5479
Classical and Approximate Theorems of Weighted Space Sampling
Ban Mohammad Hassan 1, Zainab Hasan Msheree2, Hayder Abdulameer abbas3
1Middle Technical University, Balad Technical Institute
2Middle Technical University, Technical Instructors Training Institute 3Middle Technical University, Balad Technical Institute
Article History: Received: 10 January 2021; Revised: 12 February 2021; Accepted: 27 March 2021;
Published online: 28 April 2021
Abstract: We show the same findings in this article, for the authors introduced in 2005 and 2004 that proved the classical
sampling theorems, in the space
B
pw ,1
p
and Bulzer show that three variants of the indicative analysis sampling theorem are similar in the significance that one can be shown as a corollary of one of the others. in 2014 [1] and [2], but in this work for any function in the spaceB
1,,w the space of all functions which are integral by weight function. Consequently, the two sampling theorems in the universal standard are fully identical. The outcome seems to be that our work is successful.Keyword: Sampling Theorem, Indicative Analysis, Weighted Space.
1. Introduction
With its corresponding aliasing error, the approximate sampling theorem is due to J . L .L. L. Black' (1957)' . The classic Whittaker Kotel, nikov, Shannon theorem contains these theorems, with,
,
0
,
1
,
pwp
w
Take into consideration the approximate sampling theorem.For
1
p
, define the space
p =
)
(
)
(
;
)
(
)
(
^ p pL
L
C
L
,With
1
/
p
+
1
/
p
=
1
, and forw
> 0, the Bernstein feature strap space is limited to
−
w
,
w
,
(
)
(
)
;
,=
L
C
P P w
supp.
w
,
w
^−
. Hither=
− R iudu
e
u
(
)
(
1
/
2
)
(
)
^. The Fourier transformation of ψ should be measured
in or under distributional ten, P > 2. The stipulation
(
)
^
PL
as specified in the description of
p is Always grateful for the1
P
2
2- Assertions
Theorem A: Classical Theorem on Sampling
Let
B
P,w,
1
P
,
0
then)
(
)
(
sin
)
(
)
(
=
−
t
k
wt
w
k
t
z k
--- (1)The sequence converges completely and universally on
. Sampling sequence of samples is the group in (1). ψ measurements. For this theorem's history, see e.g. [3, 4].If ψ the theorem A is no longer valid. Band not limited. Though at least it can hold on to the cap
w
→
, (see [3, 5, pp. 95, 118 – 122]).Theorem B: Approximate Sampling Rule or Standardized [6]
5480
+ − −−
=
w n w n t i z n n w it wt
e
e
d
R
(2 1) ) 1 2 ( 2)
(
)
1
(
2
1
)
(
)
(
Then)
(
)
(
)
(
)
(
sin
)
(
)
(
=
−
+
t
t
k
wt
w
k
t
w z k
--- (2)Along with the estimate of errors
)
(
)
(
2
)
(
)
(
^
t
d
t
w w
--- (3)Specifically, one has
→−
=
z k wk
wt
w
k
t
)
(
)
sin
(
)
(
lim
--- (4) Equally fort
Note that the "aliasing mistake" (
(
w
)(
t
)
).Treated in the monumental 1908 article by de La Valle Poussin (see [7] pp: 65–156 For publication of his article, and pp. 421-453 Input from a commentary by Butzer –Stens ) .
3. Main Results
For
1
p
,
0
, let ψ be any function & define
=
=
P p p Pd
L
1/ , ,.
)
(
:
{
And define
exists
p
=
:
, Desire of space
=
)
(
)
(
,
)
(
)
(
^ 1 , , ,
p
p pL
C
L
L
Where
The set of all actual numbers and then all real numbers is C (
) space defined for the set of all complex continuous, valued functions
.The "Bernstein space" of function band Limited to
−
w
,
w
,is St.
−
=
L
C
w
w
B
pw
p
(
)
(
)
,
sup
,
^ , , ,
Here=
− R u idu
e
u
)
.
.
.
(
)
2
/
1
(
)
(
^
, denotes the transformation of Fourier of ψ.In the description of ( , ( ) ^
pL
), the condition (
p,) is always fulfilled for 1 p
2
. Now we have to show that:Theorem 1 Let
B
p,,w,
1
p
,
0
, then
−
=
z kt
k
wt
Sin
w
k
t
)
(
)
(
)
.
(
)
(
--- (5) The series is)
(
)
(
sin
)
(
wt
k
t
w
k
t
z k
− =−
=
Proof5481
−
=
z kk
wt
Sin
w
k
G
t
G
(
)
(
)
(
)
Thus(
)
(
)
(
)
sin
(
wt
k
)
w
k
w
k
t
G
z k−
=
=
end this series is absolute conv. thus
−−
=
z kt
k
wt
w
k
t
)
(
)
sin
(
)
.
(
)
(
Theorem 2Let
p,,
1
P
,
0
and let
+ − −−
=
w n w n it z n wn it wt
e
e
d
(2 1) ) 1 2 ( ^ 2.
)
(
.
.
)
(
)
1
(
2
1
)
)(
(
Then)
(
)
(
)
(
)
(
)
(
sin
)
(
)
(
=
−
−+
wt
k
t
t
t
w
k
t
w z k
--- (6)In comparison to the error estimation
w wt
(
)
d
2
)
)(
(
^(
t
)
ProofLet
G
=
(
t
)
.
(
t
)
then G is bounded
+ −−
=
z n w n w n it wn it wG
t
e
G
e
d
R
) 1 2 ( ) 1 2 ( ^ 2.
.
)
1
(
2
1
)
(
)
(
+ − −
=
w n w n z n wn itv
d
v
e
ˆ ) 1 2 ( ) 1 2 ( 2)
(
)
(
)
1
(
2
1
Then
+
−
=
z k wG
t
t
R
R
k
wt
w
k
G
t
G
(
)
(
)
sin
(
)
(
)(
)
(
)
and by theorem B
d
e
e
t
it w n w n z n wn it w(
1
)
(
)
.
.
(
)
2
1
)
(
)
(
) 1 2 ( ) 1 2 ( ^ 2
+ − −−
=
+
−
=
z k wt
t
t
k
wt
Sin
w
k
t
)
(
)
(
)
.
(
)
(
)
(
)
(
)
(
Here we inquire the other way around: Does hypothesis 1 mean theorem 2, the outcome of theorem 3, but we have to prove the following outcome:
Lemma 1 Let
=
w v t id
e
t
(
)
.
.
(
)
2
1
)
(
^ 25482
)
(
).
(
)
(
)
(
)
(
2 2Sin
wt
k
t
w
k
t
S
k w
=
−
− = = 2(
)
Sin
(
wt
k
)
.
(
t
)
w
k
N N k NLim
−
− = → --- (7)For everything converging (
t
R
) and rewritable as:
+ − −
=
n w w n it z n wn it wt
e
f
e
d
t
R
S
(2 1) ) 1 2 ( ^ 2 2(
).
.
(
)
(
)
2
1
)
(
)
(
ProofThe (
t
R
) periodic extension of the function (2
w
) from the interval (
→
e
it.
(
)
) to the entire real axis R is for fixed letter−
w
,
w
gt, i.e.)
]
)
1
2
(
,
)
1
2
((
(
)
(
e
( 2 )j
w
j
w
j
z
g
t=
it wn
−
+
−
--- (8)Gt's Fourier coefficients are sin (
wt −
k
)k
Z
and, since 𝑔𝑡 is of bounded variety, At each continuity pointin its sequence of Fourier converges to (𝑔𝑡) with partial amounts bound evenly (see [5 p, p .28]
Consequently, we have
− =−
=
k w ik tSin
wt
k
t
e
g
(
)
(
)
.
(
)
.
/ (
(
2
j
+
1
)
w
,
j
Z
)
--- (9)
− =
−
k w ikC
N
n
R
C
e
t
k
wt
Sin
(
)
(
)
.
/(
,
,
0
--- (10)Now for both the series in issue, it have (9)
− = →−
N N k Nt
k
wt
Sin
w
k
Lim
2(
)
(
)
.
(
)
=
− = →−
N N k w w ikv Nk
wt
Sin
d
e
Lim
(
)
.
.
(
)
(
)
2
1
^ /
=
−
=− → w N N k w ik Nd
k
wt
Sin
e
Lim
(
)
.
(
)
.
(
)
2
1
^ / =
→ =−−
w N N k w ik Nd
k
wt
Sin
e
Lim
(
)
.
(
)
.
(
)
2
1
^ / =
w td
g
(
)
(
)
2
1
^ --- (11)Lebesgue's dominant principle of approximation justifies the trade of limit and integral (10).
)
(
)
(
)
(
).
(
.
)
(
)
(
1 ^ / ^N
n
R
L
c
e
k
wt
Sin
f
N N k w k i
−
−
− =
This illustrates the Lemma's first section. The second part is now easy to follow since, in view of (10)
=
w t wt
g
d
S
(
)
.
(
)
2
1
)
(
)
(
^ 2 =
+ − z n w n w n td
g
) 1 2 ( ) 1 2 ( ^)
(
)
(
2
1
5483
=
− − − w n w n w n it z nd
e
) 1 2 ( ) 1 2 ( ) 2 ( ^)
.
)
(
.
)
(
2
1
The final equation is correct (8)
Theorem 3
In the presumption (
F
p,1
p
), (1) implies the approximate sampling theorem theorem (2)Proof
First assume
p
2
, since
F
p,,
0
The reversal form for Fourier
−=
d
e
t
(
)
.
i t.
(
)
2
1
)
(
^ =
+
w v v w t i t id
e
d
e
2
(
)
.
.
1
)
(
.
.
)
(
2
1
^ ^ = ψ1 (t) + ψ2 (t) --- (12) Say now(
)
^R
L
p
implies
1
B
pw, , and therefore it can be extended with the classical sampling theorem (Theorem 1).
− ==
−
=
k wt
S
k
wt
Sin
w
k
t
)
(
)
(
)
.
(
)
(
)
(
)
(
1 1 1
With respect to the Lemma sample sequence f2 (1)
+ − −
=
w n w n it z n wn it wt
e
e
d
t
R
S
(2 1) ) 1 2 ( ^ 2 2(
)
.
.
(
)
(
)
2
1
)
(
)
(
)
(
)
(
.
)
(
)
(
)
(
1Sin
wt
k
t
2t
w
k
t
k
=
−
+
− = =
−
−
−
− = k wt
t
S
k
wt
Sin
w
k
)
(
)
(
)
(
)
(
)
(
2
2
--- (13)As regards the word in curly brackets, we get the definition of f2 by our Lemma
+ − −=
−
(2 1) ) 1 2 ( ^ 2 2 2(
)
.
.
(
)
2
1
)
(
)
(
)
(
n w n it z n wn it wt
t
e
e
d
S
=
d
e
w t i)
(
.
.
)
(
2
1
^
−
=
+ − −−
z n w n w n it wn itd
v
e
e
(
1
)
(
)
.
.
(
)
2
1
(2 1) ) 1 2 ( ^ 2 = - (Rw ψ) (t)If (2) and a simple approximate rest (row)(t) of the error is inserted into (13), then ((Rwψ) (t)) if (
1
p
2
), (p
2
) and the inference, ( ,(
)
^
R
p
L
p
) doesn’t normally indicate (ψ1) as a consequence of which iv(1) is not actually (
1
L
p,(
R
)
) and the reasons mentioned above are not applicable;References
A. P.L. Butzer, J.R. Higgins, R.L. Stens (classical and approximate sampling theorems, studies in the
( )
R
5484
B. P.L.Butzer , The class. And Approximate sampling theorems andTheir equivalence for entire fun. Of exponential type Journal of Approximation Theory .Volume 179 , February 20014,Pages 94- 111. C. P.L. Butzer , W. SplettstoBer , R. L. Stens , the sampling theorem and Linear predication in signal
analysis , Jahresber . Deutsch . Math. – Verein . 90 (1988) 1- 70 .
D. P. L. Butzer , G. Schmeisser , R. L. Stens , An introduction to sampling analysis , in : F. Marvasti (Ed.), Nonuniform sampling , Theory and practice , Information Technology : Transmission , Processing and Storage , kluwer Academic Publishers , Plenum Publishers , New York , 2001 , pp. 17 – 121 .
E. A. Zygmund Trigonometrical Series , Dover publications , New York , 1955 .
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