Degree of Approximation of Function ̃

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ISSN 2091-2730

Degree of Approximation of Function ̃ Class by (E,1) (C,1) Means in the Holder Metric

Santosh Kumar Sinha ¹ and U.K.Shrivastava ²

1. Deptt of Mathematics, Lakhmi Chand Institute of Technology Bilaspur (C.G.) Email – s.sinha2014@gmail.com

2. Deptt of Mathematics, Govt. E.R.R Science P.G. College Bilaspur (C.G.)

Abstract – In this paper, a theorem on degree of approximation of function ̃ class by (E,1) (C,1) means in the Holder metric has been established.

Keywords - Degree of Approximation , Summability Method , Holder Metric, (E,1) mean, (C,1) mean.

1. Introduction

The degree of approximation of a function belonging to various classes using different Summability method has been determined by many Mathematician ,Chandra [3] find the degree of approximation of function by Norlund transform .Later on Mahapatra and Chandra [4] obtain the degree of approximation in Holder metric using matrix transform .In sequal singh et.al. [ 7 ] obtain the error bound of periodic function in Holder metric again Mishra et.al. gave the generalization of result of Singh et.al. In this paper we find the degree of approximation of function ̃ by (E,1) (C,1) means in holder metric.

2. Definition

For a - periodic signal periodic integrable in the sense of Lebesgue then the Fourier series of is given by ∑ …….(2.1)

The conjugate series of Fourier series (2.1) is given by

∑ ……(2.2)

Let and denote two given modulai of continuity such that

( ) ……..(2.3)

Let denote the Banach Space of all 2 - periodic continuous function defined on [ ] under sub-norm the space [0,2 ] includes the space For some positive constant k the function space is defined by

{ | | | | } ……(2.4) With norm ‖ ‖ defined by

‖ ‖ ‖ ‖

……….(2.5) Where and are increasing function of and

‖ ‖

| | and | |

| | x …………(2.6)

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{ | | | | } ………(2.7)

Is Banach space and metric induced by norm ‖ ‖ and is said to be Holder metric clearly is a Banach space which decreases as increases that is

for ……..(2.8)

An infinite series∑ is said to be (C,1) summable to s if

∑ ……..(2.9)

The (E,1) transform is defined by

∑( ) …….(2.10)

The (E,1) transform of (C,1) transform defined ( is given by

∑( ) ……(2.11)

3.Known Results

Singh and Mahajan [7 ] established the following theorem to error bound of signal passing through (C,1)(E,1) transform.

Theorem 1 – Let defined (2.4) be such that

∫ { } …….(3.1)

∫ { } ………(3.2) Then for and we have

‖ ‖ 2(

) 3 ……..(3.3) Theorem 2 – Consider w(t) defined (2.4) and for and we have

‖ ‖ {(

)

∑ ( )

}

………(3.4)

In sequal Mishra and Khatri [11 ] gave the generalized result of above theorem . They proved the following . Theorem 3 – Let defined (2.4) be such that

∫ { }

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∫ { }

Let Np be the Norlund summability matrix generated by the non –negative {Pn} such that (n+1)pn = o(Pn) Then for ̅ we have

‖ ̅ ‖ {^{| |}

| | ( (

))

} ……..(3.5)

And if w(t) satisfies (3.1) then for ̅ we have

‖ ̅ ‖ {^{| |}

| | (

) ((

) ∑ (

))

}

s …….(3.6) 4 .Main Theorem

In this paper we have to prove a theorem on the degree of approximation of a function f(x) conjugate to a - periodic function f belonging to ̅ class by (E,1) (C,1) mean of conjugate series of its Fourier series.

Theorem 1 – Let satisfy the following condition

∫ { } …….(4.1)

∫ { } ………(4.2) Then for ̅ we have

‖ ̅ ‖ { | |

| | . (

)/

}

……….(4.3) 5 Lemma

In order to prove our main result ,we require the following lemma.

Lemma 1 - For 0

̅̅̅̅(t) = o ( ) ………(5.1) Proof - - For 0

, ( ) and | |

̅̅̅̅ ∑ { ( )

∑ ( ) ( )

}

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∑ { ( )

∑

}

= o ( )

Lemma 2 - For

̅̅̅̅(t) = o ( ) ………(5.2) Proof - For

, ( ) and | |

̅̅̅̅ ∑ { ( )

∑ ( ) ( )

}

∑ , ( )

∑ ( ) -

∑ 2 ( )

.

/ 3

. ∑ ( )

(

) Lemma 3 – If w(t) satisfies (4.1) and (4.2) then

∫ ( ) ……(5.3) Lemma 4 – If then for ̅ we get

| | | | …….(5.4)

| | (| |) …….(5.5)

6. Proof of Theorem

Let ̅ denote the partial sum of series ∑ . Then we have

̅ ̅

∫ ⁽⁾

() ….(6.1)

The (C,1) mean of ̅ is given by

̅̅̅̅̅̅ ̅ ∫

()∑ ( ) …….(6.2)

Now (E,1) (C,1) transform of ̅ is denoted by we can write as

̅ ∑ 0( ) ∫

()(

) ,∑ ( ) - 1

∫ ̅ …..(6.3)

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Where ̅ ∑ 2( ) (

) ∑ ⁽⁾

( )

3

…(6.4)

| | ∫ | | ̅

=[∫ ∫

] | | ̅

= (Say) ….(6.5)

Using (5.5) and (5.1) assume that w(t) satisfies (4.1) and (4.2); we get

∫ | | ̅

= ∫ = (

) ….(6.6)

Using (5.5) and (5.2) assume that w(t) satisfies (4.1) and (4.2); we get

∫ | | ̅

= (

) ∫

= (

) ….(6.7)

Now using (5.4) and (5.1) we get

∫ | | ̅

= | | ∫ = | |

Again using (5.4) and (5.2) we get

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∫ | | ̅

= (^{| |}

) ∫

= | | ……..(6.9)

Using the fact that we can write Combining(6.6) and(6.8) we get

.* (

)+ | | / ……(6.10) Combining(6.7) and(6.9) we get

.* (

)+ | | / ……(6.11)

Now from (2.7),(6.10) and (6.11) we have

| |

| | | |

{^{| |}

| | ( (

))

} ….(6.12)

Since ‖ ‖

| ̅ | ….(6.13)

From (6.6) and (6.7) we get

‖ ‖ ( (

)) …..(6.14)

Combining (6.12) and (6.14) we get

‖ ̅ ‖ {^{| |}

| | ( (

))

}

This complete the proof of theorem.

REFERENCES:

[1] A.Zygmund,” Trigonometric Series,Second” Edition, Combridge Univ.Press Press Combridge, 1968.

[2] K.Qureshi , “On degree of approximation to a function belonging to the class Lipα”,Indian Jour.of Pure Appl. Math., 13, No.8,PP.898 ,1982.

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[3] P.Chandra,”On the generalized Fejer means in the metric of Holder space,” Mathematische Nachrichten, vol.109,no.1,pp. 39- 45,1982.

[4] R.N.Mohapatra and P.chandra”Degree of approximation of function in Holder metric “ Acta Mathematica Hungaria,vol.41,no.1- 2,pp. 67-76,1983.

[5] P.Chandra,”Degree of approximation of functionin the Holder metric by Borel Means”,Journal of Mathematical Anal. And Applications,Vol.149,Issue 1, pp. 236 – 248,1990.

[6] G.Das,T.ghosh and B.K.Ray,”Degree of approximation of function in the Holder Metric by (e,c) means” Proceedings of the Indian Academy of Science,vol.105 pp.315-327,1995.

[7] G.Bachman,L.Narici and E.Beckenstien, “Fourier and Wavelet Analysis”, Springer, New York,NY,USA, 2000.

[8] T. Singh and P. Mahajan,”Error bound of periodic signal in the Holder metric,” International journal of mathematics and Mathematical Science, vol. 2008, article ID 495075, 9 pages,2008.

[9] Santosh Kumar Sinha and U.K.Shrivastava “Approximation of conjugate of Lip . ᶓ(t) ,p) function by Almost (N,p,q) SummabilityMethod” Int. J. Math. Sci.&Appl.,Vol.2 .No 2, PP.767-772, May2012.

[10] Santosh Kumar Sinha and U.K.Shrivastava “ Approximation of Conjugate of Function. Belonging to W(Lr ,ᶓ(t)) Class by (E,2) (C,1) Means of Conjugate Fourier Series „‟ Int. J. Sci .&Res.(IJSR), Vol 3 Issue 7, PP.17-20, July 2014.

[11] Santosh Kumar Sinha and U.K.Shrivastava “ The Almost (E,q) (N,Pn) Summability of Fourier Series” Int.

J.Math.&Phy.Sci.Research, Vol 2, Issue 1,PP.553-555, Apr – Sept 2014.

[12] Vishnu Narayan Mishra and Kejal Khatri,”Degree of Approximation of Function ̅ Class by the ( ) Means in the Holder Metric,” international journal of mathematics and Mathematical Science, vol. 2014, article ID 837408, 9 pages 2014