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Journal of Inequalities in Pure and Applied Mathematics LOCALIZATION OF FACTORED FOURIER SERIES volume 6, issue 2, article 40, HÜSEY˙N BOR I 2005. Department of Mathematics Received 20 April, 2005; Erciyes University accepted 02 May, 2005. 38039 Kayseri, Turkey Communicated by: L. Leindler EMail: bor@erciyes.edu.tr URL: http://fef.erciyes.edu.tr/math/hbor.htm Abstract Contents Home Page Go Back Close c 2000 Victoria University ISSN (electronic): 1443-5756 Quit 126-05 Abstract ¯ In this paper we deal with a main theorem on the local property of | N, pn |k summability of factored Fourier series, which generalizes some known results. 2000 Mathematics Subject Classiﬁcation: 40D15, 40G99, 42A24, 42B15. Key words: Absolute summability, Fourier series, Local property. Localization of Factored Fourier Contents Series Hüseyin Bor 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2 Known Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 3 The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 Title Page References Contents Go Back Close Quit Page 2 of 12 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au 1. Introduction Let an be a given inﬁnite series with partial sums (sn ). Let (pn ) be a sequence of positive numbers such that n (1.1) Pn = pv → ∞ as n → ∞, (P−i = p−i = 0, i ≥ 1). v=0 The sequence-to-sequence transformation Localization of Factored Fourier n Series 1 (1.2) tn = p v sv Hüseyin Bor Pn v=0 ¯ deﬁnes the sequence (tn ) of the (N , pn ) means of the sequence (sn ) generated Title Page by the sequence of coefﬁcients (pn ). Contents ¯ The series an is said to be summable N , pn k , k ≥ 1, if (see [2]) ∞ k−1 Pn (1.3) |tn − tn−1 |k < ∞. pn n=1 Go Back In the special case when pn = 1/(n + 1) for all values of n (resp. k = 1), Close ¯ ¯ , pn summability is the same as |C, 1| (resp. N , pn ) summability. Also if N Quit k ¯ we take k = 1 and pn = 1/(n + 1), summability N , pn k is equivalent to the Page 3 of 12 summability |R, log n, 1|. A sequence (λn ) is said to be convex if ∆2 λn ≥ 0 for every positive integer n, where ∆2 λn = ∆λn − ∆λn+1 and ∆λn = λn − λn+1 . J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au Let f (t) be a periodic function with period 2π and integrable (L) over (−π, π). Without any loss of generality we may assume that the constant term in the Fourier series of f (t) is zero, so that π (1.4) f (t)dt = 0 −π and ∞ ∞ Localization of Factored Fourier (1.5) f (t)∼ (an cos nt + bn sin nt) ≡ An (t). Series n=1 n=1 Hüseyin Bor It is well known that the convergence of the Fourier series at t = x is a local property of the generating function f (i.e. it depends only on the behaviour of Title Page f in an arbitrarily small neighbourhood of x), and hence the summability of the Contents Fourier series at t = x by any regular linear summability method is also a local property of the generating function f . Go Back Close Quit Page 4 of 12 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au 2. Known Results Mohanty [4] has demonstrated that the |R, log n, 1| summability of the factored Fourier series An (t) (2.1) log(n + 1) at t = x, is a local property of the generating function of f , whereas the |C, 1| summability of this series is not. Matsumoto [3] improved this result by replac- Localization of Factored Fourier Series ing the series (2.1) by Hüseyin Bor An (t) (2.2) , δ > 1. {log log(n + 1)}δ Title Page Generalizing the above result Bhatt [1] proved the following theorem. Contents −1 Theorem A. If (λn ) is a convex sequence such that n λn is convergent, then the summability |R, log n, 1| of the series An (t)λn log n at a point can be ensured by a local property. Go Back Close Quit Page 5 of 12 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au 3. The Main Result The aim of the present paper is to prove a more general theorem which includes of the above results as special cases. Also it should be noted that the conditions on the sequence (λn ) in our theorem, are somewhat more general than in the above theorem. Now we shall prove the following theorem. Theorem 3.1. Let k ≥ 1. If (λn ) is a non-negative and non-increasing sequence such that ¯ pn λn is convergent, then the summability N , pn k of the series Localization of Factored Fourier Series An (t)λn Pn at a point is a local property of the generating function f . Hüseyin Bor We need the following lemmas for the proof of our theorem. Lemma 3.2. If (λn ) is a non-negative and non-increasing sequence such that Title Page pn λn is convergent, where (pn ) is a sequence of positive numbers such that Contents Pn → ∞ as n → ∞, then Pn λn = O(1) as n → ∞ and Pn ∆λn < ∞. Proof. Since (λn ) is non-increasing, we have that m m Pm λm = λ m pn = O(1) pn λn = O(1) as m → ∞. Go Back n=0 n=0 Close m Applying the Abel transform to the sum n=0 pn λn , we get that Quit m m Page 6 of 12 Pn ∆λn = pn λn − Pm λm+1 . n=0 n=0 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au Since λn ≥ λn+1 , we obtain m m Pn ∆λn ≤ Pm λm + p n λn n=0 n=0 = O(1) + O(1) = O(1) as m → ∞. Lemma 3.3. Let k ≥ 1 and sn = O(1). If (λn ) is a non-negative and non- Localization of Factored Fourier increasing sequence such that pn λn is convergent, where (pn ) is a sequence Series of positive numbers such that Pn → ∞ as n → ∞, then the series an λ n P n Hüseyin Bor ¯ is summable N , pn k . ¯ Proof. Let (Tn ) be the sequence of (N , pn ) means of the series an λ n P n . Title Page Then, by deﬁnition, we have Contents n v n 1 1 Tn = pv ar λ r P r = (Pn − Pv−1 )av λv Pv . Pn v=0 r=0 Pn v=0 Then, for n ≥ 1, we have Go Back n Close pn Tn − Tn−1 = Pv−1 Pv av λv . Quit Pn Pn−1 v=1 Page 7 of 12 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au By Abel’s transformation, we have n−1 n−1 pn pn Tn − Tn−1 = Pv Pv sv ∆λv − Pv s v p v λv Pn Pn−1 v=1 Pn Pn−1 v=1 n−1 pn − Pv pv+1 sv λv+1 + sn pn λn Pn Pn−1 v=1 = Tn,1 + Tn,2 + Tn,3 + Tn,4 , say. Localization of Factored Fourier By Minkowski’s inequality for k > 1, to complete the proof of Lemma 3.3, it is Series sufﬁcient to show that Hüseyin Bor ∞ (3.1) (Pn /pn )k−1 |Tn,r |k < ∞, for r = 1, 2, 3, 4. n=1 Title Page Now, applying Hölder’s inequality with indices k and k , where 1 + 1 = 1 and Contents k k k > 1, we get that m+1 k−1 Pn |Tn,1 |k n=2 pn Go Back m+1 n−1 n−1 k−1 Close pn k 1 ≤ |sv | Pv Pv ∆λv Pv Pv ∆λv . Quit n=2 Pn Pn−1 v=1 Pn−1 v=1 Page 8 of 12 Since n−1 n−1 Pv Pv ∆λv ≤ Pn−1 Pv ∆λv , J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au v=1 v=1 it follows by Lemma 3.2 that n−1 n−1 1 Pv Pv ∆λv ≤ Pv ∆λv = O(1) as n → ∞. Pn−1 v=1 v=1 Therefore m+1 k−1 m+1 n−1 Pn k pn |Tn,1 | = O(1) |sv |k Pv Pv ∆λv n=2 pn n=2 Pn Pn−1 v=1 Localization of Factored Fourier Series m m+1 k pn = O(1) |sv | Pv Pv ∆λv Hüseyin Bor v=1 P P n=v+1 n n−1 m Title Page = O(1) Pv ∆λv = O(1) as m → ∞, v=1 Contents by virtue of the hypotheses of Theorem 3.1 and Lemma 3.2. Again m+1 k−1 m+1 n−1 n−1 k−1 Pn k pn k k 1 |Tn,2 | ≤ |sv | (Pv λv ) pv pv Go Back n=2 pn n=2 Pn Pn−1 v=1 Pn−1 v=1 m+1 n−1 Close pn k k = O(1) |sv | (Pv λv ) pv Quit v=2 Pn Pn−1 v=1 m m+1 Page 9 of 12 k pn k = O(1) |sv | (Pv λv ) pv v=1 P P n=v+1 n n−1 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au m pv = O(1) |sv |k (Pv λv )k v=1 Pv m = O(1) |sv |k (Pv λv )k−1 pv λv v=1 m = O(1) pv λv = O(1) as m → ∞, v=1 in view of the hypotheses of Theorem 3.1 and Lemma 3.2. Using the fact that Localization of Factored Fourier Series Pv < Pv+1 , similarly we have that m+1 k−1 m Hüseyin Bor Pn k |Tn,3 | = O(1) pv+1 λv+1 = O(1) as m → ∞. n=2 pn v=1 Title Page Finally, we have that Contents m k−1 m Pn |Tn,4 |k = |sn |k (Pn λn )k−1 pn λn n=1 pn n=1 m = O(1) pn λn = O(1) as m → ∞, Go Back n=1 Close by virtue of the hypotheses of the theorem and Lemma 3.2. Therefore, we get that Quit m k−1 Pn Page 10 of 12 |Tn,r |k = O(1) as m → ∞, for r = 1, 2, 3, 4. n=1 pn J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 This completes the proof of Lemma 3.3. http://jipam.vu.edu.au In the particular case if we take pn = 1 for all values of n in Lemma 3.3, then we get the following corollary. Corollary 3.4. Let k ≥ 1 and and sn = O(1). If (λn ) is a non-negative and non-increasing sequence such that λn is convergent, then the series nan λn is summable |C, 1|k . Proof of Theorem 3.1. Since the behaviour of the Fourier series, as far as con- vergence is concerned, for a particular value of x depends on the behaviour of the function in the immediate neighbourhood of this point only, hence the truth Localization of Factored Fourier Series of Theorem 3.1 is a consequence of Lemma 3.3. If we take pn = 1 for all val- ues of n in this theorem, then we get a new local property result concerning the Hüseyin Bor |C, 1|k summability. Title Page Contents Go Back Close Quit Page 11 of 12 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au References [1] S.N. BHATT, An aspect of local property of |R, log n, 1| summability of the factored Fourier series, Proc. Nat. Inst. Sci. India, 26 (1960), 69–73. [2] H. BOR, On two summability methods, Math. Proc. Cambridge Philos Soc., 97 (1985), 147–149. [3] K. MATSUMOTO, Local property of the summability |R, log n, 1|, Tôhoku Math. J. (2), 8 (1956), 114–124. Localization of Factored Fourier Series [4] R. MOHANTY, On the summability |R, log w, 1| of Fourier series, J. Lon- don Math. Soc., 25 (1950), 67–72. Hüseyin Bor Title Page Contents Go Back Close Quit Page 12 of 12 J. Ineq. Pure and Appl. Math. 6(2) Art. 40, 2005 http://jipam.vu.edu.au