Degree of Approximation of Continuous Functions by (E, q) (C, δ) Means

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1 Ge. Math. Notes, Vol. 11, No. 2, August 2012, pp ISSN ; Copyright ICSRS Publicatio, Available free olie at Degree of Approximatio of Cotiuous Fuctios by (E, q) (C, δ) Meas Rata Sigh 1 ad S.S. Thakur 2 1 Departmet of Applied Mathematics Gya Gaga College of Techology, Jabalpur , Idia rataverma8@gmail.com 2 Departmet of Applied Mathematics Jabalpur Egieerig College, Jabalpur , Idia samajh_sigh@rediffmail.com (Received: /Accepted: ) Abstract I this paper, we obtai a theorem o the degree of approximatio of fuctio belogig to the Lipschitz class by (E, q) (C, δ) product meas of its Fourier series. Our theorem provides the Jackso order as the degree of approximatio. Keywords: Ces ro matrix, Euler matrix, degree of approximatio. 1 Defiitio ad Notatios Let be 2π periodic ad L- itegrable over. The Fourier series of at a poit is give by A fuctio ( ) if

com 2 Departmet of Applied Mathematics Jabalpur Egieerig College, Jabalpur-482011, Idia E-mail: samajh_sigh@rediffmail.

2 13 Rata Sigh et al. (1.2). It may be observe that such fuctios are ecessarily cotiuous. The degree of approximatio of a fuctio by a trigoometric polyomial of order is defied by Zygmud [12, p-114], (1.3) Let be give ifiite series with the sequece of partial sums of its first (+1)-terms. The Euler meas of the sequece are defied by where is defied to be. If ; as, we say that or is summable ( ) to s or symbolically we write, for. See Hardy [8, p-180] ad for real ad complex values of, see Chadra [5]. if The sequece is said to be summable to limit s where are the biomial coefficiets. See Zygmud [12, p-76]. The trasform of the trasform defies the trasform of the partial sums of the series. Thus if The trasform reduces to ad respectively for ad E C = + q q q A A s s v v k k v= 0 v k= 0 ( ) ( ) ( ) 1 v δ δ 1 1 δ as. The the series is said to be summable by meas or simply summable to s.

If ; as, we say that or is summable ( ) to s or symbolically we write, for. See Hardy [8, p-180] ad for real ad complex values of, see Chadra [5].

3 Degree of Approximatio of Cotiuous 14 Let be the th partial sum of the series (1.1). The mea of, where ad, is give by (1.4) ; = 1 + ; v= 0 v k = 0 1 v 1 ( E C ) ( f x) ( q q ) q δ δ ( Av ) A s v k k ( f x δ ). We shall use the followig otatios for each : (1.5). 2 Itroductio The degree of approximatio of fuctios belogig to, by Ces ro meas ad Nörlud meas have bee discussed by a umber of researchers like Lebesgue [9], Alexits [1] ad Chadra [6]. I 1910, Lebesgue [9] proved the followig : Theorem A: If (2.1) I 1961, Alexits [1, p-301] proved the followig alog with other results. Theorem B: If (2.2) where ad is -mea of. The case was proved by Berstei [3]. I 1981, Chadra [6] proved the followig : Theorem C: If (2.3) The estimate i (2.3) was improved by Chadra [7]. I 2010, Nigam [10] obtaied the followig result o product summability method:

2 Itroductio The degree of approximatio of fuctios belogig to, by Ces ro meas ad Nörlud meas have bee discussed by a umber of researchers like Lebesgue [9], Alexits [1] ad Chadra [6].

4 15 Rata Sigh et al. Theorem D: If (2.4) ad, Tiwari ad Bariwal [11] proved the followig for meas of its Fourier series. Theorem E: If (2.5) ad I this paper we obtai a theorem o the degree of approximatio of cotiuous fuctios by meas of its Fourier series. This geeralizes the result for ad meas. Theorem: If 3 Lemmas We shall use the followig lemmas i the proof of the theorems: Lemma 1[12, p-94]: For, (3.1), where depedig o oly. Lemma 2[4]: For, Lemma 3: For, Proof: By (1.8), we have

5) ad I this paper we obtai a theorem o the degree of approximatio of cotiuous fuctios by meas of its Fourier series.

5 Degree of Approximatio of Cotiuous 16 where is mootoic decreasig it gives maximum value at k=0, by Abel s lemma This completes the proof of the Lemma. 4 Proof of the Theorem The th partial sum of the series (1.1) (see Zygmud [12, p-50]) is, The say. Now, for,, see Zygmud [12, p-91],

6 17 Rata Sigh et al. We have by Boos [2, p-104], by (1.2), we have (4.1) by, we have Case-I: for, by Lemma 1, we have by Lemma 2 ad (1.2), we get Coditio I: whe, (4.2) Coditio II: whe,

7 Degree of Approximatio of Cotiuous 18 (4.3) Combiig (4.2) ad (4.3), we have Case-II: for, by Lemma 3, we have By Lemma 3 ad (1.2), we have (4.5) Now, collectig the estimate (4.1), (4.4) ad (4.5) we get required result (2.6). Refereces [1] G. Alexits, ber die a herug eier statige fuctio druch die ces rosche mittel ihrer fourier-reihe, Math. A., 100(1928), [2] J. Boos, Classical ad Moder Methods i Summability, Oxford Uiv. Press, (2000). [3] S.N. Berstei, Sur l ordre de la meilleure approximatio des foctios cotiues par des poly mes de degré doé, Mém. Royale Acad. Belgique, 2(4) (1912), [4] P. Chadra, O the absolute Euler summability of cojugate series of a fourier series, Periodica Math. Hugarica, 3(3-4) (1973), [5] P. Chadra, O some summability methods, Bollettio della Uioe Mathematica Italiaa, 4(3)(12) (1975), [6] P. Chadra, O the degree of approximatio of cotiuous fuctios, Comm. Fac. Sci. Uiv. Akara, Sér A, 30(1981), [7] P. Chadra, Degree of approximatio of cotiuous fuctios, Riv. Mat. Uiv. Parma, 4(14) (1988), [8] G.H. Hardy, Diverget Series, Oxford, (1949). [9] H. Lebesgue, Sur la represetatio trigoométrique approchée des foctios satisfaisat à ue coditio de Lipschitz, Bull. Soc. Math. Frace, 38 (1910),

Boos, Classical ad Moder Methods i Summability, Oxford Uiv. Press, (2000). [3] S.N. Berstei, Sur l ordre de la meilleure approximatio des foctios cotiues par des poly mes de degré doé, Mém.

8 19 Rata Sigh et al. [10] H.K. Nigam, Degree of approximatio of fuctios belogig to class ad weighted class by product summability method, Surveys i Mathematics ad its Applicatios, 5 (2010), [11] S.K. Tiwari ad C. Bariwal, Degree of approximatio of fuctio belogig to the Lipschitz class by almost (E, q)(c, 1) meas of its fourier series, It. J. Math. Archive, (1)(1) (2010), 2-4. [12] A. Zygmud, Trigoometric Series, Vol. I (Secod Editio), Cambridge Uiversity Press, Lodo/New York, (1968).

Mathematics ad its Applicatios, 5 (2010), 113-122. [11] S.K. Tiwari ad C.

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