ON TAUBERIAN CONDITIONS FOR (C, 1) SUMMABILITY OF INTEGRALS

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1 REVISTA DE LA UNIÓN MATEMÁTICA ARGENTINA Vol. 54, No. 2, 213, Pages Publishe online: December 8, 213 ON TAUBERIAN CONDITIONS FOR C, 1 SUMMABILITY OF INTEGRALS Abstract. We investigate some Tauberian conitions in terms of the general control moulo of the oscillatory behavior of integer orer of continuous real functions on [, for C, 1 summability of integrals. Moreover, we obtain a Tauberian theorem for a real boune function on [,. 1. Introuction Throughout this paper we assume that f is a real value continuous function on [, an s = ft t. The symbols s = o1 an s = O1 mean that lim s = an s is boune on [,, respectively. The ientity s σ = v, 1 where v = 1 tft t an σ = σs = 1 st t, is well-known an will be use in various steps of the proofs. The classical control moulo of the oscillatory behavior of the function s is enote by ω = f, an the general control moulo of the oscillatory behavior of integer orer m 1 of the function s is efine in [1] by ω m = ω m 1 σω m 1. We note that ω m = vω m 1 for any integer m 1. Moreover, ω m can be written as ω m = vv... vω... for any integer }{{} m times m 1. For each integer m, σ m an v m are efine in [2] by 1 σ m 1 t t m > 1 σ m = σ m = 1 an 1 v m 1 t t m 1 v m =, v m = 21 Mathematics Subject Classification. 4A1, 4E5, 4G5. Key wors an phrases. C, 1 summability of integrals, improper integrals, boune functions, Tauberian conitions an theorems. 59

2 6 respectively. The relationship between the functions σ m an v m is given by the ientity σ m σ m+1 = v m. For a function s, we efine s = m m 1 s = s m 1 for any positive integer m an nonnegative, where s = s, an s = 1 s. A more useful ientity for the general control moulo of the oscillatory behavior of integer orer m 1 of s is ω m = v m 1 2 m see [2] for the proof of this ientity. A function s is sai to be C, 1 summable to a finite number l if lim σ = l. 3 It is well known that if s is C, 1 summable to l, then σ is C, 1 summable to l. But the converse is not true. For eample, let s be efine by s = 2 sin t + t cos tt. Then, σ is C, 1 summable to 1, but s is not C, 1 summable to 1. If the integral ft t = l 4 eists, then clearly s is C, 1 summable to l. The converse is not necessarily true. For eample, the function s efine by s = sin t t is C, 1 summable to 1, but it is not convergent in the orinary sense. However, 3 may imply 4 by aing some suitable conition on s. Such a conition is calle a Tauberian conition an the resulting theorem is calle a Tauberian theorem. A real value function s is slowly oscillating [3] if lim lim sup λ 1 + ma t λ st s =. 5 Note that σ is slowly oscillating for every slowly oscillating function s. On the other han, ω = O1 implies that s is an slowly oscillating function. Inee, st s = s u u fu u By the efinition of slowly oscillating function, we have lim λ 1 + lim sup ma t λ st s. u u = log t The following classical Tauberian theorem is known as Littlewoo type Tauberian theorem [4]. Rev. Un. Mat. Argentina, Vol. 54, No

3 ON TAUBERIAN CONDITIONS FOR C, 1 SUMMABILITY OF INTEGRALS 61 Theorem 1. Let s be C, 1 summable to l. If then ft t converges to l. ω = O1 Different proofs of Theorem 1 are given by Laforgia [5], Çanak an Totur [6]. Çanak an Totur [1] have prove the following theorems for C, 1 summability metho. Theorem 2. Let s be C, 1 summable to l. If s is slowly oscillating, then ft t converges to l. Theorem 3. Let s be C, 1 summable to l. If v is slowly oscillating, then ft t converges to l. The aim of this paper is to investigate some Tauberian conitions in terms of the general control moulo of the oscillatory behavior of integer orer of continuous real functions on [, for C, 1 summability of integrals. Also, we obtain a Tauberian theorem for a real boune function s. 2. A Lemma For the proofs of the main theorems in the net section we nee the following Lemma in [1] showing the ifferences between s an σλ for λ > 1 an < λ < 1. Lemma 1. i For λ > 1, s σλ = 1 λ 1 σλ σ 1 λ ii For < λ < 1, s σλ = 1 1 λ σ σλ + 1 λ 3. Main results λ λ st s t. s st t. In this section, the main theorems of the paper will be presente. Theorem 4. Let s be C, 1 summable to l. If σω m is slowly oscillating for some integer m, then ft t converges to l. Proof. Since s is C, 1 summable to l, then σ is also C, 1 summable to l. Thus, it follows from the ientity 1 that v is C, 1 summable to. By the efinition of the general control moulo of oscillatory behavior of orer m, we obtain that σω m is C, 1 summable to 6 Rev. Un. Mat. Argentina, Vol. 54, No

4 62 for some integer m 1. On the other han, since σω m is slowly oscillating, we get, by Lemma 1 i, σω m σ 2 ω m λ = 1 λ 1 σ 2ω m λ σ 2 ω m By 7 we have 1 λ λ σω m t σω m t. σω m σ 2 ω m λ 1 λ 1 σ 2ω m λ σ 2 ω m + ma t λ σω mt σω m. Taking the lim sup of both sies of 8 as, we obtain lim sup σω m σ 2 ω m λ 1 λ 1 + lim sup lim sup σ 2 ω m λ σ 2 ω m ma σω mt σω m. t λ Since σ 2 ω m converges by 6, the first term on the right-han sie of the inequality 9 vanishes. Therefore, the inequality 9 becomes lim sup σω m σ 2 ω m λ lim sup ma σω mt σω m. 1 t λ Since σω m is slowly oscillating, we have from 1 that lim σω m =. Analogously, since s is C, 1 summable to l, we obtain that σ 2 ω m 1 is C, 1 summable to. It follows from the ientity σω m = v m = m v m = m 1 σ 2ω m 1 that σ 2ω m 1 = O1. If we apply Theorem 1 to σ 2 ω m 1, we see that σ 2 ω m 1 = o1. From the ientity σω m 1 σ 2 ω m 1 = σω m we have lim σω m 1 =. Continuing in this vein, we get σω = v = o1 as. Since s is C, 1 summable to l an lim v =, lim s = l. This completes the proof. Remark 5. Theorem 4 inclues not only Theorem 3, but also Theorem 2. If σω m is slowly oscillating for some integer m, it follows from the ientity σω m+1 = vσω m that σω m+1 is slowly oscillating. However, the slow oscillation of σω m oes not imply the slow oscillation of σω m 1, since σω m = vσω m 1. If we take σω m 1 = log logt+1 t+1 t, we have that σω m 1 is not slowly oscillating, but σω m = log + 1 is slowly oscillating. If we take m = in Theorem 4, then we have Theorem 3. An equivalent efinition of slow oscillation of s given by Çanak an Totur [1] says that s is slowly oscillating if an only if v is slowly oscillating an boune. If v is slowly oscillating, then s may not be slowly oscillating. For Rev. Un. Mat. Argentina, Vol. 54, No

5 ON TAUBERIAN CONDITIONS FOR C, 1 SUMMABILITY OF INTEGRALS 63 eample, if we take f = log+1 +1, it is clear that s is not slowly oscillating. It follows from the ientity 1 that v = log + 1 is slowly oscillating. Corollary 6. If s is C, 1 summable to l an p σ 1ω m = O as p, where then lim s = l. pλ lim lim sup = 1, 11 λ 1 + p Proof. For any t λ, we have σ 1 ω m t σ 1 ω m = σ t 1ω m uu C whence we conclue that lim sup ma σ 1ω m t σ 1 ω m C log lim sup t λ Taking the limit of both sies as λ 1 +, we obtain p u pt u = C log pu p, lim lim sup ma σ 1ω m t σ 1 ω m =, λ 1 + t λ i.e. σ 1 ω m is slowly oscillating. pλ p. Corollary 7. Let s be C, 1 summable to l. If ω m is slowly oscillating for some integer m, then ft t converges to l. Proof. Since ω m is slowly oscillating, then σω m is also slowly oscillating. Hence, the conitions in Theorem 4 are satisfie. In the following theorem, we obtain convergence of s out of the C, 1 summability of σ instea of the C, 1 summability of s with the same conition as in Theorem 4. Theorem 8. Let σ be C, 1 summable to l. If σω m is slowly oscillating for some integer m, then ft t converges to l. Proof. Since σ is C, 1 summable to l, from the ientity 1, we have v 1 is C, 1 summable to. Thus, σ 2 ω m is C, 1 summable to. Since σω m is slowly oscillating for some nonnegative integer m, then we have vσω m = σω m+1 = O1. It follows from the ientity σω m+1 = σ 2ω m that we have σ 2ω m = O1. If we apply Theorem 1 to σ 2 ω m, we see that σ 2 ω m = o1. 12 Hence, σ 1 ω m is C, 1 summable to an σ 1 ω m is slowly oscillating for some nonnegative integer m. Analogously, continuing as in the proof of Theorem 4, we obtain σ 1 ω m = o1 as. It follows from the ientity σω m = σ 2ω m 1 that we have σ 2ω m 1 = O1. If we apply Theorem 1 to σ 2 ω m 1, we see that σ 2 ω m 1 = o1. 13 Rev. Un. Mat. Argentina, Vol. 54, No

6 64 By the assumption an 13 we obtain from the ientity σω m 1 σ 2 ω m 1 = σω m that σω m 1 = o1. Continuing in this vein, we obtain σ 2 ω 1 = o1. Since σ is C, 1 summable to l, we have v 2 = o1. Therefore, from the ientity σ 2 ω 1 = σ 2 ω σ 3 ω = v 1 v 2 we get v 1 = o1. Since σ is C, 1 summable to l, we have σ 2 l as. From the ientity σ σ 2 = v 1, we obtain σ l as. Thus, s is C, 1 summable to l. Since the conitions in Theorem 4 are satisfie, the proof is complete. Theorem 9. Let s be boune. If σω m is slowly oscillating for some integer m, then f converges to as. Proof. Since s is assume to be boune, σω m is boune for every nonnegative integer m. Thus, we have σ σω m = 1 σω m = σω m = o1,. Thus, we obtain that σω m is C, 1 summable to. By hypothesis, if we apply Theorem 1 to σω m, we see that σω m converges to. If we write σω m 1 instea of s in the ientity 1, we have σω m 1 σω m = σω m. So, σω m 1 = o1. Continuing in this manner, we obtain σω = v = o1 as. On the other han, bouneness of s implies bouneness of v. Finally, applying the ientity 1 to s, we have s = v + v = o1. Hence s = f = o1. This completes the proof. Acknowlegments. The authors are grateful to the referee for his/her valuable suggestions an constructive remarks in revising the paper. [1] [2] References İ. Çanak, Ü. Totur, A Tauberian theorem for Cesàro summability of integrals, Appl. Math. Lett İ. Çanak, Ü. Totur, Tauberian conitions for Cesàro summability of integrals, Appl. Math. Lett [3] F. Móricz, Z. Németh, Tauberian conitions uner which convergence of integrals follows from summability C, 1 over R +, Anal. Math [4] G. H. Hary, Divergent series, Clarenon Press Ofor, Rev. Un. Mat. Argentina, Vol. 54, No

7 ON TAUBERIAN CONDITIONS FOR C, 1 SUMMABILITY OF INTEGRALS 65 [5] A. Laforgia, A theory of ivergent integrals, Appl. Math. Lett [6] İ. Çanak, Ü. Totur, Alternative proofs of some classical type Tauberian theorems for Cesàro summability of integrals, Math. Comput. Moelling Ümit Totur Department of Mathematics Anan Meneres University Ayin 91 Turkey Tel: , Fa: utotur@yahoo.com, utotur@au.eu.tr İbrahim Çanak Department of Mathematics Ege University Izmir 351 Turkey ibrahimcanak@yahoo.com, ibrahim.canak@ege.eu.tr Receive: February 17, 212 Accepte: July 16, 213 Rev. Un. Mat. Argentina, Vol. 54, No