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APPROXIMATION OF THE CONJUGATE OF FUNCTION BELONGING TO LIP CLASS BY (E,1)(C,1) MEANS OF THE CONJUGATE SERIES OF IT’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 Lip 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, Lip class.

INTRODUCTION

A function image
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))
image
Let f be 2 periodic, integrable over (-,) in the sense of Lebesgue and belonging Lip class, then its Fourier series is given by
image and its conjugate series is
image (1)
Let imagebe the infinite series whose nth partial sum is given by image
The Cesàro means (C, 1) of sequence {Sn} is image
If image be the infinite series whose nth partial sum is given by image
The Cesàro means (C, 1) of sequence {Sn} is image
If image then sequence {Sn} or the infinite series image 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 image
If image then sequence {Sn} or infinite series image is said to be summable by Euler means method (E, 1) to S.
The (E, 1) (C, 1) transformation of {Sn}, denoted by image , is given by
image
If image then sequence {Sn} or infinite series image is said to be summable by (E, 1) (C, 1) means method to S.
If a function f is Lebesgue integrable then
image
exist for all x (Zygmund (1959), p. 131). We use following notations.
image
image

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 Lip, 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 image ,~ 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 image the conjugate of a function fLip α by (E,1) (C,1) means
imageof the conjugate series of the Fourier series (1) satisfies, for n=0, 1, 2…,
image

III. LEMMAS

We need the following lemmas for the proof of the theorem.
Lemma 1: image then
image
Proof: image
image (3)

IV. PROOF OF THE THEOREM

The nth partial sum image of conjugate series (1) is given by
image
image transform of the image is given by
image
image (4)
Using Lemma 1 and the fact thatimage, we have,
image(5)
Now, using Lemma 2, we have
image (6)
Collecting (.4), (5), (6); we have
image
or
image
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.