International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
Available online on http://www.rspublication.com/ijst/index.html
ISSN 2249-9954
Approximation of -periodic an integrable function by the Finit Fourier
Transform Via Vallee Poussin operator interms of Averaged Modulus of
Smoothness
Hussein A.H. Al-juboori and Yasemeen M. Abd-Alhasan
University of Al-Mustansiriya, College of science , Department of Mathematics
Abstract
This paper state estimate the erroe of the Finit Fourier Transform with Vallee
Poussin operator by Averaged Modulus of Smoothness, for -periodic integrale
function defined on [0,1] and state results that relation with it .
1. Introduction
Let
set of sequences of lenght N+1
.
For each N
the Finit Fourier Transform
.
This sequence can be extended to all
The inverse of
and is periodic of period N+1
is given by
.
If is afunction defined on [0,1] ,then the Rieman sum for the integral defining the
Fourier cofficient is
) .1.4)
And
-
We show that the Finit Fourier Transform partial sum
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
Available online on http://www.rspublication.com/ijst/index.html
ISSN 2249-9954
.
Is almost as agood an approximation to f as the usual partial sum
.
Apolynomial operator of degree 2n-1 is afunction of the form
.
.
Where
.
.
.
.
.
If f is an integrable ,1-periodic function defined on [0,1] then for each m
apolynomail operator
of best approximation of f such that
N,there is
[5] Hirstov used alocally global norm for bounded functions and proved that
the best one-sided approximations of 2 -periodic bounded function with
trigonometric polynomials of degree n in the norm
[6] Ivanov obtained some results about approximation of measurable and
bounded functions by bernstein polynomials in
[0,1] space
[7] C. Sanda Cismasiv ,gave some results about approximation by a new linear
positive operator of Durrmeyer type.
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
Available online on http://www.rspublication.com/ijst/index.html
ISSN 2249-9954
2. Background Material
Vallee Poussin is positive linear operator
Theorem (2.1) (de la Vallee Poussin ) [3]
For each positive integer n, there exists a trigonometric polynomial operator
degree 2n-1 with property .
We have some properties of the operators
of
:
1. is atrigonomatric polynomial operator of degree 2n-1,and
.
3.
Indeed,
We need to defined Averaged Moduius of Smoothness [5]:
=
Where
And
Such that
Which called local moduli of smoothness.
Theorem (2.2) (Korneicuk )[3]
For all
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,
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
Available online on http://www.rspublication.com/ijst/index.html
ISSN 2249-9954
If
then
cannt be true for all
and all
n=1,2,....
3. Main results
Theorem (3.1) if f(x) is an integrable ,1-periodic function on [0,1],then the
polynomial operator of best approximation satisfies
Proof
Where
is a polynomial of order M of the form
Since
From theorem (2.2)
Since
[2]
Corollary (3.1) if f is an integrable 1-periodic function on [0,1] with an integrable 1periodic derivatives then the polynomial operator of best approximation satisfies
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
Available online on http://www.rspublication.com/ijst/index.html
ISSN 2249-9954
Theorem (2.3) for
and f ,an integrable 1-periodic function defined on
[0,1],we have
.
Proof : since
is polynomial operator and
We have
Then
=
,
1
.
For the second line
.
.
So that
Corollary (3.2) suppose that f is an integrable 1-periodic function with l
integrabal 1-periodic derivatives; then
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0 an
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
Available online on http://www.rspublication.com/ijst/index.html
ISSN 2249-9954
.
From theorem(2.3)
.
And corollary (3.1)
.
Then
.
.
Corollary (3.3) suppose that f is an integrable 1-periodic function then
.
Proof: from theorem (3.2)
‘
And
Then from theorem(3.1)
.
.
.
Theorem(3.3) there is a universal constant C such that if f is an integrable ,1periodic function then
-
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
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ISSN 2249-9954
Proof: let Dirichlet kernel
.
Then
.
And
.
01
1
.
, +01
1
+
These eq. Bounded by a constant log M
Corollary (3.4) if f is an integrable 1-periodic function whose averaged modulus
satisfies
Proof :
Where o is Lando symbol
Then the (FFT) partial sum (
function derivatives and
the sequence
) convergs to f on [0,1] if f has integrabal periodic
satisfies the above estimate then for each
converges to
Proof : let be a best approximation to f ,we will use the triagle inquality to
conculude that
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International Journal of Advanced Scientific and Technical Research Issue 6 volume 1, Jan. –Feb. 2016
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ISSN 2249-9954
Apply theorem (3.3)
Since
tends to zero as M tends to infinity.
References
[1] Charles L-Epsten "How will does the finit fourier transform approximation the
fourier transform " (2004)Wiley periodicals, Inc
[2] Popov,V.A (1981) " Averaged local moduli and their applications in
approximation and functions spaces",PWN-polish scientific publishers, Warsaw
[3] Lorent G.G (1966) "Approximation of function" ,printed in the (USA)
[4] Jacson,Dunham ,The theory of approximation ,Amer .Math .Soc
[5] Hristov, V.H. (1989), "Best one-sided approximation and mean approximation by
interpolation polynomials of periodic functions" Math.Balk.,New Series,3,418-428.
[6] Ivanov ,K.G.(1984),"Approximation by Bernstein polynomials in L_p-metric
constructive theory" ,Bulg. Acad. Of Sci, 421-429.
[7] C.Sanda Cismasiv, "A new linear positive operator of Durrmeyer type associated
with Bleimann-Butzer-Hahn operator ",Vol. 6(55),No. 1 (1-8) ,(2013).
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