then applying the discrete
Jensen inequality, we obtain
X
p
|yk ||s(wt − k)|
k
≤
R
defining the operators
K
(Sw
f )(t)
p
1 X
|s(wt − k)| m(wt)|yk |
m(wt)
k
X
= m(wt)p−1
|yk |p |s(wt − k)|
k
∞ Z
X
:=
w q(wu − k)f (u)du s(wt − k), (9)
≤ m0 (s)p−1
X
k
R
k=−∞
where q : R → R is a bounded kernel function with properties
(3) and (4). The sampling series (6) with (yk ) in (8) has been
considered in [4] and [14]. If in (8) q(u) = χ[0,1] (u), the
characteristic function of the interval [0, 1], then we get the
Kantorovich-type operator (5).
In fact, the Kantorovich-type operators [5] were introduced
for irregular sampling scheme, based on a strictly monotone
increasing sequence (tk )k∈Z , instead of the integers k ∈ Z. In
this case
Z tk+1 /w
w
f (u)du.
yk =
tk+1 − tk tk /w
A non-linear setting of the Kantorovich operators, related with
a Lipschitz condition on the kernel, was introduced in [15].
For the simplicities of notations we restrict ourselves with the
linear operators and regular samples tk = k ∈ Z.
II. B OUNDEDNESS OF SEMI - DISCRETE OPERATORS IN Lp
The results of this section are mainly not new, maybe
except Proposition 5. Our idea is to consider the semi-discrete
operators (6) separately from generalized sampling operators
(2) and the Kantorovich-type operators (5).
Let, first, consider the semi-discrete operators (6), where the
kernel function s satisfies the condition m0 (s) < ∞, hence
s ∈ L1 (R). Note that the condition (4) is not needed at this
stage, this has to be satisfied for the convergence statement
lim kSw f − f kp = 0.
w→∞
To consider the convergence statements of (6) we have
to suppose yk = Fk,w (f ), where (Fk,w ) is a sequence of
functionals with certain restrictive conditions.
Proposition 1: If y = (yk )k∈Z ∈ lp (Z), p ≥ 1, and
m0 (s) < ∞, then Sw : lp (Z) → Lp (R) satisfies the
boundedness inequality
1/p
kSw ykp ≤ ksk1 m0 (s)(p−1)/p w−1/p kyklp (Z) .
P ROOF: By definition of (6) we write
Z X
p
p
kSw ykp ≤
|yk ||s(wt − k)| dt.
R
|yk |p |s(wt − k)|.
k
After integration we obtain
kSw ykpp ≤ m0 (s)p−1
X
|yk |p
k
Z
|s(wt − k)|dt
R
= m0 (s)p−1 ksk1
1 X
|yk |p ,
w
k
which yields the statement.
Remark 1: Proposition 1 explains why in [2] it was important to introduce the subspace Λp ⊂ Lp (R) for 1 ≤ p < ∞
by
1 X
k 1/p
Λp := {f ∈ M (R) : kf klp (Σw ) :=
|f ( )|p
< ∞},
w
w
k
M (R) being the space of all Lebesgue measurable and
bounded functions f : R → C (see [2], Def. 10, or [6], Def.
k
4). Indeed, if f ∈ Λp and yk = f ( w
), then from Proposition 1
p
p
one deduce that Sw : Λ → L (R) and
1/p
kSw f kp ≤ ksk1 m0 (s)(p−1)/p kf klp (Σw )
(see [10], Prop. 3.2). Moreover, the last inequality can not be
valid for every f ∈ Lp (R), since can happen that kSw f0 kp =
∞ for some f0 ∈ Lp (R) (see [5], Introduction).
The situation is completely different for (yk ) of (8).
Proposition 2 ([14]): Let f ∈ Lp (R), 1 ≤ p < ∞. For
K
the generalized Kantorovich-type operators Sw
: Lp (R) →
p
L (R) defined by (6) and (8) there holds
1−1/p
1/p
K
kSw
f kp ≤ ksk1 m0 (s)1−1/p kqk1
m0 (q)1/p kf kp .
The proof is similar to the proof of Proposition 1.
By Proposition 2 in the case q = s in (8) follows
Corollary 1: For the generalized Kantorovich-type operaK
tors Sw
: Lp (R) → Lp (R) defined in (6) by
Z
yk = w s(wu − k)f (u)du,
R
there holds
K
kSw
f kp ≤ ksk1 m0 (s)kf kp .
Consider the (semi-discrete) Whittaker operator
X
sinc
Sw
y (t) :=
yk sinc(wt − k) (t ∈ R, w > 0).
k
introduced in [2]. There was proved
Proposition 3 ([2], Theorem 28): For 1 < p < ∞
sinc
the Whittaker operator Sw
: lp (Z) → Lp (R) satisfies the
boundedness inequality
sinc
kSw
ykp ≤ Cw−1/p kyklp (Z)
(w > 0),
the constant C > 0 being independent of w.
Remark 2: Note that by Proposition 1 the boundedness
inequality for the generalized sampling operator (2) holds for
1 ≤ p < ∞.
Proposition 3 yields as a corollary the following
Proposition 4 ([2], Corollary 29): For every f ∈ Λp , 1 <
p < ∞, there holds
sinc
kSw
f kp ≤ Ckf klp (Σw )
(w > 0).
A new observation for (yk ) in (8) by Proposition 3 reads
as follows
Proposition 5: For every f ∈ Lp (R), 1 < p < ∞, there
holds
1−1/p
K,sinc
kSw
f kp ≤ Ckqk1
m0 (q)1/p kf kp
(w > 0).
∈
P ROOF: According to proof of Proposition 2 for f
Lp (R), 1 ≤ p < ∞, one has
1−1/p
w−1/p kyklp (Z) ≤ kqk1
m0 (q)1/p kf kp .
Applying Proposition 3 yields the result.
III. B OUNDED VARIATION , KERNELS , DIFFERENCES
Considering functions f ∈ BV (R), the space of functions
of bounded variation, we could expect that a boundedness
inequality may be held, i.e.,
VR [Sw f ] ≤ C(Sw ) VR [f ],
(10)
where the constant C = C(Sw ) depends on the operator Sw :
Lp (R) → Lp (R), 1 ≤ p ≤ ∞, and the total variation is defined
as
Z
VR [f ] :=
|f 0 (u)|du (f 0 ∈ L1 (R)).
(11)
Proposition 6 ([13], Prop. 1): Suppose for 0 < m, n ≤ 1
µm (u) :=
λ(u)
,
sinc(mu)
µm,n :=
λ(u)
sinc(mu) sinc(nu)
are bounded on [0, 1]. Define kernels related to the kernel (12)
as
Z 1
λ(u)
cos(πtu)du,
(13)
sm (t) :=
0 sinc(mu)
and
Z
1
sm,n (t) :=
0
λ(u)
cos(πtu)du.
sinc(mu) sinc(nu)
(14)
Then the kernels (12) and (13) have the representations
Z m
1
sm (t + x)dx
(15)
s(t) =
2m −m
Z n
Z m
1
=
sm,n (t + x + y)dy, (16)
dx
4mn −m
−n
Z n
1
sm (t) =
sm,n (t + y)dy.
(17)
2n −n
Since (15) and (17) are valid, we have the following
Corollary 2 ([13], Cor. 1): If sm,n ∈ L1 (R), then s, sm ∈
1
L (R) and
ksk1 ≤ ksm k1 ≤ ksm,n k1 .
Therefore, all our kernels s, sm , sm,n belong to the
Bernstein class Bπ1 .
To consider the boundedness of the n-th derivative of (6)
we construct a kernel related to the kernel (12) similarly to
(14). We have the following
Proposition 7 ([13], Prop. 2): Suppose
µ(u) :=
λ(u)
sinc(m1 u) · · · sinc(mn u)
(12)
for 0 < m1 , ..., mn ≤ 1 is bounded on [0, 1]. Define the kernel
related to the kernel (12) as follows
Z 1
λ(u)
sm1 ,...,mn (t) :=
cos(πtu)du.
0 sinc(m1 u) · · · sinc(mn u)
(18)
Then
Z m1
1
s(t) = n
dx1
2 m1 · · · mn −m1
Z mn
...
sm1 ,...,mn (t + x1 + · · · + xn )dxn . (19)
where an even window function λ ∈ C[−1, 1] satisfies the
boundary conditions λ(0) = 1, λ(u) = 0 (|u| > 1). The kernel
in (12) belongs to the Bernstein class Bπ1 , the set of all entire
functions on C of exponential type π, which belong to L1 (R),
when restricted to the real axis R.
The main idea to prove the boundedness of (6) and its
first derivative is to construct some kernels related to the
kernel (12). For the kernels defined by (12) hold the following
assertions.
We define the absolute continuity of functions on R as
follows.
Definition 1 ([9], p. 6): We say that f ∈ AC(R), the space
of all absolutely continuous functions on R, Riff f (x) admits
x
for every x ∈ R the representation f (x) = −∞ g(v)dv for
1
some g ∈ L (R).
To investigate the boundedness inequalities for arbitrary
derivatives of (6) we need the following definition.
Definition 2 ([9], p. 6): We say that f ∈ AC r−1 (R), r =
2, 3, ..., the space of all (r − 1)-times absolutely continuous
R
In [13] (also [1]) the boundedness inequality (10) was called
as the variation detracting property.
In this section we have to specify the kernel function s ∈
L1 (R) of (6) in the form
Z1
s(t) :=
λ(u) cos(πtu) du,
0
−mn
functions on R, iff f (x) admits for every x ∈ R the representation
Z ur−1
Z x
Z u1
g(ur )dur
du1
du2 ...
f (x) =
−∞
−∞
−∞
1
for some g ∈ L (R) and each of the iterated integrals, possibly
apart from the first, is defining a function in L1 (R).
Let us now study some properties of f ∈ AC r−1 (R).
Proposition 8 ([13], Prop. 5): If f ∈ AC r−1 (R), (r =
2, 3, ...), then f ∈ AC(R), and there exist f (j) ∈ L1 (R) ∩
AC(R), j = 1, ..., r − 1, f (r) ∈ L1 (R), and
VR [f (j) ] = kf (j+1) k1 ,
j = 0, 1, ..., r − 1.
We introduce for a sequence (yk )k∈Z differences (l ∈
Z, n ∈ N)
4l yk := yk+l − yk ,
4nl yk
:=
4l (4n−1
yk ),
l
4l1 4l2 ...4ln yk := 4l1 (4l2 ...4ln yk )
We need the following result to prove the boundedness in
variation for the first derivative of (6).
Theorem 2: Let the operators (6) be defined by the kernel
sm,n ∈ Bπ1 ⊂ L1 (R) in (14) with m, n ∈ {1/2, 1}. Then the
derivative (Sw y)0 ∈ BV (R) and
VR [(Sw y)0 ] ≤ wksm,n k1
provided the last series is convergent.
P ROOF: By (16) we get
Z m
Z n
1
00
dx
s00m,n (t + x + y)dy.
s (t) =
4mn −m
−n
Using the formula (cf. [17], Ch. II, §19, (4))
Z h1
Z h2
δh1 δh2 g(x) =
dt1
g 00 (x + t1 + t2 )dt2
δh1 δh2 ...δhn f (x) := δh1 (δh2 ...δhn f (x)),
where h, hi ∈ R, n ∈ N.
The next proposition gives the estimate for the sum of (n +
1)-th differences via the total variation of the n-th derivative.
Proposition 9 ([13], Prop. 7): Let f ∈ AC n (R), yk =
k
, k ∈ Z. Then
f (tk + a) and let a ∈ R, w > 0, tk = w
s00 (t) =
l
=
=
(20)
l
The boundedness inequalities in variation of operators (6)
will be considered for AC(R), the absolutely continuous
functions on R, defined by Definition 1. Then we get
Theorem 1: Let the operators (6) be defined by the kernel
sm ∈ Bπ1 ⊂ L1 (R) in (13) with m ∈ {1/2, 1}. Then Sw y ∈
BV (R) and
∞
X
VR [Sw y] ≤ ksm k1
|41 yk |,
k=−∞
provided the last series is convergent.
We omit the proof, since this is analogous to the proof of the
next theorem.
For the proof of Theorem 2 we need the following lemmas.
Lemma 1: Let lim|l|→∞ ul = 0 and let (vl )l∈Z be bounded.
If for p, q ∈ N the series U below is convergent, then
ul 4p 4q vl =
l=−∞
∞
X
− sm,n (wt − l − m + n)
+ sm,n (wt − l − m − n) .
ksk1 ≤ sup
t∈R
i=1 j=1
421 yk+i+j−2 .
∞
X
|s(t − k)| ≤ (1 + π)ksk1 ,
(23)
k=−∞
valid for any s ∈ Bπ1 . Therefore, since yl is bounded, the
series (22) is absolutely and uniformly convergent. Under
assumptions on parameters m, n ∈ {1/2, 1} we can find a
number b ∈ {0, 1/2} such that ±m ± n − b ∈ Z. Therefore,
denoting zl := sm,n (wt − l + b) and using Lemma 1 we get
(Sw y)00 (t) =
w2 X
yl 42m 42n zl+b−m−n
4mn
l
l=−∞
M X
N
X
(22)
Term by term differentiation is justified by an explanation as
follows. Since sm,n ∈ Bπ1 ⊂ L1 (R), for sm,n the condition
m0 (sm,n ) < ∞ is satisfied by Nikolskii’s inequality (see [12],
Th. 6.8)
vl 4p 4q ul−p−q .
Lemma 2 ([13], Lemma 2): For M, N ∈ N we have
4M 4N y k =
w2 X
yl δm δn sm,n (wt − l)
4mn
l
w2 X
yl sm,n (wt − l + m + n)
4mn
− sm,n (wt − l + m − n)
IV. B OUNDEDNESS INEQUALITIES IN VARIATION
∞
X
1
δm δn sm,n (t).
4mn
Therefore, by (6) and m0 (s) < ∞ we have
X
(Sw y)00 (t) = w2
yl s00 (wt − l)
k=−∞
U :=
(21)
−h2
we obtain
δh f (x) := f (x + h) − f (x − h), δhn f (x) := δh (δhn−1 f (x)),
∞
X
n+1 4
yk ≤ VR [f (n) ].
1
|421 yk |,
k=−∞
−h1
and central differences of a function f : R → R as
wn
∞
X
=
=
w2 X
zl+b−m−n 42m 42n yl−2m−2n
4mn
l
w2 X
zk 42m 42n yk−m−n−b .
4mn
k
(24)
Integrating over R, the application of Lemma 2 gives
X
w
k(Sw y)00 k1 ≤
ksm,n k1
|42m 42n yk |
4mn
k
≤
2m X
2n X
X
w
ksm,n k1
4mn
i=1 j=1
|421 yk+i+j−2 |
k
= wksm,n k1
X
|421 yk |
k
k
Let yk = f ( w
) and f ∈ AC 1 (R). The application of
Proposition 9 yields
X
w
|421 yk | ≤ VR [f 0 ],
k
and we have
Corollary 3 ([13], Theorem 2): Let f ∈ AC 1 (R), then
under assumptions of Theorem 2 (Sw f )0 ∈ BV (R) and there
holds
VR [(Sw f )0 ] ≤ ksm,n k1 VR [f 0 ].
Similar corollary holds for the Kantorovich-type operators.
Let us consider the case (8), hence
Z
u+k
)du.
yk =
q(u)f (
w
R
We have the following
Corollary 4: Let f ∈ AC 1 (R), then under assumptions of
K
f )0 ∈ BV (R) and
Theorem 2, (Sw
K
VR [(Sw
f )0 ] ≤ ksm,n k1 kqk1 VR [f 0 ].
P ROOF: We have to use Theorem 2. Denote
u+k
).
zk := f (
w
Then we have
Z
421 yk =
q(u)421 zk du.
R
Summing up and estimating we get
Z
X
X
2
w
|41 yk | ≤ w |q(u)|
|421 zk |du.
R
k
k
u
Now, using Proposition 9 (a = w
) we obtain
X
w
|421 zk | ≤ VR [f 0 ].
k
Analogously to Theorem 2 the boundedness in variation for
arbitrary derivatives of (6) can be proved.
Theorem 3: Let the operators (6) be defined by the kernel
sm1 ,...,mn+1 ∈ Bπ1 ⊂ L1 (R) in (18) with m1 , ..., mn+1 ∈
{1/2, 1}. Then (Sw y)(n) ∈ BV (R) and
VR [(Sw y)(n) ] ≤ wn ksm1 ,...,mn+1 k1
∞
X
k=−∞
provided the last series is convergent.
|41n+1 yk |,
V. C ONCLUSION
In the present note we defined the semi-discrete operators,
which as the special cases produce the Shannon operators
or Kantorovich-type operators, respectively. The idea was to
present an unified approach to consider some boundedness
inequalities. It appeared that if certain series of differences of
given sequence is convergent, then the semi-discrete operator
defines a function of bounded variation. Analogous statements
are valid also for derivatives.
ACKNOWLEDGMENT
This research was partially supported by the Estonian Sci.
Foundation, grant 8627. The authors would like to thank the
referee for suggestions to supplement references and improvement of the language of the paper.
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