A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES

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1 Bulleti of Mathematical Aalysis ad Applicatios ISSN: , URL: Volume 8 Issue 42016), Pages A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES ŞEBNEM YILDIZ Abstract. The aim of this paper is to geeralize a mai theorem dealig with local property of Fourier series to the A, θ k summability. Also some ew ad kow results are obtaied dealig with some basic summability methods. 1. Itroductio Let a be a gie ifiite series with partial sums s ), ad let p ) be a sequece of positie umbers such that P = p p as. 1.1) The sequece-to-sequece trasformatio T = 1 P s 1.2) defies the sequece T ) of the Riesz mea or simply the N, p ) mea of the sequece s ) geerated by the sequece of coefficiets p ) see [6]). The series a is said to be summable N, p, θ k, k 1, if see [9]) =1 =0 T T 1 k <. 1.3) I the special case whe θ = P p ad θ =, we obtai N, p k see [1]) ad R, p k see [3]) summabilities, respectiely. Also, if we take θ = ad p = 1 for all alues of, the we get C, 1 k summability see [5]). Let f be a periodic fuctio with period 2π, ad Lebesgue itegrable oer π, π). Without loss of geerality, we may assume that the costat term of the Fourier 2000 Mathematics Subject Classificatio. 26D15, 40D15, 40F05, 40G99, 42A24. Key words ad phrases. Summability factors, absolute matrix summability, ifiite series, Fourier series, Hölder iequality, Mikowski iequality, local property. c 2016 Uiersiteti i Prishtiës, Prishtië, Kosoë. Submitted Noember 6, Published December 20, Commuicated by Huseyi Bor. 91

2 92 ŞEBNEM YILDIZ series of f is zero, that is =1 π π ft)dt = 0, ft) a cost + b sit) = C t). 1.4) A sequece λ ) is said to be coex if 2 λ 0 for eery positie iteger, where λ = λ λ +1. Gie a ormal matrix A = a ), we associate two lower semimatrices Ā = ā ) ad  = â ) as follows: =1 ad ā = a i,, = 0, 1,... a = a a 1, a 1,0 = 0 1.5) i= â 00 = ā 00 = a 00, â = ā = ā ā 1,, = 1, 2, ) It may be oted that Ā ad  are the well-kow matrices of series-to-sequece ad series-to-series trasformatios, respectiely. The, we hae ad A s) = a s = =0 A s) = ā a 1.7) =0 â a. 1.8) Let A = a ) be a ormal matrix, i.e., a lower triagular matrix of ozero diagoal etries. The A defies the sequece-to-sequece trasformatio, mappig the sequece s = s ) to As = A s)), where A s) = =0 a s, = 0, 1, ) =0 Let θ ) be ay sequece of positie real umbers. The series a is said to be summable A, θ k, k 1, if see [8]) where =1 θ k 1 A s) k <, 1.10) A s) = A s) A 1 s). 1.11) Remark. If we take θ = P p ad a = p P, the we get N, p k summability. Also, if we take θ = ad a = p P, the we get R, p k summability.

3 A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES The Kow Results Some kow results hae bee proed dealig with local property of Fourier series see [2], [11]). Furthermore, i [4], Bor has proed the followig result. Theorem 2.1. Let k 1 ad p ) be a sequece satisfyig the coditios P = Op ) 2.1) P p = Op p +1 ). 2.2) If θ ) is ay sequece of positie costats such that m θ 1 P λ ) k = O1), 2.3) m θ λ = O1), 2.4) P m θ 1 P λ +1) k = O1), 2.5) ) k 1 θ ) ) k 1 θ p p 1 = O, 2.6) P P 1 =+1 P the the summability N, p, θ k of the series C t)λ P /p at a poit ca be esured by local property, where λ ) is coex sequece such that 1 λ is coerget. By usig the aboe result, Sarıgöl has obtaied the followig theorem see [7]). Theorem 2.2. Let k 1 ad let p ) be a sequece satisfyig the coditios =1 P P /p ) = O1/). 2.7) Let λ ) be a coex sequece such that 1 λ is coerget. sequece of positie costats such that =+1 m m θ p P P P If θ ) is ay k λ < 2.8) ) k λ < 2.9) θ ) ) k 1 1 p P P 1 = O P P, 2.10) the the summability N, p, θ k of the series C t)λ P /p at a poit ca be esured by local property of f. I [10], Sulaima has proed the followig theorem coerig all the results before this. =1

4 94 ŞEBNEM YILDIZ Theorem 2.3. Let k 1, ad let the sequeces p ), θ ), λ ) ad ϕ ) where θ > 0, are all satisfyig the followig coditios 1 =1 θ 1 1/k ϕ =+1 p P =1 λ +1 = O λ ), 2.11) ) k λ k ϕ k <, 2.12) λ k ϕ k <, 2.13) P the the summability N, p, θ k of the series esured by local property of f. ) 1/k) 1 λ <, 2.14) ) k 1 θ ) ) k 1 θ p p 1 = O, 2.15) P P 1 =1 3. The Mai Result P P C t)λ ϕ at a poit ca be The aim of this paper is to geeralize Theorem 2.3 for A, θ k summability factors of Fourier series i the followig form. Theorem 3.1. Let k 1 ad let A = a ) be a positie ormal matrix such that a o = 1, = 0, 1,..., 3.1) a 1, a, for + 1, 3.2) a = O p ), P 3.3) 1 a â,+1 = Oa ). 3.4) If the coditios 2.11)-2.14) of Theorem 2.3 are satisfied ad θ ) holds the followig coditios, θ a â,+1 = O { θ a }, 3.5) =+1 =+1 θ a a = O { θ a a }, 3.6) the the series C t)λ ϕ is summable A, θ k, k 1. Proof of Theorem 3.1 Proof. Let I ) deotes the A-trasform of the series =1 C t)λ ϕ. The, by 1.7) ad 1.8), we hae I = â a λ ϕ.

5 A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES 95 Applyig Abel s trasformatio to this sum, we get that 1 I = â λ ϕ ) a r + â λ ϕ r=1 1 = â λ ϕ )s + â λ ϕ s = 1 1 a λ ϕ s + = I,1 + I,2 + I,3 + I,4. a 1 â,+1 λ ϕ s + â,+1 λ +1 ϕ s + a λ s ϕ To complete the proof of Theorem 3.1, by Mikowski s iequality, it is sufficiet to show that =1 I,r k <, for r = 1, 2, 3, ) First, by applyig Hölder s iequality with idices k ad k, where k > 1 ad = 1, we hae that 1 k + 1 k θ k 1 I,1 k = 1 k a λ ϕ s 1 a λ k ϕ k s k O the other had, sice by 3.1) ad 3.2), we hae { 1 } k 1 a 1 a a 3.8) 1 Therefore, usig coditio 2.12), 3.6) ad 3.8), we get = O1) = O1) = O1) { 1 I,1 k = O1) θ a m λ k ϕ k =+1 m θ a a λ k ϕ k m θ a a a k ϕ k λ k = O1) as m, a λ k ϕ k }

6 96 ŞEBNEM YILDIZ by irtue of the hypotheses of Theorem 3.1. Now, usig Hölder s iequality ad the usig coditio 2.14) we hae that { 1 = O1) I,2 k { 1 θ 1 1/k ϕ = O1) = O1) m { 1 â,+1 λ ϕ s â,+1 k λ ϕ s k θ 1 1 k )1 k) { 1 θ k 1 a k 1 { 1 ) 1/k) 1 λ } k 1 } k â,+1 k 1 â,+1 ϕ λ θ 1 1 k )1 k) ϕ λ θ 1 1 k )1 k) â,+1 ϕ λ θ 1 1 k )1 k) )1 1 k ) =+1 )1 1 k )} )1 1 k )} )1 1 k )} θ a â,+1 The elemets â 0 for each,. I fact, it is easily see from the positieess of the matrix, 3.1) ad 3.2), that â 00 = 1, 1 â = ā 0 ā 1,0 + a 1,i a i ) i=0 1 = a 1,i a i ) 0 for ) i=0 So, usig the coditios 2.14) ad 3.5), we get = O1) I,2 k = O1) m θ 1 1/k ϕ m ) 1 ϕ λ θ 1 1 k )1 k) k ) 1 λ = O1) as m, )1 1 k ) θ a by irtue of the hypotheses of Theorem 3.1. Furthermore, usig the coditios 2.11), 2.13), 3.4)-3.5), ad 3.9), we hae that θ k 1 I,3 k = O1) { 1 â,+1 { 1 p 1 a k 1 p â,+1 ϕ λ +1 s ϕ k λ +1 k s k } â,+1 ϕ k λ k } k { 1 â,+1 P } k 1

7 A NEW NOTE ON LOCAL PROPERTY OF FACTORED FOURIER SERIES 97 = O1) = O1) = O1) m p m m ϕ k λ k =+1 θ a ϕ k λ k θ a â,+1 ϕ k λ k = O1) as m, by irtue of the hypotheses of Theorem 3.1. Fially, usig the coditios 2.12) ad 3.3), we hae that m m m θ k 1 I,4 k a k λ k s k ϕ k = O1) =1 =1 =1 a k λ k ϕ k = O1) as m, by irtue of hypotheses of the Theorem 3.1. Sice the behaiour of the Fourier series cocers the coergece for a particular alue of x depeds o the behaiour o the fuctio i the immediate eighborhood of this poit oly, this justifies 1.4) ad alid. This completes the proof of Theorem 3.1. Ackowledgmets. The authors would like to thak the aoymous referee for his/her commets that helped us improe this article. Refereces [1] H. Bor, O two summability methods, Math. Proc. Cambridge Philos Soc ) [2] H. Bor, Local property of N, p k summability of the factored Fourier series, Bull. Ist. Math. Acad. Siica ) [3] H. Bor, O the relatie stregth of two absolute summability methods, Proc. Amer. Math. Soc ) [4] H. Bor, O the local property of Fourier series, Bull. Math. Aal. Appl ) [5] T. M. Flett, O a extesio of absolute summability ad some theorems of Littlewood ad Paley, Proc. Lod. Math. Soc ) [6] G. H. Hardy, Dierget Series, Oxford Ui. Press Oxford 1949). [7] M. A. Sarıgöl, O the local property of the factored Fourier series, Bull. Math. Aal. Appl ) [8] M. A. Sarıgöl, O the local properties of factored Fourier series, Appl. Math. Comp ) [9] W. T. Sulaima, O some summability factors of ifiite series, Proc. Amer. Math. Soc ) [10] W. T. Sulaima, O local property of factored Fourier series, Bull. Math. Aal. Appl ) [11] Ş. Yıldız, A ew theorem o local properties of factored Fourier series, Bull. Math. Aal. Appl ) 1-8. Şebem Yıldız Mathematics Departmet, Ahi Era Uiersity, Kırşehir,Turkey address: sebemyildiz@ahiera.edu.tr; sebem.yildiz82@gmail.com