On approximation properties
of Kantorovich-type sampling operators
Olga Orlova, Gert Tamberg 1
Department of Mathematics,
Tallinn University of Technology,
E STONIA
Modern Time-Frequency Analysis
Strobl 2014
AUSTRIA
June 5, 2014
1
This is a joint work with Andi Kivinukk (Tallinn University)
This research was supported by the Estonian Science Foundation, grant 9383 and by the Estonian Min. of Educ. and Research, project SF0140011s09.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
1 / 34
Shannon Sampling Series
Sampling theory
The aim of this talk is to study approximation properties of a
generalized Shannon sampling operators
(Sw f )(t) :=
∞
X
k =−∞
f(
k
)s(wt − k )
w
with a bandlimited kernel s, which is a Fourier transform of a certain
even window function λ ∈ C[−1,1] , λ(0) = 1, λ(u) = 0 (|u| > 1), i.e.
Z1
s(t) := s(λ; t) :=
λ(u) cos(πtu) du.
0
We give the conditions for the kernel s, which allow us to have the
estimate of order of approximation via modulus of smoothness in form:
kf − Swr f k 6 Mr ω2r (f ;
O. Orlova, G. Tamberg (TUT)
1
).
w
Kantorovich-type Sampling Operators
Strobl 2014
2 / 34
Shannon Sampling Series
Preliminary results
Def
The Bernstein class Bσp for σ > 0 and 1 6 p 6 ∞ consists of those
bounded functions f ∈ Lp (R) which can be extended to an entire
function f (z) (z ∈ C)of exponential type σ, i. e.,
|f (z)| 6 eσ|y | kf kC
(z = x + iy ∈ C).
The class Bσp is a Banach space if one takes the norm of Lp (R).
Lemma
We have
Bσ1 ⊂ Bσp ⊂ Bσr ⊂ Bσ∞ ,
Bαp ⊂ Bβp ,
O. Orlova, G. Tamberg (TUT)
1 6 p 6 r 6 ∞,
06α6β<∞
Kantorovich-type Sampling Operators
Strobl 2014
3 / 34
Shannon Sampling Series
Whittaker-Kotelnikov-Shannon sampling series
Whittaker-Kotelnikov-Shannon sampling theorem
Theorem (Whittaker-Kotelnikov-Shannon)
p
If g ∈ BπW
, 1 6 p < ∞, or g ∈ Bσ∞ for some 0 6 σ < πW , then
g(t) =
∞
X
k =−∞
g(
k
sinc
) sinc(Wt − k ) =: (SW
g)(t),
W
(1)
the series being uniformly convergent on each compact subset of R.
p
The Bernstein class BπW
forms the set of fixed points of the sampling
operators (1). Now it is natural to ask what happens in WKS Theorem
p
if instead of g ∈ BπW
we consider uniformly continuous and bounded
functions f ∈ C(R)? M. Theis [Theis’19] showed that the equality in (1)
does not hold for any g ∈ C(R).
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
4 / 34
Shannon Sampling Series
Whittaker-Kotelnikov-Shannon sampling series
Whittaker-Kotelnikov-Shannon sampling theorem
Theorem (Whittaker-Kotelnikov-Shannon)
p
If g ∈ BπW
, 1 6 p < ∞, or g ∈ Bσ∞ for some 0 6 σ < πW , then
g(t) =
∞
X
k =−∞
g(
k
sinc
) sinc(Wt − k ) =: (SW
g)(t),
W
(1)
the series being uniformly convergent on each compact subset of R.
p
The Bernstein class BπW
forms the set of fixed points of the sampling
operators (1). Now it is natural to ask what happens in WKS Theorem
p
if instead of g ∈ BπW
we consider uniformly continuous and bounded
functions f ∈ C(R)? M. Theis [Theis’19] showed that the equality in (1)
does not hold for any g ∈ C(R).
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
4 / 34
Shannon Sampling Series
Generalized sampling series
Generalized Shannon sampling series
We get uniform convergence
kf − Sw f kC → 0 (w → ∞)
for any f ∈ C(R) if, instead of the cardinal sine,
P we use different kernel
1
1
functions s ∈ L (R) ∩ C(R) (sinc 6∈ L (R)), k ∈Z s(u − k ) = 1, u ∈ R.
This approach gives us the generalized sampling series for t ∈ R,
w >0
∞
X
k
(2)
(Sw f )(t) :=
f ( )s(wt − k ).
w
k =−∞
Whereas a generalized sampling series with a particular kernel were
already considered by M. Theis in 1919, a study of those series for
arbitrary kernel functions was initiated at RWTH Aachen University
(P. L. Butzer et al) and widely studied there since 1977 and in Perugia
since 2005.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
5 / 34
Shannon Sampling Series
Generalized sampling series
Generalized Shannon sampling series
We get uniform convergence
kf − Sw f kC → 0 (w → ∞)
for any f ∈ C(R) if, instead of the cardinal sine,
P we use different kernel
1
1
functions s ∈ L (R) ∩ C(R) (sinc 6∈ L (R)), k ∈Z s(u − k ) = 1, u ∈ R.
This approach gives us the generalized sampling series for t ∈ R,
w >0
∞
X
k
(2)
(Sw f )(t) :=
f ( )s(wt − k ).
w
k =−∞
Whereas a generalized sampling series with a particular kernel were
already considered by M. Theis in 1919, a study of those series for
arbitrary kernel functions was initiated at RWTH Aachen University
(P. L. Butzer et al) and widely studied there since 1977 and in Perugia
since 2005.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
5 / 34
Shannon Sampling Series
Generalized sampling series
0.4
1.0
0.2
0.5
-4
-2
2
4
-4
-2
2
-0.5
-0.2
-1.0
-0.4
4
Take a function f and compute the corresponding sampling series
(S2 f )(t) :=
∞
X
k =−∞
k
f
s(2t − k ),
2
(W = 2)
and the approximation error S2 f − f .
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
6 / 34
Shannon Sampling Series
Generalized sampling series
0.4
1.0
0.2
0.5
-4
-2
2
-4
4
-2
2
-0.5
-0.2
-1.0
-0.4
4
Now we compute the sampling series, taking W = 6,
(S6 f )(t) :=
∞
X
k =−∞
k
f
s(6t − k )
6
and the approximation error S6 f − f .
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
6 / 34
Shannon Sampling Series
Generalized sampling series
0.4
1.0
0.2
0.5
-4
-2
2
-4
4
-2
2
-0.5
-0.2
-1.0
-0.4
4
At last we take W = 16, getting
(S16 f )(t) :=
∞
X
f
k =−∞
k
16
s(16t − k )
and the approximation error S16 f − f .
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
6 / 34
Shannon Sampling Series
Sampling operators in Lp case
Since in practice signals are however often discontinuous, this talk is
also concerned with the convergence of Sw f to f in the Lp (R)-norm for
1 6 p < ∞.
The sampling series Sw f of an arbitrary Lp -function f may be
divergent. Take f ∈ Lp (R), f (k /w) = 1 for a fixed w > 0, then
(Sw f )(t) ≡ 1 6∈ Lp (R).
Also operators Sw f do not always provide good results in applications,
e.g. in Signal Processing. Generalized sampling operators depend on
exact values f (k /w), whilst in practice we often have to deal with the
so called "jitter errors" i.e. the impossibility to determine the exact
values at the nodes. Replacing the exact value f (k /w) with an average
of f around k /w might lead to smaller errors and therefore be useful in
applications.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
7 / 34
Shannon Sampling Series
Sampling operators in Lp case
Z. Burinska, K. Runovski and H. J. Schmeisser introduced in [BRS’06]
for 0 < p 6 ∞ additional shifts λ:
(Sw;λ f )(t) :=
∞
X
k =−∞
f(
k
+ λ)s(w(t − λ) − k )
w
and estimated the order of approximation via modulus of smoothness.
C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV’06] introduced
a suitable subspace of Lp (R). They estimated the order of
approximation for the classical (Whittaker-Kotel’nikov-Shannon)
operator for 1 < p < ∞ and for sampling operators with time-limited
kernels for 1 6 p < ∞ in [Butzer, Stens’08] and [BBSV’10] via
averaged modulus of smoothness.
We used the same approach and estimated the order of approximation
for sampling operators with band-limited kernels for 1 < p < ∞ in
[T’13] via the averaged modulus of smoothness and for 1 6 p < ∞ in
[Kivinukk,T’14] and [T’14] via the classical modulus of smoothness.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
8 / 34
Shannon Sampling Series
Sampling operators in Lp case
Z. Burinska, K. Runovski and H. J. Schmeisser introduced in [BRS’06]
for 0 < p 6 ∞ additional shifts λ:
(Sw;λ f )(t) :=
∞
X
k =−∞
f(
k
+ λ)s(w(t − λ) − k )
w
and estimated the order of approximation via modulus of smoothness.
C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV’06] introduced
a suitable subspace of Lp (R). They estimated the order of
approximation for the classical (Whittaker-Kotel’nikov-Shannon)
operator for 1 < p < ∞ and for sampling operators with time-limited
kernels for 1 6 p < ∞ in [Butzer, Stens’08] and [BBSV’10] via
averaged modulus of smoothness.
We used the same approach and estimated the order of approximation
for sampling operators with band-limited kernels for 1 < p < ∞ in
[T’13] via the averaged modulus of smoothness and for 1 6 p < ∞ in
[Kivinukk,T’14] and [T’14] via the classical modulus of smoothness.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
8 / 34
Shannon Sampling Series
Sampling operators in Lp case
Z. Burinska, K. Runovski and H. J. Schmeisser introduced in [BRS’06]
for 0 < p 6 ∞ additional shifts λ:
(Sw;λ f )(t) :=
∞
X
k =−∞
f(
k
+ λ)s(w(t − λ) − k )
w
and estimated the order of approximation via modulus of smoothness.
C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV’06] introduced
a suitable subspace of Lp (R). They estimated the order of
approximation for the classical (Whittaker-Kotel’nikov-Shannon)
operator for 1 < p < ∞ and for sampling operators with time-limited
kernels for 1 6 p < ∞ in [Butzer, Stens’08] and [BBSV’10] via
averaged modulus of smoothness.
We used the same approach and estimated the order of approximation
for sampling operators with band-limited kernels for 1 < p < ∞ in
[T’13] via the averaged modulus of smoothness and for 1 6 p < ∞ in
[Kivinukk,T’14] and [T’14] via the classical modulus of smoothness.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
8 / 34
Shannon Sampling Series
Sampling operators in Lp case
C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV’07] introduced
Kantorovich-type sampling operators


(k +1)/w
Z
∞
X 

f (u) du  s(wt − k )
(SwK f )(t) =
w
k =−∞
k /w
and proved the convergence for 1 6 p < ∞ case.
We estimated the order of approximation for Kantorovich-type
sampling operators


(2nk +1)/2nw
Z
∞
X


I
(Sw,n
f )(t) =
f (u) du  s(wt − k )
nw
k =−∞
(2nk −1)/2nw
with band-limited kernels for 1 6 p 6 ∞ in [Orlova,T’14] via the
classical modulus of smoothness.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
9 / 34
Shannon Sampling Series
Sampling operators in Lp case
C. Bardaro, P. L. Butzer, R. Stens and G. Vinti in [BBSV’07] introduced
Kantorovich-type sampling operators


(k +1)/w
Z
∞
X 

f (u) du  s(wt − k )
(SwK f )(t) =
w
k =−∞
k /w
and proved the convergence for 1 6 p < ∞ case.
We estimated the order of approximation for Kantorovich-type
sampling operators


(2nk +1)/2nw
Z
∞
X


I
(Sw,n
f )(t) =
f (u) du  s(wt − k )
nw
k =−∞
(2nk −1)/2nw
with band-limited kernels for 1 6 p 6 ∞ in [Orlova,T’14] via the
classical modulus of smoothness.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
9 / 34
Shannon Sampling Series
Singular integrals
Singular integrals
In order to proceed with Kantorovich-type sampling operators we need
to define the singular integral.
If χ ∈ L1 (R) is such that
Z∞
χ(u)du = 1,
−∞
then the convolution integral of f with χρ (u) := ρχ(ρu), namely
(Iρχ f )(x)
Z∞
:= (f ∗ χρ )(x) =
f (u)ρχ(ρ(x − u))du
(x ∈ R)
(3)
−∞
is called the singular (convolution) integral with kernel χ.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
10 / 34
Shannon Sampling Series
Kantorovich-type sampling operators
Kantorovich-type sampling operators
As a final step we replace the exact value f (k /w) in (2) with an average
χ
of f around k /w, using the singular integral Inw
f (n ∈ N) with k /w as
an argument. Thus for f ∈ Lp (R) (1 6 p 6 ∞) the Kantorovich-type
sampling operators are given by (t ∈ R; w > 0; n ∈ N)

 ∞
Z
∞
X
k
K

(4)
(Sw,n
f )(t) :=
f (u)nwχ(nw( − u))du  s(wt − k )
w
k =−∞
−∞
with kernels s, χ ∈ L1 (R),
R∞
−∞
χ(u)du = 1,
P
k ∈Z s(u
− k ) = 1 (u ∈ R).
Remark. Bardaro, Butzer, Stens and Vinti considered the case
χ = χ[0,1] , n = 1 only.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
11 / 34
Shannon Sampling Series
Bandlimited kernels
Bandlimited kernels
In this talk we consider an even band-limited kernel s, i.e. s ∈ Bπ1 ,
defined by an even window function λ ∈ C[−1,1] , λ(0) = 1, λ(u) = 0
(|u| > 1) by the equality
Z1
s(t) := s(λ; t) :=
λ(u) cos(πtu) du.
(5)
0
In fact, this kernel is the Fourier transform of λ ∈ L1 (R),
r
π ∧
λ (πt).
s(t) =
2
(6)
Lemma (Butzer, Splettstößer, Stens’88)
If s ∈ Bπ1 , then for f ∈ C(R)
kSws f − f kC kIws f − f kC .
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
12 / 34
Shannon Sampling Series
Bandlimited kernels
Many kernels can be defined by
Z1
s(t) :=
λ(u) cos(πtu) du,
0
e.g.
1) λ(u) = 1 defines the sinc function;
2) λ(u) = 1 − u defines the Fejér kernel sF (t) = 12 sinc 2 2t (cf. [13]);
3) λj (u) := cos π(j + 1/2)u, j = 0, 1, 2, . . . defines the Rogosinski-type
kernel (see [7]) in the form
1
1 1
sinc(t + j + ) + sinc(t − j − )
(7)
rj (t) :=
2
2
2
4) λH (u) := cos2
πu
2
= 12 (1 + cos πu) defines the Hann kernel (see [8])
sH (t) :=
O. Orlova, G. Tamberg (TUT)
1 sinc t
;
2 1 − t2
Kantorovich-type Sampling Operators
(8)
Strobl 2014
13 / 34
Shannon Sampling Series
Bandlimited kernels
5) the general cosine window
λC,a (u) :=
m
X
ak cos k πu
(9)
k =0
defines the Blackman-Harris kernel (see [9])
m
1X ak sinc(t − k ) + sinc(t + k )
sC,a (t) :=
2
(10)
k =0
provided (here and following bxc is the largest integer less than or
equal to x ∈ R)
m
m+1
bX
bX
2c
2 c
1
a2k =
a2k −1 = .
(11)
2
k =0
k =1
We get Hann kernel (8) if we take m = 1 in (10).
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
14 / 34
Shannon Sampling Series
Bandlimited kernels
6) powers of the Hann window (see [6], formula(25a))
πu λH,m (u) := cosm
2
m m
1 X m
cos (k − )πu ,
= m
2
k
2
(12)
(13)
k =0
give a general Hann kernel in the form
sH,m (t) = 2−m
Γ(1 +
m
2
Γ(1 + m)
− t)Γ(1 +
m
2
+ t)
(14)
.
Comparing the window function λH,m in (13) and the general cosine
window λC,a in (9) we see that the general Hann kernel in case of
m = 2n (n ∈ N) is a special case of the Blackman-Harris kernel.
Indeed, sH,2n = sC,a∗ , where the parameter vector a∗ ∈ Rn+1 has
2n
1
∗
components a0∗ = 212n 2n
n and ak = 22n−1 n−k for k = 1, 2, . . . , n.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
15 / 34
Shannon Sampling Series
Bandlimited kernels
7) the general Rogosinski-type window
λR,a (u) :=
m
X
aj λj (u) =
m
X
aj cos π(j + 1/2)u
(15)
j=0
j=0
defines the general Rogosinski-type kernel
m
1X 2j + 1
2j + 1 aj sinc(t −
) + sinc(t +
)
ra (t) :=
2
2
2
(16)
j=0
provided
m
X
aj = 1.
(17)
j=0
Comparing the window function λH,m in (13) and the general
Rogosinski-type window λR,a in (15) we see that the general Hann
kernel in case of m = 2n + 1 (n ∈ N0 ) is a special case of the general
Rogosinski-type kernel. Indeed, sH,2n+1 = sR,a∗ , where the parameter
vector a∗ ∈ Rn+1 has components aj∗ = 212n 2n+1
n−j for j = 0, 1, . . . , n.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
16 / 34
Shannon Sampling Series
Kernels for singular integrals
Kernels for singular integrals
In this talk we consider for the singular integrals an even kernel
χ ∈ L1 (R) with absolute moment
X
m0 (χ) := sup
|χ(u − k )| < ∞,
u∈R k ∈Z
defined by an even window function µ, µ(0) = 1 by the equality
Z∞
χ(t) := χ(µ; t) :=
µ(u) cos(πtu) du.
(18)
0
In fact, this kernel is also a kernel for generalized sampling operators,
when we additionally require µ(2k ) = 0, k ∈ N.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
17 / 34
Shannon Sampling Series
Kernels for singular integrals
Many kernels can be defined by
Z∞
χ(t) := χ(µ; t) :=
µ(u) cos(πtu) du
0
e.g.
1) µ(u) = sinck
u (k ∈ N) defines the B-spline kernel Bk −1 , in
2
particular case k = 1 we get the indicator function χ[−1/2,1/2] ;
u m−1
P
defines a spline kernel of degree m if
2) µa (u) =
ak sinck +1
2
k =0
P
ak = 1;
u 2
3) µ(u) = e−π( 2 ) defines a Gauss-Weierstraß kernel
2
χGW (t) = e−πt .
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Operator norms
Norms of the Kantorovich-type sampling operators
Norms of the Kantorovich-type sampling operators
Theorem
K = S s (I χ ) we have for
For Kantorovich-type sampling operator Sw,n
w nw
1 6 p 6 ∞ an estimate of the operator norm (1/p + 1/q = 1)
1
1 p
q
K
kχk1 m0 (s)
kSw,n
kp→p 6 nksk1 m0 (χ)
Remark. We see that in the case of generalized sampling operators
Sw (i.e. n → ∞) this estimate is not bounded for 1 6 p < ∞.
Remark. Take χ = χ[−1/2,1/2] . We proved that in this case for L1 (R) we
K k
K
have kSw,n
1→1 = nksk1 and for C(R) we have kSw,n kC→C = m0 (s).
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
19 / 34
Operator norms
Nikolskii’s inequality
Nikolskii’s inequality
Theorem (Nikolskii’s inequality)
Let 1 6 p 6 ∞. Then, for every s ∈ Bσp ,
∞
X
(
kskp 6 sup
u∈R
)1/p
|s(u − k )|
p
6 (1 + σ)kskp .
k =−∞
Our kernels s ∈ Bπ1 . By Nikolskii’s inequality we have
ksk1 6 m0 (s) = kSw k,
which gives the estimate
1
1
p
K
kSw,n
kp→p 6 kSw k n m0 (χ) kχk1q
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
Theorem
If we have for sampling operator Sws
1
)p
w
kSws f − f kp 6 M1 ωk (f ;
and for the singular integral
χ
kInw
f − f kp 6 M2 ω` (f ;
1
)p ,
w
K = S s (I χ )
then for Kantorovich-type sampling operator Sw,n
w nw
K
kSw,n
f − f kp 6 M3 ωr (f ;
1
)p ,
w
where r := min{k , `}.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
21 / 34
Order of approximation
Corollary
K = S s (I χ ), where s is
For Kantorovich-type sampling operator Sw,n
w nw
Rogosinski-type, Hann, Blackman-Harris or B-spline kernel and χ is a
indicator function or Gauss-Weierstraß kernel we have
K
kSw,n
f − f kp 6 Mω2 (f ;
1
)p .
w
Indeed, for Rogosinski-type, Hann, Blackman-Harris and B-spline
kernels we have the estimates of order of approximation for
corresponding generalized sampling operators via modulus of
smoothness order 2. For singular integrals with indicator function and
Gauss-Weierstraß kernels we have the estimates of the order of
approximation via modulus of smoothness order 2.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
22 / 34
Order of approximation
The main theorem
Theorem
K = S s (I χ ) (w > 0, n ∈ N)
Let Kantorovich-type sampling operator Sw,n
w nw
be defined by the kernel s ∈ Bπ1 with window λ and by the kernel
χ ∈ L1 with window µ. If for some r ∈ N
νn (u) := λ(u)µ
u n
=1−
∞
X
cj u 2j ,
∞
X
|cj | 6 ∞.
(19)
j=r
j=r
Then for f ∈ Lp (R) (1 6 p 6 ∞)
K
kSw,n
f − f kp 6 Mr ω2r (f ;
1
)p .
w
(20)
The constants Mr are independent of f and w.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
The main theorem
Idea of the proof
Lemma (cf. Butzer, Stens’85)
p
1 (α + β = 2), then
If g ∈ Bαπw
and s ∈ Bβπ
Sws g = Iws g.
p
p
χ
K = S s (I χ )
Take g ∈ Bπw/n
, then Inw
g ∈ Bπw
and for Sw,n
w nw
χ
K
K
kSw,n
f − f kp 6 kSw,n
kp→p kf − gkp + kIws (Inw
g) − gkp + kg − f kp .
χ
g) = sw ∗ χnw ∗ g = (s ∗ χn )w ∗ g = ϕw ∗ g = Iwϕ g = Swϕ g
Iws (Inw
where ϕ ∈ Bπ1 is defined
Z1
Z1
νn (u) cos πut du =
ϕ(t) :=
0
O. Orlova, G. Tamberg (TUT)
λ(u)µ
u n
cos πut du.
0
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
The main theorem
The space Λp
Definition (Bardaro, Butzer, Stens, Vinti’ 2006)
Let Σ := (xj )j∈Z ⊂ R be an admissible partition of R, i.e.
0 < inf ∆j 6 sup ∆j < ∞, ∆j := xj − xj−1 and let the discrete
j∈Z
j∈Z
`p (Σ)-seminorm of a sequence of function values fΣ on Σ of a function
f : R → C be defined for 1 6 p < ∞ by
kf k`p (Σ) :=

X

|f (xj )|p ∆j
j∈Z
1/p

.

The space Λp for 1 6 p < ∞ is defined by
Λp := {f ∈ M(R); kf k`p (Σ) < ∞ for each admissible sequence Σ}.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
The main theorem
Denote for 1 6 p < ∞ X p (R) := Λp and X ∞ (R) := C(R). Bσp ⊂ X p (R).
Theorem (Kivinukk,T’14)
Let sampling operator Swr (w > 0) be defined by the kernel sr ∈ Bπ1
with λ = λr and for some r ∈ N let
λr (u) := 1 −
∞
X
j=r
cj u 2j ,
∞
X
|cj | 6 ∞.
(21)
1
)p .
w
(22)
j=r
Then for f ∈ X p (R) (1 6 p 6 ∞)
r
kSW
f − f kp 6 Mr ω2r (f ;
The constants Mr are independent of f and w.
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
The main theorem
p
We have for g ∈ Bπw/n
K
kSw,n
g − gkp = kSwϕ g − gkp 6 Mr ω2r (g;
6 Mr ω2r (f ;
1
)p
w
1
1
1
)p + Mr ω2r (g − f ; )p 6 Mr ω2r (f ; )p + Mr 22r kg − f kp .
w
w
w
Theorem (Jackson-type theorem)
Given f ∈ C(R) or f ∈ Lp (R) (1 6 p < ∞). Then there exists a gσ∗ ∈ Bσp
(1 6 p 6 ∞) and a constant Ck > 0 (depending only on k ∈ N) such
that
1
kf − gσ∗ kp 6 Ck ωk (f , )p .
σ
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
Examples
Examples
Theorem
χ
I
Let Cw,a
= Sws (Inw
) be a Kantorovich-type sampling operator whith
s = sC,a (a ∈ Rm+1 ) and χ = χ[−1/2,1/2] , then for f ∈ Lp (R)
(1 6 p 6 ∞)
1
I
kCw,a,n
f − f kp 6 Ma,2 ω2 (f ; )p .
w
Moreover, if there holds
m
X
k =1
ak k 2 = −
1
,
(22 − 1)22 n2
then
I
kCw,a,n
f − f kp 6 Ma,4 ω4 (f ;
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
1
)p .
w
Strobl 2014
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Order of approximation
Examples
The Bibliographic References I
Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Approximation error
of the Whittaker cardinal series in terms of an averaged modulus
of smoothness covering discontinuous signals.
J. Math. Anal. Appl. 316, 269–306 (2006)
Bardaro, C., Butzer, P.L., Stens, R.L., Vinti, G.: Prediction by
samples from the past with error estimates covering discontinious
signals.
IEEE Transactions on Information Theory 56, 614–633 (2010)
Burinska, Z., Runovski, K., Schmeisser, H.J.: On the
approximation by generalized sampling series in Lp -metrics.
Sampling Theory in Signal and Image Processing 5, 59–87 (2006)
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
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Order of approximation
Examples
The Bibliographic References II
Butzer, P.L., Splettstößer, W., Stens, R.L.: The sampling theorems
and linear prediction in signal analysis.
Jahresber. Deutsch. Math-Verein 90, 1–70 (1988)
Butzer, P.L., Stens, R.L.: Reconstruction of signals in Lp (R)-space
by generalized sampling series based on linear combinations of
B-splines.
Integral Transforms Spec. Funct. 19, 35–58 (2008)
Harris, F.J.: On the use of windows for harmonic analysis.
Proc. of the IEEE 66, 51–83 (1978)
Kivinukk, A., Tamberg, G.: On sampling series based on some
combinations of sinc functions.
Proc. of the Estonian Academy of Sciences. Physics Mathematics
51, 203–220 (2002)
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
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Order of approximation
Examples
The Bibliographic References III
Kivinukk, A., Tamberg, G.: On sampling operators defined by the
Hann window and some of their extensions.
Sampling Theory in Signal and Image Processing 2, 235–258
(2003)
Kivinukk, A., Tamberg, G.: On Blackman-Harris windows for
Shannon sampling series.
Sampling Theory in Signal and Image Processing 6, 87–108
(2007)
Kivinukk, A., Tamberg, G.: Interpolating generalized Shannon
sampling operators, their norms and approximation properties.
Sampling Theory in Signal and Image Processing 8, 77–95 (2009)
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
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Order of approximation
Examples
The Bibliographic References IV
Kivinukk, A., Tamberg, G.: On window methods in generalized
Shannon sampling operators.
In: New Perspectives on Approximation and Sampling TheoryFestschrift in honor of Paul Butzer’s 85th birthday. Birkhauser
Verlag (2014 (to appear)).
In press
Tamberg, G.: Approximation error of generalized shannon
sampling operators with bandlimited kernels in terms of an
averaged modulus of smoothness.
Dolomites Research Notes on Approximation 6, 74–82 (2013)
Theis, M.: Über eine interpolationsformel von de la
Vallee-Poussin.
Math. Z. 3, 93–113 (1919)
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
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Order of approximation
Examples
Thank you!
O. Orlova, G. Tamberg (TUT)
Kantorovich-type Sampling Operators
Strobl 2014
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Order of approximation
Examples
[4],[13],[1],[2],[11], [5], [12],[10], [3]
O. Orlova, G. Tamberg (TUT)
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