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Approximation of Conjugate Function Belonging to Lip(ξ(t),r)
Class by (C,1)(E,q) Means
Kalpana saxena
1, Sheela Verma
21,2_{Department of Mathematics, Govt. M.V.M. Barkatullah University, Bhopal India }
_{ }
Abstract: In this paper, introduce the concept of (C,1)(E,q) product operators and establishes a new theorems on degree of approximation of a function ̅, conjugate to a 2π periodic function f, belonging to Lip(ξ(t),r) Class by (C,1)(E,q) product means conjugate fourier series have been established.
Keywords And Phrases: Degree of approximation, Lip(ξ(t),r) class of function, (C,1) Summability, (E,q) Summability, (C,1)(E,q) product Summability, fourier series, conjugate fourier series, Lebesgue integral.
I. INTRODUCTION
The degree of approximation of the function belonging to Lipα by Cesaro means and norlund means has been discussed by a number of researches like Alexits[6], Sahney and Goel[9], Chandra[1],\Qureshi[8], Rhoades[4], Hare Krishna Nigam[14] obtained the degree of approximation of the function belonging to Lipα by (C,1)(E,1) product means of its fourier series.
In the present we have extended this result by obtaining the degree of approximation of function f belonging to a generalized class Lipα.. Mittal and singh [7], Rhoades etal. [8], mishra etal. [9,10] and mishra and mishra [11] for function of various classes Lipα, Lip( ,r), Lip(ξ(t),r) and W(Lr,ξ(t)),(r 1) by using various Summability methods. But till now, nothing seems to have been done as far to obtain the degree of approximation of conjugate of a function (C,1)(E,q) product Summability method of its conjugate series of fourier series. In this paper, we obtain a new theorem on the degree of approximation of a function
̃, conjugate to a periodic function f Lip(ξ(t),r) Class , by (C,1)(E,q) product Summability means.
Definition and notation: Let f(x) be periodic with period 2π and integrable in the sense of Lebesgue. The fourier series of f(x) is given by
( ) ∑_{ }( ) (1.1)
With nth partial sums ( )
The conjugate series of the fourier series (1.1) is given by
∑_{ }( ) (1.2)
With nth partial sums ̅̅̅ (f;x) and we shall call it as conjugate fourier series through out of paper
 norm of a function f:R→R is defined by ‖ ‖ * ( ) +
 Norm is defined by ‖ ‖ (∫  ( ) ) , (1.3)
The degree of approximation of a function f:R→R by a trigonometric polynomial tn of order n under Sup norm ‖ ‖ is
defined by Zygmund [13] and is given as
‖ ‖ * ( ) ( ) + (1.4)
And ( ) is given by
( ) ‖ ‖ (1.5)
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f(x+t) ― f(x) = O(  ) for 0 1 (1.6)
f(x) ( ) if
Let ∑ be a given infinite series with the sequence of its nth partial sums * +
The (C,1) transform is defined as the nth partial sum of (C,1) Summability
∑ n (1.10)
Then the series ∑_{ } is summable to S by (C, 1) method
( ) _{( )} {∑_{ }( ) } n (1.11)
Then the infinite series ∑_{ } is said to be summable (E,q) to the definite number S (Hardy[3]),
The (C,1) transform of the (E,q) transform defines (C,1)(E,q) transform of the partial sum of series ∑ and we denote
it by ( ) .
Thus if ( )
∑
=
∑ *( ) ∑ ( ) +
Where denotes the (E,q) of and denotes (C,1) transform of , then the series ∑ is said to be Summable
to(C,1)(E,q) method or summable (C,1)(E,q) to a definite number S.
We use the following notations:
( ) ( ) ( )
= Integral part of * +
̅ ( )
( )∑ [( ) ∑ ( )
( _{ )}
]
II. MAIN THEOREMS
A good amount of work has been done on degree of approximation of functions belonging to Lip ( ) (ξ( ) ) classes using Cesaro, N ̈rlund and generalized N ̈rlund single Summability methods by a number of researchers like Alexits [6], Sahney and Goel DS [9], Qureshi [8], Chandra [1] and Rhoades[4].
But till now nothing seems to have been done so far in the direction of present work. Therefore ,in present paper, a theorem on degree of approximation of a function ̅, conjugate to periodic function f, belonging to
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2.1 Theorem:If f is a periodic function, Lebesgue integrable on [0,2π], belonging to the (ξ( ) ) class then its degree of approximation by ( )( )̅̅̅̅̅̅̅̅̅̅̅̅̅̅ summability means of its conjugate fourier series is given by
‖( ̅̅̅̅) ‖ *( ) (_{ })+ (2.1)
Provided ξ(t) satisfies the following condition:
{∫ (  ( ) ( ))
_{}} _{ (}
) (2.2)
And
{∫ (  ( ) ( ) )
} {( ) }
(2.3)
Where is an arbitrary number such that
Condition (2.2) and (2.3) hold uniformly in and ̅̅̅̅̅̅ ( )( )̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ means of the series (1.2) provided
∑_{ }( ) ( ( ) ) (2.4)
And ̅( )
∫ ( )
III. LEMMAS
For the proof of our theorem following lemmas are required:
Lemma 1:̅̅̅( ) ( )
Proof: For
( ) ( )  
 ̅̅̅( )
( )∑ [( ) ∑ ( )
( _{ ) }
(_{ )}
]

( )∑ [( ) ∑ ( )
⁄
]
( )∑ [( ) ∑ ( )
]
=_{ ( )}
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Lemma2For and any n, we have̅̅̅( ) .
( )/ (( )( ) ∑( )
)
Proof: For _{ } ( ⁄ ) ( ⁄ )
 ̅̅̅( )
( )∑ [( ) ∑ ( )
( _{ )}
(_{ )}
]

( )∑ [( ) {∑ ( )
( _{ ) }
}]

( )∑ [( ) {∑ ( )
}]
  
( )∑ [( ) {∑ ( )
}]

( )∑ [( ) {∑ ( )
}]

( )∑ [( ) {∑ ( )
}]

Now,
 
( )∑ [( ) {∑ ( )
}]

( )∑ [( ) {∑ ( )
}]
  _{}
( )∑ [( ) {∑ ( )
}]
( )∑
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( )
(
( )) (3.2)
Now
Considering second form of (3.1) and using Abel’s Lemma
 
( )∑ [( ) {∑ ( )
}]

( )∑ ( ) ∑ ( )

( )( ) ∑( )
⌈
( )( ) ∑ ( ) ⌉ (3.3)
Combining (3.1),(3.2) and (3.3), we get
̅̅̅( ) (
( )) (( )( ) ∑ ( )
) (3.4)
IV. PROOF OF THE THEOREM:
Let ̅ ( ) denote S, the nth partial sum of the series (1.2). Then following Lal[5], we have,
̅ ( ) ̅( )
∫ ( )
( _{ ) }
Therefore using (1.2). the (E,q) transform of ̅ ( ) is given by
̅̅̅̅ ̅( )
( ) ∫ ( ) {∑ ( )
( _{ ) }
(_{ )}
}
Now denoting ( )( )̅̅̅̅̅̅̅̅̅̅̅̅̅̅̅ transform of ̅ by ( ) ,
We write
( )
̅̅̅̅̅̅̅̅ ̅( )
( )∑( )
∫ ( )
[∑ ( )
_{ ( } _{) }
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∫ ( )̅̅̅( )
Where
̅̅̅( )
( )∑ [( ) ∑ ( )
( _{ ) }
(_{ )} ]
[∫ ∫
] ( ) ̅̅̅( )
(say) (4.1)
We consider,
  ∫  ( ) ̅̅̅( )
Using H ̈lders’s inequality and the fact that
( ) ( ( ) )
Approximation of conjugate function belonging to ( ( ) ) class
[∫ 2  ( )
( ) 3
] [∫ 2 ( ) ̅̅̅( ) 3 ]
(
) [∫ ,
( ) ̅̅̅̅( )

_{]}_{ by (2.2) }
(_{ }) [∫ , ( ) _{]} _{ by lemma1 }
Since ( )is a positive increasing function and using second mean value theorem for integrals,
,(_{ }) (_{ }) [∫ _{ }_{]}_{ }_{for some }_{ }
= ,(
) ( ) 0,
1
= O,(
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= O,( ) () (4.2)
Using H ̈lder’s inequality,
  ∫  ( ) ̅ ( )
dt
  [∫  ( ) ( )
]
+O[∫  ( )
( )( )
∑ ( )
_{ }
] (by lemma 2) (4.3)
=O( ) ( ) say
Using H ̈lder’s inequality and lemma 2
  (_{ }) [∫ 2
_{ ( )}
( ) 3
] [∫ 2 ( )_{ }3
]
= O,( )_{ }  [∫ , ( )_{ }
] by (2.3)
Now putting t=
2
( )
3 [∫ {
(_{ )} ( ) }
_{ }
]
Since ( )is a positive increasing function and using second mean value theorem
For integrals,
,( )_{ } (_{ }) *∫ ( )
+ for some
= O,( )
( ) *∫
( )
+ for some
= O,( )
( ) 0{
( )
( ) }
1
= O,( )
( ) *( ) ( ) _{+}
{ (
)} {( )
_{}}
,( ) (
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0( ) 1 (4.4)
Similarly
  (_{ }) [∫ 2
_{ ( )}
( ) 3
] [∫ 2 ( )( ) ∑ ( )
3
]
= O(_{ }) [∫ , _{ ( )} ( )
] [∫ ,
( )( )

] by 2.4
= O[∫ , ( ) ( )
] [∫ ,
( )

]
= O{( ) } [∫ , ( )_{ }
]
by (2.3)
Now putting t=
{( ) } [∫ {
(_{ )} ( ) }
]
Since ( )is a positive increasing function and using second mean value theorem
For integrals,
,( ) (_{ }) *∫ ( )
+ for some
= ,( ) ( ) *∫
( )
+ for some
= ,( ) ( ) 0{
( )
( ) }
1
{( ) (
)} [( )( ) ]
{ (
)} [( )
_{]}
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Combining (4.1) (4.2) (4.3) (4.4) & (4.5)( )̅̅̅̅̅̅̅̅ ̅( ) {( ) ( )}
Now Using  norm we get,
‖( ̅̅̅̅) ( )‖ 2∫ ‖( ̅̅̅̅) ( )‖
3
= [{∫ ,( ) (
)
} ]
,( ) (_{ }) 0,∫  1
= ,( ) (
)This completes the proof of theorem.
Applications: The following corollaries can be derived from our main theorem:
Corollary 1: If ( ) = , 0 1 , then the class Lip( ( ) ), r 1,
Reduces to the class Lip( ) and the degree of approximation of the function ̅, conjugate to a periodic function f
Lip( ), 1 , is given by( )̅̅̅̅̅̅̅̅ ̅ .
( ) /
Proof: ‖( ̅̅̅̅) ̅‖ ,∫ ‖( ̅̅̅̅) ̅‖
Or
,( ) (_{ }) = ,∫ ‖( ̅̅̅̅) ̅‖
Or
O(1) = ,∫ ‖( ̅̅̅̅) ̅‖  .O{
( ) (_{ })}
Hence
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For if not the right hand side will be O(1) , therefore( )̅̅̅̅̅̅̅̅ ̅ {(
) ( ) }
= O.
( ) /
Corollary 2: If r in Corollary 1, then the class Lip( ) reduces to the Lip class and the degree of approximation of a
function ̅, Conjugate to a periodic function f Lip , 0 1 is given by
‖( ̅̅̅̅) ̅‖ O( ( ) )
Remark: An independent proof of above corollaries 1 can be obtained along the same lines of our theorem.
Acknowledgement
Author is thankful to her family their encouragement and support to this work.
REFERENCES
[1] Chandra, Holder degree of Approximation of function in the metric, Journal of the Indian Maths Soc. Vol. 53(1988) . 99114
[2] Lal S. and kushwaha J. Degree of approximation of Lipschitz function by product Summability method, Int. Math. Forum, 4, 2009, No. 43, 21012107.
[3] Rhoades B.E, On degree of approximation of a function belonging to Lipschitz class, (2003).
[4] Zygmund A. Trignometrical series, Cambridge University Press(1960)
[5] Alexits G., Uber die annaherung einer metigen function durch die Cesaro Schen mittle ihrer Fourierreihe, Math. Ann., 100(1928), 264277.
[6] Qureshi K., On degree of approximation of a function belonging to the Lip class, Indian J. of Pure & Applied maths, 13(1982), 898. [7] Sahney B.N. and God D.S. On degree of approximation of a
function, Ranchi Univ., Math J. 4(1973).
[8] Approximation of Conjugate Function belonging to Lip( ( ) ) Class by (C,1)(E,1) Means.
BIOGRAPHY
Dr. Kalpana Saxena received Ph.D. in Mathematics from University of APS Rewa 1999. Dr. Kalpana Saxena is presently posted as a professor at GOVT. M.V.M. College Bhopal. Her main interests are Product Summability of Fourier series and sequence.
Sheela Verma perusing Ph.D. in