Hasan, Sabir

Zaini, S. Zahid Ali

English

Zaini S. Zahid Ali

Hasan Sabir

University of Szeged

Acta Sci. Math., 49 (1985), 295—281 Integrability of Rees—Stanojevic sums S. ZAHID ALI ZAINI and SABIR HASAN 1. A sequence {¿¡„} of positive numbers is called quasi-monotone if n~paniO for some fi, or equivalently, if a„+1 ^ an(l +a/n). RAM [3] defined that a sequence (AJ of numbers satisfies condition (S*) if 0 as fe-» and there exists a sequence {Ak} such that {Ak} is quasi-monotone, (1) . and (2) \Aak\ S Ak for all k. Condition (S*) is weaker than the condition (S) of Sidon introduced in [4]. REES and STANOJEVI6 [2] (see also GARRETT and STANOJEVI6 [1]) introduced the modified cosine sums (1.1) g„(*) = d / 2 ) 2 Aak+2 2 Aajcoskx fc=0 k=>lj=k and obtained a necessary and sufficient condition for the integrability of the limit of (1.1). RAM [3] proved the following theorem in which he showed that cotiditiort (S*) is sufficient for the integrability of the limit of (1.1). T h e o r e m A. Let the sequence {ak} satisfy condition (S*). Then g(x) = lira 2 [(1/2)4ak -I- 2 ¿"s cos kx] " - * ~ J k = l J=k exists for x€(0, n], and g(x)ÇL(0, n). Received January 24, 1983. 296 S. Zahid A l i Zain i and S. Hasan We say that a sequence {<?„} of numbers satisfies condition (S**) if {a„} is a null sequence and (3) nAa„ = 0(l) («->«.). We claim that our condition (S**) includes a more general class of sequences {o„} than that of Ram's condition (S*). E x a m p l e . The sequence ( _ 1 y+ i does not satisfy the conditions (S*) of Ram as \Aa„\^{n log (n+l)-1) and so 2 \Aa„\ = oo. This in fact contradicts conditions (1) and (2) of (S*). On the other hand this sequence satisfies the condition (3) of (S**). The object of this paper is to show that condition (S**) is sufficient for the .integrability of the limit of (1.1). T h e o r e m . Let the sequence {a„} satisfy condition (S**). Then g(x)= Hm ¿ [ ( l / 2 ) J a t + 2 Aaj cos kx] k=l j=k exists for x£(0, ?:], and g(x)(:L(0, n). 2. We require the following lemma for the proof of our theorem. L e m m a . Let {o„} be a null sequence and nAa„=0( 1), «—Then P r o o f . Applying Abel's transformation, we find 2Aam= 2 2\m + l)A2amHn + l)Aan, m = 0 m=0 m =0 n and since (n+l)Aa„—0 but 2 ^am=a0-a„-<-a0 as an—0, « — t h e n m=0 n—1 2 (m + l)A 2am — a0> m = 0 OO i.e. the series 2 converges. m — 0 3. P r o o f of t he T h e o r e m . We have g„ (x) = 2 [(1/2) M + 2 Aas cos kx] = k=1 j=k = 2(l/2)Aak + 2 Aak cos kx—a„+1D„(x)+(l/2)an+1. t=i t=i ln tegrabi l i ty o f Rees—Stanojevic sums 297 Using Abel's transformation, we obtain gn(x) = 2 0/2)M+ Aak(Dk(x)+\l2) + an(Dn(x)+\/2)~ k=l k =1 (4.1) И-1 - < i , + 1 ] ) 1 W - f l 1 + ( l / 2 ) i „ + 1 = 2 AakDk(x) + anDn(x)-an+1D„(x). /t=i Applying again Abel's transformation, we have (4.2) ft,(x)= "2 (k + \)An-akFk{x)^nAan_YFn^(x)+anDn{x)-a^1Dn(x), k=1 where D„(x) and F„(x) denotes the Dirichlet and Fejér kernels respectively. If x ^ O (mod 2k), then since a„—0, the last two terms of the right hand side of (4.2) tends to zero as И — 0 0 . Moreover, at X ? É O (mod27t) F„(x) always remains finite as n — °° and since nAa„ —0 therefore nAa„-1F„_1(x)-*0 as и— Since Fk(x)=o(l/(k+ l)x2) if x ^ O and 2 ( k + ] ) A 2 a k is convergent then 0 0 the series 2 Vе+\)А2ак Fk(x) converges. Hence for x ^ O (mod 2n) k= 1 g ( x ) = lim g„(x) = 2(k + \)A*akFk(x). The integrability of g(x) follows from the lemma; indeed, we have / g(x)dx = 2(k + l)A2akJ Fk(x)dx = (я/2) J (k + \)A*ak < It since J Fn(x) Jx=7r/2. 0 Coro l l a ry . Le/ {a„} be a null sequence and nAan=o(\), n—«>. ГЛел (1/x) 2 A ak sin (/c + l/2)x = h(x)/x converges for x£(0, л], a/id /i(x)/x€L(0, л). P roo f . From (4.1) we have g(x) = 2 /laftZ)t(x) = I 2 M sin (k + l/2)x]/2 sin (x/2) = h (x)/2 sin (x/2). k = l 4 = 1 / According to the theorem, g(x) exists for x^O, and #(x)€L[0, rt] if nAan—o(\), which implies our result. 298 S. Zah id A U Za in i and S. Hasan: Integrabi l i ty o f Rees—Stanojevic sums References [1] J. W. GARRETT and C. V. STANOJEVIC, On L1 convergènce of certain"cosine sums, Proc. Amer. Math. Soc., 54 (1976), 101—105. [2] C. S . REES and C. V. STANOJEVIÉ, Necessary and sufficient condition for integrability of cosine sums, J. Math. Anal. Appt., 43 (1973), 579—586. [3] B. RAM, Integrability of Rees—Stanojevié sums, Acta Sei. Math., 42 (1980), 153—155. [4] S. SIDON, Hinreichende Bedingungen für den Fourier-Charakter einer trigonometrischen Reihe, J. London Math. Soc., 14 (1939), 158—160. DEPARTMENT OF MATHEMATICS ALIGARH MUSLIM UNIVERSITY ALIGARH—202001, INDIA 

Integrability of Rees-Stanojević sums

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Integrability of Rees-Stanojević sums

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