ISSN ONLINE(2319-8753)PRINT(2347-6710)

Binod Prasad Dhakal Associate professor, Central Department of Mathematics (Education), Tribhuvan University, Nepal |

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## Keywords |

Degree of approximation, (E,1) (C,1) Summability, Fourier series, LipÃ¯ÂÂ¡ class. |

## INTRODUCTION |

A function |

The degree of approximation En(f) of a function f: R→ R by a trigonometric polynomial tn of degree n is defined by (Zygmund (1959)) |

Let f be 2Ã¯ÂÂ° periodic, integrable over (-Ã¯ÂÂ°,Ã¯ÂÂ°) in the sense of Lebesgue and belonging LipÃ¯ÂÂ¡ class, then its Fourier series is given by |

and its conjugate series is |

(1) |

Let be the infinite series whose nth partial sum is given by |

The Cesàro means (C, 1) of sequence {Sn} is |

If be the infinite series whose nth partial sum is given by |

The Cesàro means (C, 1) of sequence {Sn} is |

If then sequence {Sn} or the infinite series is said to be summable by Cesàro means (C,1) to S. (Hardy (1913), p.96) |

The Euler means (E, 1) of sequence {Sn} is |

If then sequence {Sn} or infinite series is said to be summable by Euler means method (E, 1) to S. |

The (E, 1) (C, 1) transformation of {Sn}, denoted by , is given by |

If then sequence {Sn} or infinite series is said to be summable by (E, 1) (C, 1) means method to S. |

If a function f is Lebesgue integrable then |

exist for all x (Zygmund (1959), p. 131). We use following notations. |

## II. MAIN THEOREM |

There are several results, for example, Alexits (1965), Chandra (1975), Sahney & Goel (1973) and Alexits & Leindler (1965) for the degree of approximation of functions f Ã¯ÂÂLipÃ¯ÂÂ¡, but most of these results are not satisfied for n= 0, 1 or α = 1.Therefore, this deficiency has motivated to investigate degree of approximation of functions belonging to Lip α considering cases 0< α <1 and α = 1 separately. Considering theses specific cases separately, we have obtained better and sharper estimate of ,~ conjugate of Lip α than all previously known results as follows, |

Theorem: If f: R → R is 2π periodic, Lebesgue integrable function in (-π,π) and belonging to Lip α, 0< α ≤1, then the degree of approximation of the conjugate of a function fÃ¯ÂÂLip α by (E,1) (C,1) means |

of the conjugate series of the Fourier series (1) satisfies, for n=0, 1, 2…, |

## III. LEMMAS |

We need the following lemmas for the proof of the theorem. |

Lemma 1: then |

Proof: |

(3) |

## IV. PROOF OF THE THEOREM |

The nth partial sum of conjugate series (1) is given by |

transform of the is given by |

(4) |

Using Lemma 1 and the fact that, we have, |

(5) |

Now, using Lemma 2, we have |

(6) |

Collecting (.4), (5), (6); we have |

or |

This completes the proof of theorem. |

## V. CONCLUSION |

In this paper a new theorem on degree of approximation of conjugate function f conjugate to a function f belonging to Lipα class has been established by (E,1) (C,1) summability of conjugate series of a Fourier series. |

## References |

[1] Alexits G. and Leindler L.,“ Über die Approximation im starken Sinne (German),” Acta Math. Acad. Sci. Hungar., 16, 27-32, 1965. [2] Alexits G.,“ Über die Annäherung einer stetigen Funktion durch die Cesàroschen Mittel ihrer Fourierreihe,” Math. Ann., 100, 264-277, 1928. [3] Chandra Prem, “On the degree of approximation of functions belonging to the Lipschitz class,” Nanta Math., 8(1), 88 – 91, 1975. [4] Hardy G. H., “On the summability of Fourier series,” Proc. London Math. Soc., 12, 365-372, 1913. [5] Sahney Badri N. and Goel D. S., “ On the degree of approximation of continuous functions,” Ranchi Univ., Math. J., 4, 50-53, 1973. [6] Zygmund A., “Trigonometric series,” Cambridge University Press, 1959. |