REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS

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1 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS JACK ENYEART Contents Introduction 1 1 The Basics 2 2 Examples of summation techniques 4 3 Abelian theorems 13 4 Tauberian theorems 15 References 2 Introduction The concepts of convergence and divergence, while not defined explicitly until early 19th century, have been studied since the third century BC Throughout this time, mathematicians have largely focused upon convergent sequences and series, having seemingly little analytical reason to study series that diverged However, as the area of mathematical analysis developed over the past few centuries, studying properties of divergent series and methods that sum divergent series has produced results that are very useful in the areas of harmonic analysis, number theory, and combinatorics/probability theory This paper is essentially an investigation of such methods called summation methods and includes classical examples, as well as proofs of general properties about these methods The first section of the paper contains definitions and basic terminology used throughout the paper, as well as a brief commentary on summation methods and the divergent series to which they are applied Since all the summation methods within this paper will all essentially involve the process of averaging, a definition of weighted averaging is also provided Within section two of the paper, we present classical examples of summation methods, starting with the identity summation method Next, the summation method involving standard averaging by arithmetic means, called the Cesàro method, is discussed After this point, more involved summation methods are discussed at length, including Hölder methods, higher order Cesàro means, and the Euler method For each of these summation methods, its corresponding collection of averaging weights can be organized as an infinite matrix, and is listed explicitly as such when possible These matrices are afterwards referred to as averaging matrices In the latter part of this section, we present three theorems pertaining to averaging matrices the first states that the product of any two averaging matrices will also be an averaging matrix The second theorem gives a necessary and sufficient condition that ensures a matrix will be regular that is, it will not affect the limit of series that is convergent in 1

2 2 JACK ENYEART the normal sense a property that is typically desirable in summation methods Finally we offer a theorem that proves the product of two regular averaging matrices is also a regular averaging matrix The section is concluded with a discussion of a summation method unlike those mentioned before This method, called the Abel method, is a less obvious averaging method whose collection weights cannot be organized within an infinite matrix Rather, to each real number in the interval (, 1 there corresponds a sequence of averaging weights thus, we have a continuous, uncountable collection of weights for the Abel method The third section of the paper includes two Abelian theorems An Abelian theorem relates the relative strengths of two specific summation methods we say that a summation method X is stronger than another method Y, if it sums all Y-summable series and possibly more The first Abelian theorem included in the section states that the Abel method is stronger than the Cesàro method, and the second that the Cesàro method is stronger than the identity method (convergence in the normal sense The final section of the paper contains two Tauberian theorems, which are partial converses to the Abelian theorems That is, given summation methods X and Y with X stronger than Y, a Tauberian theorem specifies what additional properties a Y-summable series must possess to be X-summable as well The two Tauberian theorems included in this section are exactly the partial converses to the two Abelian theorems listed in section three The proofs of the Tauberian theorems require the most sophisticated mathematics seen within this dissertation, and are quite involved For further information regarding this topic, the reader is encouraged to turn to the text that was central in the writing of this paper, GH Hardy s Divergent Series I would like to take this opportunity to thank the University of Notre Dame Math Department for their consistent support throughout the writing of this thesis, and my entire time as an undergraduate at Notre Dame I am grateful to the many kind and encouraging faculty members within Hayes-Healy that have taught and guided me over the past four years I would like to especially thank Prof Liviu Nicolaescu, who over the past two years has sacrificed much of his time to work with me to sharpen my mathematical skills and develop the very paper that you are currently reading Throughout my correspondence with him, Liviu always remained enthusiastic, interested, and generous, and I am thankful to have had him as an advisor 1 The Basics We begin by briefly covering basic concepts involving sequences and series, as well as defining general terminology Given any series, k a k, we say that such a series converges if the sequence of partial sums s n = n k a k converges to some limit as n approaches infinity For every set A we denote by R A the set of functions A R We define Seq := R N, where N denotes the set of non-negative integers Denote by Seq c the subspace of Seq consisting of converging sequences We regard the limit of a convergent sequence as a linear map lim : Seq c R Definition 11 A summation method is a quadruple (T, X, U, u where X is a subspace of Seq, U is a topological space, u U,

3 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 3 T is a linear operator T : X R U, U = U \ {u } such that the following hold: (a Seq c X (b For any sequence s X, the functions Ts : U R has a finite limit at u, Typically, we will refer to the (T, X, U, u method as the T-method The vector space X is called the space of T-convergent sequences The method T is said to be regular if s Seq c we have lim n s = lim u u Ts(u For any summation method T discussed within this paper, consisting of the quadruple (T, X, U, u as stated above, the topological space U will be one of the following two types In the first case, we take U to be the compactification U = N := N { } the transformation T takes a sequence, say s, to another sequence, call it s In this case, we are interested in investigating the limit of the transformed sequence at infinity Thus, for any summation method with U = N, u =, (ie lim u u Ts(u = lim u Ts(u In the other case, where we have a summation method T with topological space U = [, 1], the transformation T takes a sequence s to a function Ts : [, 1 R In this case, u = 1, and we investigate the lim u 1 Ts(u As indicated above, since T is a summation method, if s converges to a finite limit at infinity, Ts also converges to a finite limit at u Moreover, s may be divergent, yet Ts may converge to a finite limit at u Naturally, we tend to think of sequences which converge to a finite limit as well-behaved or tame, while we think of divergent sequences as disordered or wild Since a summation method has the ability to transform a divergent sequence into a function that possesses a finite limit at infinity, while not disrupting the finite limit of a sequence that is already convergent, it is natural to think of T as a taming -transformation Remark 12 Note that there exists a canonical bijection between any sequence, say a n, and its series k a k Namely to the series k a k we associate the sequence of partial sums n s n = a k, n Conversely, to the sequence s(n we associate the series n a k = s n, a = s, a n = s n s n 1, n 1 k= k= And so, while each method actually operates upon a sequence, we will think of that sequence as the sequence of partial sums of a series Thus we see the reason for calling such processes summation methods All of the summation-transformations within this paper will be obtained by averaging For every fixed u U we will consider a sequence of weights: w(u = w (u, w 1 (u,, w n (u, As a true weighted-average, the sequence satisfies the following two properties: w n (u, n, (

4 4 JACK ENYEART w n (u = 1, u n Given any sequence s = s, s 1,, s n,, we form its averages ( Ts(u = w (us + w 1 (us w n (us n + = n w n (us n If lim u u Ts(u = c, where c is finite, then we say that s is T-convergent and a k is T-summable, with T-limit c While within this paper we will apply summation methods directly to given sequences, it is important that the reader keep in mind the relationship between sequences and series, and the series that the sequence originates from Notation Let f,g be real functions f g if and only if f(n lim n g(n = 1 f(n = O(g(n as n if and only if there exists M > and N R such that f(n = o(g(n if and only if f(n M g(n f(n lim n g(n = n > N 2 Examples of summation techniques When U = N, the collection of weights {w(u} u U can be organized as an infinite matrix The collection {w(u} occupies the u th row of the infinite matrix Definition 21 (Averaging Matrix Let M be an infinite square matrix, with entries M (ij = m ij where i corresponds to row, and j to column We call M an averaging matrix if it satisfies the following properties m ij, i, j N m ij = 1, i N j (21a (21b Note: row/column indices of matrices will start with, and increase as usual We will now give several examples of standard summation methods that can be represented by averaging matrices We refer [1] for a wealth of many other examples Example 22 (The Identity summation method Let I be the identity map Seq Seq Here we take U to be the compactification U = N := N { } Then u =, U = N We define the family of weights used in this trivial method as follows: {, if n u; w n (u = 1, if n = u

5 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 5 Therefore, I is represented by the infinite identity matrix: 1 1 I = 1 To calculate Is, we write s as an infinite column vector and multiply: 1 1 Is = 1 Clearly Is = s, and thus, lim u Is(u = lim u s u, so the I-convergent sequences are sequences that converge in the normal sense, just as we would expect Example 23 (Cesàro method The Cesàro method uses standard averaging, and is the most simple non-trivial summation method The weights are evenly distributed, and are defined as follows: w n (u = s s 1 s 2 { 1 u+1, if n u;, if n > u which can be represented by the following infinite averaging matrix, C = n n n Applying the Cesàro method to any sequence s yields the following transformed sequence So in general: Cs = s, s + s 1, 2 Cs(u = 1 u + 1 s + s 1 + s 2, 3 If the sequence Cs is convergent, then the sequence s is called (C, 1-convergent and the limit is called the (C, 1-value of the sequence Additionally, the series k= a k that bijects with the sequence s is called (C, 1-summable u i= s i

6 6 JACK ENYEART Example 24 (Hölder means Hölder methods are a generalization of the Cesàro method The k th Hölder method, referred to as the H k method, is described by applying the Cesàro method k successive times to a given sequence Thus, the infinite matrix that describes the weights for the k th Hölder method is given by C k The general expression for the weights of the k th Hölder method will not be listed explicitly, as this expression is rather involved However, the following theorem ensures that all the rows of the matrix C i will satisfy the two necessary weighting properties, (21a and (21b Theorem 25 If A and B are both lower-triangular averaging matrices, then AB will be a lower-triangular averaging matrix as well Proof Since A (ij = a ij and B (ij = b ij i, j N, it is obvious that AB (ij i, j N It is also clear that lower-triangularity is preserved in matrix multiplication Thus, we need only show that AB satisfies (21b Since A and B satisfy property (21b individually, and are lower triangular we deduce a ij = j By matrix multiplication: i j= a ij = j AB (ij = b ij = i b ij = 1 j= i a ik b kj Since AB is lower-triangular, the sum of the elements on the i th row of AB, for any i, is expressed as: AB (ij = j i AB (ij = j= i j= k= k= This shows that AB is also an averaging matrix i i ( i ( i a ik b kj = a ik b kj = 1, i Thus, according to the previous theorem, the space of lower-triangular averaging matrices is closed under multiplication There are other interesting properties of averaging matrices that are preserved under multiplication, and we will discuss these later in the paper Example 26 (Cesàro means Multiplication by the matrix C above is equivalent to taking a standard average, which essentially involves two steps: a summation, and then a division In this way, the Hölder method can be thought of process of repetitive averaging that is, a summation, followed by a division, followed by a summation, followed by a division, etc While the Hölder method involves alternating between the process of summation and division, the Cesàro method involves a finite amount of summations performed successively, followed by only a single division Although perhaps not as apparent as in the case of the Hölder method, the Cesàro method is indeed an averaging method a fact that can be verified through investigation of its weighting sequences To introduce the general Cesàro method, we will first define a few terms Given any sequence s = s, s 1,, s n,, we define k= S 1 n = s + s 1 + s s n j=

7 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 7 and define S k n by the follow recursion relation: Sn k = S k 1 + S1 k Sn k 1 We now go through the following steps to describe the weighting sequence of the Cesàro method and explain the need to define the above terms Consider the generating functions: n S 1 nx n = (1 + x + x 2 + n s n x n = (1 x 1 n s n x n (1 x n S 1 nx n = n s n x n Similarly, so that, (1 x Snx 2 n = Snx 1 n, n n (1 x 2 Snx 2 n = s n x n n n Iterating the above procedure we obtain the following identity (1 x k n S k nx n = n s n x n n S k nx n = (1 x k n s n x n (22 Taking the (k 1 st derivative of the equality (1 x 1 = n x n x < 1 we deduce ie, (k 1! (1 x k = n! (n k + 1! xn k+1, n 1 (1 x k = n ( n x n k+1 k 1 Since the terms in the above sum are for n < (k 1, we begin the the summation at n = (k 1, which yields 1 (1 x k = ( n + k 1 x n (23 k 1 n Thus, by (22 and (23: Hence n S k nx n = (1 x k n s n x n = n S k n = m= ( n + k 1 k 1 n ( n m + k 1 s m k 1 x n n s n x n

8 8 JACK ENYEART Thus, we can see how to partially construct a matrix to model the weights of the k th Cesàro method: ( δ k 1 k 1 ( δ k ( 1 k 1 δ k 1 1 k 1 ( δ k+1 ( 2 k 1 δ k ( 2 k 1 δ k 1 2 k 1 ( δ n+k 1 ( n+k 2 ( n k 1 δn k 1 δ k 1 n k 1 Now we just need to define the coefficient δ n, for each row, to ensure that each row sums to 1 The sum of the n th row is: n ( { ( ( ( } i + k 1 k 1 k n + k 1 δ n = δ n k 1 k 1 k 1 k 1 i= From the classical identity n ( ( i + k 1 n + k = k 1 k i= we conclude that if the sum of the nth is 1 if and only if δ n := 1 ( n+k k It is worth noting that the denominator of each coefficient δ n is exactly the number of terms of our original sequence that have been summed in each particular row For example, when n = 2, we have summed ( ( k+1 k 1 copies of s(, k ( k 1 copies of s(1, and k 1 copies of s(2, for a total of ( k+2 k terms thus, we divide by this number since this is an averaging method That being said, we now we have a legitimate sequence of weights for the Cesàro method that we can write explicitly The weighting sequence for the k th Cesàro methodis, { ( δ u n+k 1 u w n (u = k 1, if n u;, if n > u where δ u is defined as above Note that when k = 1, we obtain the sequence of weights represented by the matrix C given before thus the 1 st Cesàro method is equivalent to the 1 st Hölder method, as we expected Example 27 (The Euler method The weights of Euler method also involve binomial coefficients, yet the distribution of the weights is concentrated toward the middle, rather than the beginning (as was the case for Cesàro method For the Euler method, we define the following weighting system: ( u w n (u = ε u, where ε u := 1 n 2 u

9 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 9 Thus, the transformation matrix of the Euler Method is the following: ( ε ( ε 1 ( 1 ε ( ε 2 ( 2 ε 2 ( 2 1 ε E = ( ε u ( u ( u εu 1 ε u u u It is clear that the sum of any row of E is 1, since ( u u ( 1 u ε u = n 2 u = 1 u ( u n 2 u = 1 u n n n= Given any sequence s, if the lim u Es(u exists and is finite, then we say that s is E convergent and its Euler Sum is (lim u Es(u At this point in the paper, we have seen many examples of averaging methods that were able to be represented by infinite matrices, and we claimed that these were all legitimate summation methods That is, we claimed that they were able to induce convergence in some divergent sequences, while not affecting the limit of already convergent sequences We will now prove this claim, and in fact, elucidate the key properties that determine whether or not a matrix will be a summation method Theorem 28 (Characterization of regular averaging matrices Suppose T is a summation method given by an averaging matrix (c mn m,n Then the summation method is regular if and only if lim c mn =, n (24 m n= Proof First we will prove sufficiency of (24 Suppose s is a convergent sequence with limit s Define t m := c mn s n We want to show that lim m t m = s Thus, we need to show that n= n= ε >, M > : t m c m s < ε, m > M Observe that t m s (21b = c mn s n c mn s = c mn (s n s n= n= c mn s n s (25 Since s n s as n, N > such that s n s < ε, n > N (26 2K n=

10 1 JACK ENYEART We now express the infinite sum on the right hand side of the inequality (25 as follows: N c mn s n s = c mn s n s + c mn s n s (26 n= N n= n= c mn s n s + ε 2K n>n c mn (i n>n N c mn s n s + ε 2 Now consider: N c mn s n s = c m s s + c m1 s 1 s + + c mn s N s Set Thus, n= n= A N := max{ s s, s 1 s,, s N s } N c mn s n s ( c m + c m1 + + c mn A N n= But by (24, lim m c mi i Thus for any i =, 1,, N, there exists M i > such that m > M i : ε c mi < 2(N + 1A N Let M := max{m, M 1,, M N } Thus, if m > M then N N ε c mn s n s < A N = ε 2(N + 1A N 2 Therefore t m c m s n= c mn s n s = n= n= N c mn s n s + n= c mn s n s < ε 2 + ε 2 = ε This shows that lim m t m = s We have thus shown the sufficiency of (24 To prove the necessity we argue by contradiction Thus we assume that T is regular yet (24 is not satisfied Then there exists N such that the sequence of terms on the N-th column does not converge to zero Consider the sequence s = s, s 1,, s n,, where {, n N; s n = 1, n = N Clearly lim n s n = Thus, since T is regular, lim m Ts = On the other hand, we have Ts(m = c mn s n = c mn s N = c mn n= Thus lim Ts = lim c m,n = lim m m This contradiction implies that (24 is necessary n s n n>n (Contradiction!

11 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 11 The reader is now encouraged to revisit the aforementioned averaging method matrices, and verify that each one is indeed regular that is, each one satisfies the conditions of Theorem 28 Now that we specific conditions for regularity, we can give a more general proof of Theorem 25 Theorem 29 If A and B are both regular averaging matrices, then AB will be a regular averaging matrix as well Proof AB satisfies (21a and (21b automatically, by Theorem 25 We need only show that it satisfies (24 listed above After matrix multiplication, the entry AB (ij is given by: We want to show that: i AB (ij = a i,k b k,j k= lim AB (ij = j N i Let j be some fixed column, and let ε > be arbitrarily small Since B satisfies (iii, we know that N > such that b i,j < ε 2 for every i > N Since N is finite, we know that N k= b k,j is bounded, so K such that: N b k,j < K Also, A satisfies (24, so Set Now consider: For all i > M : AB (ij = k= M : a i, < ε 2K i > M M 1 : a i,1 < ε 2K i > M 1 M N : a i,n < ε 2K i > M N M := max{m, M 1,, M N } i a i,k b k,j = k= N a i,k b k,j k= ε 2K N a i,k b k,j + k= N b k,j k= i a i,k b k,j k>n ε 2K K ε 2 Employing how we ve defined N, as well as using the fact that j a i,j = 1: i i k>n a i,k b k,j ε 2 k>n a i,k ε 2

12 12 JACK ENYEART Thus, i N i AB (ij = a i,k b k,j = a i,k b k,j + a i,k b k,j ε 2 + ε 2 = ε k= k= k>n lim i AB (ij = Example 21 (Abel Summation In this case, U = [, 1], u = 1 For any u [, 1, we let In this case, the u average of a sequence s w n (u = (1 uu n, n (1 u n s n u n The sequence is called Abel convergent or A convergent if the limit lim n 1 (1 u s n u n n exists and is finite When s n are the partial sums of the series n a n, then the notion A convergence can be rephrased in a very intuitive way Denote by a(t a generating series of the sequence (a n n a(t = n Then the generating series of the sequence s n is Hence, n s n t n = (1 + t + t 2 + n (1 u n a n t n s n u n = n a n t n = 1 1 t a n u n a n t n n We see that the series n a n is A-convergent if the following two things happens: The series n a nu n is convergent for any u [, 1 The lim u 1 n a nu n exists and it is finite This limit is called the Abel sum of the series For example, the alternating series n ( 1n is A-convergent and its Abel sum is 1 2 Indeed, n ( 1 n u n = ( u n = 1 n 1 + u, u [, 1

13 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 13 3 Abelian theorems We now prove a theorem that demonstrates how the Abelian Method is indeed stronger than any Cesàro Method, in the sense that any series that is Cesàro summable is immediately Abel summable, but not vice-versa Theorem 31 If the series n a n is Cesàro summable to S then it is also Abel summable, with the same sum, S In other words, if a n = S (C, 1, then a n = S (A Proof The proof of the theorem will follow immediately form the following result Lemma 32 If d n >, d n =, d n x n is convergent when x < 1 and c n Sd n, where S, then C(x = c n x n SD(x = S d n x n, as x 1 Let us first explain how we can deduce Theorem 31 from Lemma 32 Without loss of generality, we may assume that S Set Because of the Cesàro convergence we have s n := a + + a n, S 1 n := s + + s n S 1 n Sn as n In particular, this implies that the series n S1 nx n is absolutely convergent for x < 1 We conclude that the series f(x = n a n x n = (1 x 2 n is absolutely convergent for x < 1 Using the identity we deduce S 1 nx n (1 x 2 = d dx (1 x 1 = nx n 1 = (n + 1x n a n x n 1 S = (1 x 2 Snx 1 n 1 = n x n (n + 1x n n n But recall a n = S(C, 1, which implies S 1 n ns (so of course S 1 n (n + 1S Thus, by Lemma 32 f(x S as x 1, so a n = S(A Proof of Lemma 32 Without loss of generality, we may suppose S = 1 Because c n d n, for any ε >, there exists N such that n > N ensures that We obtain and similarly, C(x = N c n x n + n= 1 ε < c n d n < 1 + ε c n x n (1 + εd(x + n>n C(x (1 εd(x N d n x n n= N c n x n n= N c n x n n=

14 14 JACK ENYEART Since d n > we see that, Thus, D(x N d n x n n= lim D(x lim x 1 x 1 N N d n x n Since D(x as x 1, when we divide both bounding inequalities of C(x by D(x we see that C(x lim sup x 1 D(x 1 + ε, C(x lim inf x 1 D(x 1 ε, and so C(x D(x We now know that any Cesàro summable series is automatically Abel-summable Thus, to show that the Abel method is stronger than Cesàro, we need only produce a series that is Abel-summable, but not Cesàro-summable Example 33 Consider a n = ( 1 n+1 n As usual, we define n k= a k = s n We can see n a n clearly by looking at the sequence of partial sums, s = s, s 1, s 2,, s n, =, 1, 1, 2, 2, 3, 3, 4, n= Thus, we can write s n explicitly as, { s n = n 2, n even; n+1 2 n odd which obviously diverges as n Additionally, a n is not Cesàro-summable (C, 1, ie Cs diverges Recall that Cs(u = 1 u s i u + 1 Now we investigate the sequence partial sums of Cs: i= N Cs = Cs(, Cs(1, Cs(n =, 1 2,, 1 2,, {, n even; Cs(n = 1/2 n odd Thus, Cs diverges as claimed, and a n is not Cesàro-summable method to the same series n( u n n a n u n = n ( 1 n+1 nu n = n Now apply the Abel But we can quickly derive the following identity for the expression on the right-hand side: ( u n = 1 u [, u n Differentiating both sides, and multiplying by ( u we obtain, n( u n u = (1 + u 2 u [, 1 n

15 Therefore, we have and REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 15 a n u n = n lim u 1 n u (1 + u 2 u [, 1 a n u n = lim u 1 u (1 + u 2 = 1 4 so n a n is Abel-summable Thus, we have found a series that is Abel-summable, but not Cesàro-summable Using this fact, along with the statement of Theorem 31, we see how the Abel method is stronger than the Cesàro method (ie it can sum all of the Cesàro-summable series, and more 4 Tauberian theorems We have so far proved the following sequence of implications usual convergence Cesàro summability Abel summability In this section we want to describe conditions when the opposite implications hold Such conditions are known in the literature as Tauberian conditions Before starting, we briefly explain notation that will be used x n = O L (y n K : x n > Ky n n Our first result The following theorem lists conditions that guarantee that an Abel-summable series will be Cesàro-summable as well Theorem 41 If a n = s(a, a n is real, and s n = O L (1, then a n = s(c, 1 To prove 41 we follow the strategy in [1] We first observe that Theorem 41 is a consequence of the following more general result Theorem 42 If f(x = a n x n C 1 x when x 1, and a n is real, and a n = O L (1, then (a + a a n = s n Cn In fact, we can easily show that Theorem 41 is a corollary of Theorem 42 conditions of Theorem 41 are satisfied, then s n x n = 1 a n x n s 1 x 1 x, n n (41 If the and s n = O L (1 Thus, assuming Theorem 42 is true, we can apply it to s n (rather than a n, as used in Theorem 42 to obtain (s + s s n sn s + + s n s n and so a n = s(c, 1 Thus Theorem Now we progress to the substantial proof of Theorem 42 (and consequently 41 We will prove this as a special case of the theorem that involves Stieltjes integrals Theorem 43 If α(t increases with t, I(y = e yt dα(t (42 is convergent for y >, and I(y Cy 1 where C, when y, then α(t Ct

16 16 JACK ENYEART Proof We will need two lemmas to complete this proof Lemma 44 If g(x is a real function, and Riemann integrable in (, 1, then there are polynomials p(x and P (x such that p(x < g(x < P (x and 1 (P (x p(xdx = g(x = e t (P (e t p(e t dt < ε (43 { 1, a x b;, otherwise where a < b 1 Let ε > be arbitrary Although g is discontinuous, we can certainly find a continuous function h(x such that g h, 1 (h gdx < ε/6 By Stone-Weierstrass theorem, there exists polynomial Q(x such that h Q < ε/6 for all x [, 1] If P(x = Q + ε/6, then g h < P and since 1 1 P g dx P g P Q + Q h + h g 1 P Q dx + 1 Q h dx + 1 h g dx < ε/2 We can go through an entirely symmetric process for p such that p < g to ensure that g p dx < ε/2 This being true, along with the fact that p < g < P implies 1 P g dx + 1 g p dx = 1 (P gdx + 1 (g p dx = 1 (P pdx < ε It follows from the previous argument that the Lemma applies to any step function containing a finite number of jumps We know that if g is a Riemann integrable function, then there are finite step-functions g 1 and g 2 such that g 1 g g 2, 1 (g 2 g 1 dx < ε/3 We now associate polynomials p 1, P 1 with g 1, and p 2, P 2 with g 2, in the way that the Lemma suggests Then So 1 1 (P 2 p 1 dx = (P 2 g 2 dx < ε/3 1 1 (g 1 p 1 dx < ε/3 [(P 2 g 2 + (g 2 g 1 + (g 1 p 1 ]dx < 3ε We can carry out a simple substution in the above integral, with x = e t, 1 (P 2 (x p 1 (xdx = e t (P 2 (e t p 1 (e t dt

17 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 17 Lemma 45 Suppose that α(t increases with t, that I(y is convergent for y, that I(y Cy 1, and that g(x is of bounded variation in (,1 Then χ(y = e yt g(e yt dα(t exists for all positive values of y except values τ/ω, where ω is a discontinuity of α, and τ a discontinuity of g(e t ; and χ(y C e t g(e t dt (44 y when y through any sequence of positive values which excludes these exceptional values Proof Since τ and ω are at most countable individually, their intersection is also countable (ie the points that we throw away, so we exclude at most a countable set of values, call them y k, of y, which will not affect the value of the integral Because a function of bounded variation is Riemann integrable, by Lemma 44 we can choose polynomials p and P such that p < g < P, e t [P (e t p(e t ]dt < ε So e t p(e t dt < e t g(e t dt < e t P (e t dt; and since α increases with t, e yt p(e yt dα(t e yt g(e yt dα(t e yt P (e yt dα(t, when y y k and thus e yt e nyt dα(t = e (n+1yt dα(t e yt P (e yt dα(t C y C (n + 1y = C y e t P (e t dt e t e nt dt, Hence, if y through any sequence of not including values y k, we have lim sup y y e yt g(e yt dα(t lim sup y y e yt P (e yt dα(t = C e t P (e t dt < e t g(e t dt + Cε In a completely similar process involving p, we arrive at the following relationship, lim inf y e yt g(e yt dα(t > C e t g(e t dt Cε; y Which essentially proves (44

18 18 JACK ENYEART Now we are able to prove Theorem 43 Suppose, without loss of generality, that α( = We take { x 1, if e 1 x 1; g(x = if x < e 1 ; So clearly g(e t = e t for t < 1 and g(e t = for t > 1 Then we have, And χ(y = e yt g(e yt dα(t = e t g(e t dt = 1 y dα(t = α( 1 y dt = 1 Thus, by Theorem 43, α(y 1 Cy 1 when y, or equivalently, α(t Ct when t, with the exception being made in either case of the countable sets of discontinuity In this specific case we need only worry about one countable set of discontinuities, name the values of α, thus, α(t Ct as t through points of continuity of α(t And since α increases with t, it is true without special stipulation Now we will demonstrate that Theorem 43 implies Theorem 42 Let α(t be a stepfunction with jumps a n for t = n, so I(y = S(y = a n e ny, and S(y Cy 1, which implies s n Cn This is very close to the statement of Theorem 43 We just need the same conclusion for a n = O L (1 Assume the conditions of Theorem 43 are satisfied, and there exists H such that a n > H Then let b n = a n + H > Then bn x n = a n x n + H 1 x C + H 1 x Thus, assuming what we ve just shown in the previous paragraph, b + b b n (C + Hn, and so n k= a k = s n Cn, which is Theorem 42 We will now state and prove the second of the two major Tauberian theorems discussed in this paper This theorem will describe conditions that, when satisfied, ensure a Cesàroconvergent sequence is convergent in the normal sense Theorem 46 If a n = s(c, 1, for a n real, and a n = O L (n 1, then a n converges Proof We follow the elegant approach of [2] First, we let s n = a + + a n S n = s + s s n We consider the following discrete analog of Taylor s formula for h >, where a ξ is a number such that S n+h = S n + hs n h(h + 1a ξ, (45 min a k a ξ max a k for n < k n + h

19 REGULARIZATION OF DIVERGENT SERIES AND TAUBERIAN THEOREMS 19 To prove (45, consider S n+h S n = (s + + s n+h (s + + s n = (s n+1 + s n s n+h S n+h S n = h(a + a 1 + a n + (ha n+1 + (h 1a n+2 a n+h The bound on a ξ follows from S n+h S n = hs n + (ha n+1 + (h 1a n a n+h 1 h(h + 1 min 2 a k (ha n+1 + (h 1a n a n+h 1 h(h + 1 n<k n+h 2 max a k n<k n+h The remaining expression (45 follows from the Intermediate Value Theorem We may now assume that a n = (C, 1 since, if a n = L(C, 1, then we can replace a by a L and every partial sum will be decreased by L So we need only consider an = (C, 1, which implies lim S s + s s n n = lim n n n We can re-write equation (45 in a form that isolates s n, Since h >, we may divide to give hs n = 1 2 h(h + 1a ξ + S n+h S n = We now use a n K n s n = S n+h S n h 1 2 (h + 1a ξ for some K > and S n nε s n 2n + h ε + K h + 1 h 2n Since this inequality holds for all h, we take h 2n ε/k, which yields the following arbitrarily small positive upper bound on s n s n < 3 Kε Now we go through a similar process to obtain a lower bound for s n, in which we use an analogous form of equation 45 for h < For the sake of ease, let h = i < Through an entirely similar process as performed above, we see that or equivalently, S n S n i = s n + s n s n i+1 S n S n i = s n + (s n a n + (s n (a n + a n (s n (a n + a n 1 + a n i+2 We can now group all the s n terms together to give, n is n = S n S n i + (ν (n i + 1a ν which can be further simplified to ν=n i+1 s n = S n S n i i (i 1a γ

20 2 JACK ENYEART where min a j a γ max a j for n i + 1 j n Now, we once again use that to give the lower bound, a n K n s n for some K > and S n nε (2n + i ε K i 1 i 2(n i and taking i 2n ε/k we obtain a lower bound on s n, (This may need an added step/additional justification s n > 3 Kε So together with the expression for the upper bound we have 3 Kε < s n < 3 Kε and since ε was arbitrary, we conclude that s n Remark 47 One can prove (see [3, I5] that if n is Abel convergent and a n = O L (n 1 then s n = O L (1 Theorem 41 then implies that n a n is Cesaro convergent Invoking Theorem 46 we deduce that n a n is convergent in the usual sense References [1] G H Hardy: Divergent Series, AMS Chelsea Publishing, 1991 [2] H D Kloosterman: On the convergence of series summable (C, r and on the magnitude of the derivatives of a function of a real variable, J London Math Soc, 15(194, [3] J Korevaar: Tauberian Theory A Century of Developments, Springer Verlag, 24 University of Notre Dame