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A Tauberian Theorem for a General Summability Method

A Tauberian Theorem for a General Summability Method We investigate conditions under which M summability implies Abel summability and give the generalized Littlewood Tauberian theorem for M summability method. Mathematics Subject Classification 2010: 40E05, 40G10. Key words: Tauberian theorems, power series methods, slow oscillation, Abel summability method, M summability method. 1. Introduction Let A denote the space of analytic functions in 0 < x < 1. To each f (a) = f (a, x) = an xn in A we associate a series an of Taylor n=0 n=0 an is said to be Abel summable to s if coefficients of f . A series n=0 n - x converges for 0 < x < 1 and tends to s as x 1 . Abel's con tinuity theorem (see [1]) for power series states that if converges, then lim f (a, x) = an . It is known that the converse of this imn=0 plication is not true in general. However, we have a conditional converse of Abel's continuity theorem known as the generalized Littlewood Tauberian theorem (see [10]) for the Abel summability method asserting that if is Abel summable to s and slowly oscillating, then = s. We remind the reader that a series an is said to be slowly oscillating n=0 (see [7]) if ak = o(1), n > m , n/m 1. k=m+1 For a different proof of the generalized Littlewood Tauberian theorem (see [10]), we refer the reader to Landau [6], Schmidt [7], and Canak [2]. ¸ Denote the class of kernels of the integral transforms of functions in A by . We now need the following properties of functions in : 1. There exists a number 0 = 0 () (0, 1) such that every is analytical in [0 , 1). 2. For every , (x) , x 1- . 3. Each is zero-free in [0 , 1). 4. For each m 1, m (x) = o(1), x 1- where 0 = and m (x) = m-1 (x) x m-1 (t)dt. 0 For every f in A and we define M (f, ) = M (f, , x) = if x = 0 and limx0 M (f, , x) = f (0 ) if x = 0 . A series an is said to be M summable to s if n=0 (1.1) x 0 f (t)(t)dt 1 (x) lim M (f, , x) = s. Example 1.1. The series an whose general term is the Taylor n=0 coefficient of the function g defined by g(x) = f (1 (x)), where f is bounded in A is M summable. It is plain that every Abel summable series is M summable, but the converse statement is not always true. The M summability method is regular with respect to the Abel summability method. Indeed, since 1 (x) = (x), it easily follows that Abel summability of an implies M summability n=0 of an . However, if a function f satisfies the condition n=0 x 0 = o(1), x 1- , then the converse statement is also true. This follows from the identity (1.2) f (x) = M (f, , x) + x 0 An important subclass of is the class of following functions m (x) = 1 m = 1, 2, ... and 0 = 0. The function m (x) = (1-x)m defines a summability method (A, m) which is regular with respect to Abel summability method. For the more information about the M and (A, m) summability methods, we refer to [8]. Recently, a number of authors including 1 (1-x)m , A TAUBERIAN THEOREM FOR A GENERAL SUMMABILITY METHOD 125 Canak et al. [3, 5], Canak [4], and Totur [9] have given Tauberian ¸ ¸ theorems for the (A, m) summability method. The (A, 1) summability method as a special case of M summability method is of considerable interest. A series an is said to be (A, 1) n=0 summable to s if (1.3) lim (1 - x) x x 0 f (t) dt = s. (1 - t)2 It is clear that Abel summability of an implies (A, 1) summability n=0 an . That the converse is not true in general follows from the of n=0 series an whose general term is the Taylor coefficient of the function n=0 f defined by f (x) = sin((1 - x)-1 ) on 0 < x < 1. The identity (1.2) becomes f (x) = (1 - x) x x 0 f (t) (1 - x) dt + 2 (1 - t) x x 0 f (t) dt 1-t 1 for (x) = (1-x)2 . Since a series which is Abel summable to s is (A,1) summable to s, we have (1.4) (1 - x) x x 0 f (t) dt = o(1), x 1- . 1-t In the case where the condition (1.4) holds, every (A,1) summable series is Abel summable. An example of an (A, 1) summable series can be obtained by applying x f (t) integration by parts to (1-x) o (1-t)2 dt. x 1 Example 1.2. If n=0 n+1 is (A, 1) summable to 0. n k=0 ak is Abel summable to s, then It is natural to ask under which conditions M summability implies Abel summability of an . In this paper we both answer this question d give the generalized Littlewood Tauberian theorem for M summability method. 2. A Tauberian theorem By the following theorem, we prove that every M summable series is Abel summable under certain conditions. Theorem 2.1. Let (2.1) x 0 be M summable to s. If = o(1), x 1- f (0 ) 2 (0 ) f (t)2 (t)dt 1 (x) and f has a zero of order three at 0 such that limit f limx0 2(x) , (x) is defined to be the then is Abel summable to s. Proof. Applying integration by parts to M (f, , x), we have (2.2) where T (f, , x) = T (f, , x), we have (2.3) M (f, , x) = f (x) - T (f, , x), x 0 . Applying again integration by parts to x 0 2 (x) T (f, , x) = f (x) - 1 (x) f (t)2 (t) dt 1 (x) Combining (2.2) with (2.3), we obtain (2.4) M (f, , x) = g(x) + x 0 f (t)2 (t) dt 1 (x) where g(x) = f (x) - 2 (x) f (x). Since 1 (x) have by (2.1) and (2.4) that (2.5) is M summable to s, we lim g(x) = lim M (f, , x) = s. Multiplying g(x) by 1 (x) and dividing by 2 (x), we arrive at 2 (2.6) 2 (x) g(x) = - 2 1 (x) f (x) 2 (x) f (x) 2 (x) f Since 2(00)) is defined to be the limit limx0 ( zero of order three, we obtain x at 0 where f has a (2.7) f (x) = -2 (x) g(t) 1 (t) dt. 2 (t) 2 A TAUBERIAN THEOREM FOR A GENERAL SUMMABILITY METHOD 127 Since lim g(x) = s, we have by L'Hospital's rule that (2.8) lim f (x) = s. This completes the proof. 1 Taking (x) = (1-x)2 in Theorem 2.1, we have the following Tauberian theorem for the (A, 1) summability method. Corollary 2.2. Let (2.9) (1 - x) 0 x be (A, 1) summable to s. If ln(1 - t)f (t)dt = o(1), x 1- f (x) -x-ln(1-x)) and f has a zero of order three at 0 such that be the limit f (x) limx0 -x-ln(1-x) , at 0 is defined to then is Abel summable to s. We note that Theorem 2.1 can be given for the composition of M and M defined by (2.10) (M M )(f (a), x) = M (M (f (a)), , x) = x 0 M (f, , t)(t)dt 1 (x) where , . The following theorem is a consequence of the generalized Littlewood Tauberian theorem for Abel summability method. Theorem 2.3. Let an be M summable to s, (2.1) be satisfied and n=0 f f have a zero of order three at 0 such that 2(00)) is defined to be the limit ( limx0 to s. f (x) 2 (x) . If is slowly oscillating, then converges Proof. We have by Theorem 2.1 that an is Abel summable to s. n=0 Since an is slowly oscillating, we have from the generalized Littlewood n=0 Tauberian theorem that an converges to s. n=0 If the condition (2.1) is replaced by a stronger condition, we have the following result. Theorem 2.4. Let (2.11) and be M summable to s. If (x) 2 (x) , x 1- f (x) = o is slowly oscillating, then converges to s. Proof. We easily see that the condition (2.11) implies (2.1). It follows by Theorem 2.1 that an is Abel summable to s. Since an n=0 n=0 is slowly oscillating, we have from the generalized Littlewood Tauberian theorem that an converges to s. n=0 http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Annals of the Alexandru Ioan Cuza University - Mathematics de Gruyter

A Tauberian Theorem for a General Summability Method

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Abstract

We investigate conditions under which M summability implies Abel summability and give the generalized Littlewood Tauberian theorem for M summability method. Mathematics Subject Classification 2010: 40E05, 40G10. Key words: Tauberian theorems, power series methods, slow oscillation, Abel summability method, M summability method. 1. Introduction Let A denote the space of analytic functions in 0 < x < 1. To each f (a) = f (a, x) = an xn in A we associate a series an of Taylor n=0 n=0 an is said to be Abel summable to s if coefficients of f . A series n=0 n - x converges for 0 < x < 1 and tends to s as x 1 . Abel's con tinuity theorem (see [1]) for power series states that if converges, then lim f (a, x) = an . It is known that the converse of this imn=0 plication is not true in general. However, we have a conditional converse of Abel's continuity theorem known as the generalized Littlewood Tauberian theorem (see [10]) for the Abel summability method asserting that if is Abel summable to s and slowly oscillating, then = s. We remind the reader that a series an is said to be slowly oscillating n=0 (see [7]) if ak = o(1), n > m , n/m 1. k=m+1 For a different proof of the generalized Littlewood Tauberian theorem (see [10]), we refer the reader to Landau [6], Schmidt [7], and Canak [2]. ¸ Denote the class of kernels of the integral transforms of functions in A by . We now need the following properties of functions in : 1. There exists a number 0 = 0 () (0, 1) such that every is analytical in [0 , 1). 2. For every , (x) , x 1- . 3. Each is zero-free in [0 , 1). 4. For each m 1, m (x) = o(1), x 1- where 0 = and m (x) = m-1 (x) x m-1 (t)dt. 0 For every f in A and we define M (f, ) = M (f, , x) = if x = 0 and limx0 M (f, , x) = f (0 ) if x = 0 . A series an is said to be M summable to s if n=0 (1.1) x 0 f (t)(t)dt 1 (x) lim M (f, , x) = s. Example 1.1. The series an whose general term is the Taylor n=0 coefficient of the function g defined by g(x) = f (1 (x)), where f is bounded in A is M summable. It is plain that every Abel summable series is M summable, but the converse statement is not always true. The M summability method is regular with respect to the Abel summability method. Indeed, since 1 (x) = (x), it easily follows that Abel summability of an implies M summability n=0 of an . However, if a function f satisfies the condition n=0 x 0 = o(1), x 1- , then the converse statement is also true. This follows from the identity (1.2) f (x) = M (f, , x) + x 0 An important subclass of is the class of following functions m (x) = 1 m = 1, 2, ... and 0 = 0. The function m (x) = (1-x)m defines a summability method (A, m) which is regular with respect to Abel summability method. For the more information about the M and (A, m) summability methods, we refer to [8]. Recently, a number of authors including 1 (1-x)m , A TAUBERIAN THEOREM FOR A GENERAL SUMMABILITY METHOD 125 Canak et al. [3, 5], Canak [4], and Totur [9] have given Tauberian ¸ ¸ theorems for the (A, m) summability method. The (A, 1) summability method as a special case of M summability method is of considerable interest. A series an is said to be (A, 1) n=0 summable to s if (1.3) lim (1 - x) x x 0 f (t) dt = s. (1 - t)2 It is clear that Abel summability of an implies (A, 1) summability n=0 an . That the converse is not true in general follows from the of n=0 series an whose general term is the Taylor coefficient of the function n=0 f defined by f (x) = sin((1 - x)-1 ) on 0 < x < 1. The identity (1.2) becomes f (x) = (1 - x) x x 0 f (t) (1 - x) dt + 2 (1 - t) x x 0 f (t) dt 1-t 1 for (x) = (1-x)2 . Since a series which is Abel summable to s is (A,1) summable to s, we have (1.4) (1 - x) x x 0 f (t) dt = o(1), x 1- . 1-t In the case where the condition (1.4) holds, every (A,1) summable series is Abel summable. An example of an (A, 1) summable series can be obtained by applying x f (t) integration by parts to (1-x) o (1-t)2 dt. x 1 Example 1.2. If n=0 n+1 is (A, 1) summable to 0. n k=0 ak is Abel summable to s, then It is natural to ask under which conditions M summability implies Abel summability of an . In this paper we both answer this question d give the generalized Littlewood Tauberian theorem for M summability method. 2. A Tauberian theorem By the following theorem, we prove that every M summable series is Abel summable under certain conditions. Theorem 2.1. Let (2.1) x 0 be M summable to s. If = o(1), x 1- f (0 ) 2 (0 ) f (t)2 (t)dt 1 (x) and f has a zero of order three at 0 such that limit f limx0 2(x) , (x) is defined to be the then is Abel summable to s. Proof. Applying integration by parts to M (f, , x), we have (2.2) where T (f, , x) = T (f, , x), we have (2.3) M (f, , x) = f (x) - T (f, , x), x 0 . Applying again integration by parts to x 0 2 (x) T (f, , x) = f (x) - 1 (x) f (t)2 (t) dt 1 (x) Combining (2.2) with (2.3), we obtain (2.4) M (f, , x) = g(x) + x 0 f (t)2 (t) dt 1 (x) where g(x) = f (x) - 2 (x) f (x). Since 1 (x) have by (2.1) and (2.4) that (2.5) is M summable to s, we lim g(x) = lim M (f, , x) = s. Multiplying g(x) by 1 (x) and dividing by 2 (x), we arrive at 2 (2.6) 2 (x) g(x) = - 2 1 (x) f (x) 2 (x) f (x) 2 (x) f Since 2(00)) is defined to be the limit limx0 ( zero of order three, we obtain x at 0 where f has a (2.7) f (x) = -2 (x) g(t) 1 (t) dt. 2 (t) 2 A TAUBERIAN THEOREM FOR A GENERAL SUMMABILITY METHOD 127 Since lim g(x) = s, we have by L'Hospital's rule that (2.8) lim f (x) = s. This completes the proof. 1 Taking (x) = (1-x)2 in Theorem 2.1, we have the following Tauberian theorem for the (A, 1) summability method. Corollary 2.2. Let (2.9) (1 - x) 0 x be (A, 1) summable to s. If ln(1 - t)f (t)dt = o(1), x 1- f (x) -x-ln(1-x)) and f has a zero of order three at 0 such that be the limit f (x) limx0 -x-ln(1-x) , at 0 is defined to then is Abel summable to s. We note that Theorem 2.1 can be given for the composition of M and M defined by (2.10) (M M )(f (a), x) = M (M (f (a)), , x) = x 0 M (f, , t)(t)dt 1 (x) where , . The following theorem is a consequence of the generalized Littlewood Tauberian theorem for Abel summability method. Theorem 2.3. Let an be M summable to s, (2.1) be satisfied and n=0 f f have a zero of order three at 0 such that 2(00)) is defined to be the limit ( limx0 to s. f (x) 2 (x) . If is slowly oscillating, then converges Proof. We have by Theorem 2.1 that an is Abel summable to s. n=0 Since an is slowly oscillating, we have from the generalized Littlewood n=0 Tauberian theorem that an converges to s. n=0 If the condition (2.1) is replaced by a stronger condition, we have the following result. Theorem 2.4. Let (2.11) and be M summable to s. If (x) 2 (x) , x 1- f (x) = o is slowly oscillating, then converges to s. Proof. We easily see that the condition (2.11) implies (2.1). It follows by Theorem 2.1 that an is Abel summable to s. Since an n=0 n=0 is slowly oscillating, we have from the generalized Littlewood Tauberian theorem that an converges to s. n=0

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Annals of the Alexandru Ioan Cuza University - Mathematics – de Gruyter

Published: Jan 1, 2015

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