A CENTURY OF TAUBERIAN THEORY. david borwein

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1 A CENTURY OF TAUBERIAN THEORY david borwein Abstract. A narrow path is cut through the jungle of results which started with Tauber s corrected converse of Abel s theorem that if a n = l, then a n x n l as x 1. 1

2 Just over a century ago, in 1897, Tauber proved the following corrected converse of Abel s theorem: Theorem T. If a n x n l as x 1, and (T ) na n = o(1), then a n = l. Subsequently Hardy and Littlewood proved numerous other such converse theorems, and they coined the term Tauberian to describe them. In summability language Theorem T can be expressed as: If a n = l (A), where A denotes the Abel summability method, and if the Tauberian condition (T ) holds, then a n = l. The simplest example of an Abel summable series that is not convergent is given by a n := ( 1) n, for which a n = 1 2 (A). 2

3 Tauber s innocent looking theorem was the start of a veritable Tauberian jungle of results which Korevaar, in a recent book, made a very worthwhile effort to organize and present in a coherent manner. The book s 483 pages are densely packed and there are around 8 references. Rather than attempting the impossible task of giving such a comprehensive description of the jungle in the course of a short talk, I will cut a reasonably narrow path through part of it, touching on some of the key results. In 1914 Hardy and Littlewood proved the following generalization of Theorem T in which the strong two-sided Tauberian condition (T ) is replaced by the much weaker one-sided condition (T 1 ): Theorem H-L. If a n x n l as x 1, and (T 1 ) na n C, a positive constant, then a n = l. Note that by changing sign throughout, the Tauberian condition (T 1 ) could be expressed as na n C. 3

4 An interesting, and non-trivial, illustration of the potency of Theorem H-L, is a proof that the series n=1 1 n z, which is absolutely convergent and defines the Riemann zeta function ζ(z) when Rz >1, is not Abel summable on the line z =1+it. This amounts to observing that n=1 1 n 1+it cannot be Abel summable, for if it were Theorem H-L (or even a weaker two-sided version of it) would imply that the series is actually convergent, which it cannot be since Hardy has shown that, for fixed t, N n=1 1 n 1+it = i + l + o(1) as N, tn it where l is finite and independent of N. In fact l turns out to be ζ(1+it). 4

5 Karamata simplified Hardy and Littlewood s proof of Theorem H-L in 193, and in 1952 Wielandt elegantly modified Karamata s proof as follows: Suppose, without loss in generality, that a n x n asx 1. Let F be the linear space of real functions f for which a n f(x n ) asx 1. Then every real polynomial p with p() = is in F. Let g := χ [1/2,1], the characteristic function of [1/2, 1]. Given ε >, there are real polynomials p 1, p 2 with p 1 () = p 2 () = and p 1 (1) = p 2 (1) such that p 1 (x) g(x) p 2 (x) for x 1, and 1 p 2 (t) p 1 (t) dt < ε t(1 t) C. Then, by (T 1 ), a n g(x n ) a n p 1 (x n ) C = C n=1 n=1 p 2 (x n ) p 1 (x n ) n x n q(x n ), x n (1 x n ) q(x n ) C (1 x) n where q(x) := p 2(x) p 1 (x) x(1 x) 5 =: m b k x k. k=

6 Further, as x 1, (1 x) x n q(x n )= m k= b k 1 x 1 x m k+1 k= b k k +1 = 1 q(t) dt < ε C. Hence and likewise lim sup x 1 lim inf x 1 a n g(x n ) <ε, a n g(x n ) > ε. It follows that g F, and therefore, for N = log 2/ log x, a n g(x n )= N a n asx 1. 6

7 Another proof of Theorem H-L is by means of Wiener s powerful Tauberian theorem involving Fourier transforms which he published in 1932: Theorem W. If K L(, ), φ L (, ), e itx K(t) dt x (, ), and then, H L(, ), K(x t)φ(t) dt = o(1) as x, (1) H(x t)φ(t) dt = o(1) as x. 7

8 To prove Theorem H-L with l = by means of Theorem W, let s(x) := a n, and F (x) := a n x n. n x Then, by hypothesis, F (x) =o(1) as x 1, and it follows (fairly easily) from this and (T 1 ) that s(x) =O(1), and hence that, for t>, F (e t )= a n e nt = e tx ds(x) =t e tx s(x) dx. Now take φ(x) :=s(e x ) and K(x) := exp( x e x ). Then K(x t)φ(t) dt = F ( exp( e x ) ) = o(1) as x, and, x (, ), e itx K(t) dt = u ix e u du = Γ(1 + ix). Further, φ(x) = O(1), and it follows from (T 1 ) that, given δ>, x such that φ(y) φ(x) 2δ for x x y x + δ. Taking H := δ 1 χ [,δ] and then H := δ 1 χ [ δ,] in (1), we obtain respectively lim sup x φ(x) 2δ and lim inf x 8 φ(x) 2δ,

9 from which it follows that φ(x) and hence that s(x) as x. Wiener s theorem yields Tauberian theorems for many standard summability methods. Karamata proved various Tauberian theorems, the most famous being the following one about Laplace transforms which he proved in 1931: 9

10 Theorem K. Let A be a non-decreasing, unbounded function on [, ) with A(), and let L be a slowly varying function (i.e., t >, L(xt)/L(x) 1 as x ). Then, for σ, B(x) := e t/x da(t) x σ L(x) as x (i.e., B is regularly varying with index σ) if and only if A(x) xσ L(x) as x. Γ(1 + σ) From this theorem Karamata derived: Theorem K 1. Let A be a non-decreasing, unbounded and regularly varying function on [, ) with A(), and let the function s be continuous and bounded below on [, ). If (2) then (3) e yt s(t) da(t) l 1 A(x) x e yt da(t) as y +, s(t) da(t) l as x. This is also a Tauberian theorem since (3) (2) without the one-sided boundedness condition on s. It follows from a theorem established by Korenblum in 1955 that the condition in Theorem K 1 that A be regularly varying can be replaced by the weaker condition A(y) (4) A(x) 1 when y 1, y>x, x (i.e., log A(x) is slowly oscillating). From this extension of Theorem K 1, I was able to prove: 1

11 Theorem DB. Let A be a non-decreasing, unbounded and function on [, ) with A(), and let the function s be continuous [, ). If (2) and (4) are satisfied, and in addition (5) lim inf{s(y) s(x)} when y x 1, y>x, then s(x) l as x. The proof uses a variant of a method developed by Vijayaraghavan in 1926 to first deduce that s(x) is bounded. 11

12 Theorem DB can be specialized by taking A(x) :=n for n x<n+1,, 1,..., and s(n) :=s n := n a k, k= to obtain as a corollary the following result which Schmidt established in 1925: Corollary. If (6) and s n e ny l e ny = l as y + 1 e y (7) lim inf(s m s n ) when m n 1, m>n (i.e., s n is slowly decreasing), then s n l. Note that (6) is equivalent to (1 x) s n x n = a n x n l as x 1, and that na n > C (7), so that the Corollary generalizes Theorem H-L. 12

13 Another classical Tauberian result concerns the Cesàro method C α,α> 1, and the Borel method B defined by: if 1 ( n+α n ) k= a n = l (C α ), n ( ) k + α a n k l as n ; k a n = l (B), or s n l (B), if e x s n x n n! l as x, where s n := n a k. k= 13

14 Theorem B. If a n = l (B), and (T 2 ) then nan n r C, r, a n = l(c 2r ). In 196 Rajagopal proved a version of this result with a weaker Tauberian condition than (T 2 ). The case r = of the result with (T 2 ) replaced by the stronger two-sided condition na n = O(1) was proved by Hardy and Littlewood in In 1925 Schmidt showed in the case r = that (T 2 ) can be relaxed to lim inf(s m s n ) when < m n, n. Recently Kratz and I established a quantatitive version of Vijayaraghavan s classical 1926 result and used it to give a short proof of the case r = of Theorem B. It is worth noting that though summability C (i.e., convergence) implies summability B, summability C α with α> does not in general imply summability B. Various Tauberian theorems have been used in assorted proofs of the prime number theorem. A particularly interesting one is the following one proved in 1931 by Ikehara, a student and colleague of Wiener s: 14

15 Theorem I-W. Suppose that the function F has the following properties: (i) For Rz >1, F(z) = e zt A(t) dt, where A is a nondecreasing function with A(). (ii) For Rz >1, z 1,F(z) =G(z) + 1, where G(z) z 1 is continuous on the half-plane Rz 1. Then e t A(t) 1 as t. The prime number theorem can be proved with the aid of Theorem I-W as follows: Let A(t) :=ψ(e t ), where ψ(x) := log p. p n x The p s in the sum defining the Chebyshev function ψ are the odd primes, and it is known that the prime number theorem, viz., π(x) := x as x, log x p x is equivalent to ψ(x) x as x. For Rz >1, we have that F (z) = e zt A(t) dt = 1 u z 1 ψ(u) du = ζ (z) zζ(z) = G(z)+ 1 z 1, the function G satisfying the requirements of Theorem I-W since the Riemann zeta function ζ(z) has no zeros in the half plane Rz 1 and is holomorphic in the whole plane, except for a simple pole at z = 1 with residue 1. Hence, by Theorem I-W, e t ψ(e t ) 1ast and so ψ(x) x as x. 15

16 References 1. D. Borwein, Tauberian theorems concerning Laplace transforms and Dirichlet series, Arch. Math. (Basel) 53 (1989), D. Borwein and W. Kratz, A one-sided Tauberian theorem for the Borel Summability method, J.Math. Analysis and Applications 293 (24), G.H. Hardy, Divergent Series, Oxford, G.H. Hardy and J.E. Littlewood, Tauberian theorems concerning power series and Dirichlet s series whose coefficients are positive, Proc. London Math. Soc. (2) 13 (1914), G.H. Hardy and J.E. Littlewood, Theorems concerning the summability of series by Borel s exponential method, Rend. Palermo 41 (1916), S. Ikehara, An extension of Landau s theorem in the analytic theory of numbers, J. Math. and Phys. M.I.T. (2) 1 (1931), J. Karamata, Über die Hardy-Littlewoodschen Umkehrungen des Abelschen Stetigkeitssatzes, Math. Z. 32 (193), J. Karamata, Neuer Beweis und Verallgemeinerung der Tauberschen Sätze, welche die Laplacesche Transformation betreffen, Math. Z. 164 (1931), B. Korenblum, On the asymptotic behaviour of Laplace integrals near the boundary of a region of convergence (Russian), Dokl. Akad. SSSR (NS) 14 (1955), J. Korevaar, Tauberian Theory, a century of developments, Springer,

17 11. R. Schmidt, Über divergente Folgen und lineare Mittelbildungen, Math. Z. 22 (1925), R. Schmidt, Umkersätze des Borelschen Summierungsverfahren, Schriften Köningsberg 1 (1925), A. Tauber, Ein Satz aus der Theorie der uneindliche Reihen, Monatsh. Math. u. Phys. 8 (1897), T. Vijayaraghavan, A Tauberian theorem, J. London Math. Soc. (1) 1 (1926), T. Vijayaraghavan, A theorem concerning the summability of series by Borel s method, Proc. London Math. Soc. (2) 27 (1928), D.V. Widder, The Laplace Transform, Princeton, H. Wielandt, Zur Umkehrung des Abelschen Stetigkeitssatzes, J. Reine Angew Math. 56 (1952), N. Wiener, Tauberian theorems, Annuls of Math. 33 (1932),