APPROXIMATION OF THE CONJUGATE OF
FUNCTION BELONGING TO LIPÃÂ¯ÃÂÃÂ¡ CLASS BY
(E,1)(C,1) MEANS OF THE CONJUGATE
SERIES OF ITÃÂ¢Ãâ¬Ãâ¢S FOURIER SERIES

Binod Prasad Dhakal

APPROXIMATION OF THE CONJUGATE OF FUNCTION BELONGING TO LIPÃÂ¯ÃÂÃÂ¡ CLASS BY (E,1)(C,1) MEANS OF THE CONJUGATE SERIES OF ITÃÂ¢Ãâ¬Ãâ¢S FOURIER SERIES

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

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Abstract

In this In this paper a new estimate on degree of approximation of conjugate function ~ f conjugate to a function f belonging to LipÃ¯ÂÂ¡ class has been determined by (E,1) (C,1) summability of conjugate series of a Fourier series..

Keywords

Degree of approximation, (E,1) (C,1) Summability, Fourier series, LipÃÂ¯ÃÂÃÂ¡ 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 2ÃÂ¯ÃÂÃÂ° periodic, integrable over (-ÃÂ¯ÃÂÃÂ°,ÃÂ¯ÃÂÃÂ°) in the sense of Lebesgue and belonging LipÃÂ¯ÃÂÃÂ¡ 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

be the infinite series whose nth partial sum is given by

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

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

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

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 ÃÂ¯ÃÆÃÅ½LipÃÂ¯ÃÂÃÂ¡, 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 fÃÂ¯ÃÆÃÅ½Lip 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

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

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

APPROXIMATION OF THE CONJUGATE OF FUNCTION BELONGING TO LIPÃÂ¯ÃÂÃÂ¡ CLASS BY (E,1)(C,1) MEANS OF THE CONJUGATE SERIES OF ITÃÂ¢Ãâ¬Ãâ¢S FOURIER SERIES