Quantitative Tauberian theorems omas Powell University of Bath Mathematical Logic: Proof Theory, Constructive Mathematics MFO (delivered online) 11 November 2020 ese slides are available at https://t-powell.github.io/talks Introduction Abel’s theorem Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever 1 X ai = s i=0 it follows that F(x) ! s as x % 1: is is a classical result in elementary analysis called Abel’s theorem (N.b. it also holds in the complex setting). You can use it to e.g. prove that 1 X (−1)i = log(2): i + 1 i=0 Does the converse of Abel’s theorem hold? NO. For a counterexample, define F :(−1; 1) ! R by 1 1 X F(x) = = (−1)ixi 1 + x i=0 en 1 F(x) ! as x % 1 2 but 1 X (−1)i does not converge i=0 Tauber’s theorem fixes this Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever F(x) ! s as x % 1 AND jnanj ! 0 it follows that 1 X ai = s i=0 is is Tauber’s theorem, proven in 1897 by Austrian mathematician Alfred Tauber (1866 - 1942). Tauberian theorems e basic structure of Tauber’s theorem is: 1 X i Let F(x) = aix i=0 en if we know (A) Something about the behaviour of F(x) as x % 1 (B) Something about the growth of fang as n ! 1 en we can conclude P1 (C) Something about the convergence of i=0 ai. is basic idea has been considerably generalised e.g. for Z 1 F(s) := a(t)t−s dt 1 and has grown into an area of research known as Tauberian eory. ere is now a whole textbook (published 2004, 501 pages) Tauberian theorems have an interesting structure convergence + growth condition ) convergence Can we give a quantitative interpretation of these theorems e.g. rate of convergence + quantitative growth condition ) rate of convergence In many cases, this would seem to pose a real challenge, as the proofs of Tauberian theorems are typically based on complicated analytic techniques. Remainder of the talk: 1 Some very simple results (published) 2 Some slightly less simple results (unpublished) 3 Some rough ideas for the future (speculation) Simple results Cauchy variants of Abelian and Tauberian theorems P1 i Pn From now on, fang is a sequence of reals, F(x) := i=0 aix and sn := i=0 ai. e following may well feature in some form elsewhere in the literature... eorem (Abel’s theorem, Cauchy variant) Suppose that • fsng is Cauchy, • fxmg 2 [0; 1) satisfies limm!1 xm = 1. en limm;n!1 jF(xm) − snj = 0. eorem (Tauber’s theorem, Cauchy variant) Suppose that • 1 fF(vm)g is Cauchy, where vm := 1 − m , • an = o(1=n). en limm;n!1 jF(vm) − snj = 0. A finitization of Abel’s theorem eorem (Finite Abel’s theorem, P. 2020) Let fang and fxkg be arbitrary sequences of reals, and L 2 N a bound for fjsnjg. Fix some " 2 Q+ and g : N ! N. Suppose that N1; N2 2 N and p ≥ 1 are such that " 1 " jsi − snj ≤ and ≤ 1 − xm ≤ 4 p 8LN1 for all i; n 2 [N1; maxfN + g(N); lg] and all m 2 [N2; N + g(N)] where l 8Lp m N := maxfN ; N g and l := p · log 1 2 " en we have jF(xm) − snj ≤ " for all m; n 2 [N; N + g(N)]. Simple application of Gödel’s functional interpretation to textbook proof of Abel’s theorem. Nothing deep Corollary: A quantitative Abel’s theorem Suppose that jsnj ≤ L for all n 2 N and moreover (A) φ is a rate of Cauchy convergence for fsng, or equivalently ! n X 8" > 0 8m; n ≥ φ(") ai ≤ " i=m (B) is a rate of convergence for xm % 1. en a rate of convergence for limm;n!1 jF(xm) − snj = 0 is given by " ΦL,φ,ψ(") := max φ("=4); 8Lφ("=4) A finitization of Tauber’s theorem eorem (Finite Tauber’s theorem, P. 2020) 1 Let fang be an arbitrary sequence of reals, and L a bound for fjanjg. Define vm := 1 − m , and fix some " 2 Q+ and g : N ! N. Suppose that N1; N2 2 N are such that " " ijaij ≤ and jF(vm) − F(vn)j ≤ 8 4 for all i 2 [N1; l] and all m; n 2 [N2; N + g(N)] where 2LN2 l 4Lp m N := max 1 ; N and p · log for p := N + g(N) " 2 " en we have jF(vm) − snj ≤ " for all m; n 2 [N; N + g(N)]. Corollary: A quantitative Tauber’s theorem Suppose that janj ≤ L for all n 2 N and moreover (A) φ is a rate of Cauchy convergence for njanj ! 0 1 (B) is a rate of convergence for fF(1 − m )g 1 en a rate of convergence for limm;n!1 jF(1 − m ) − snj = 0 is given by 2Lφ("=8)2 ΦL,φ,ψ(") := max ; ("=4) " Again, nothing deep in any of the above results, but they can be used in conjunction with known results from proof mining to generate concrete quantitative lemmas: Lemma Let fang be a sequence of positive reals whose partial sums fsng are bounded above. en 1 limm;n!1 jF(1 − m ) − snj = 0. Note: If fsng is a Specker sequence then there is no computable rate of convergence 1 for limm;n!1 jF(1 − m ) − snj = 0. Lemma Let fang be a sequence of positive reals and L a bound for the partial sums fsng. en for any " 2 Q+ and g : N ! N we have 1 9N ≤ ΓL("; g) 8m; n 2 [N; N + g(N)] (jF(1 − m ) − snj ≤ ") for ΓL("; g) given as follows: l 8Lf (d4L="e)(0) m • Γ ("; g) := L " l m • 8Lpa f (a) := pa · log " l m • 8La pa := ~g " for ~g(x) := x + g(x). ese results (and more) appear in: Less simple results Tauber’s theorem was first extended by Littlewood (1911) Littlewood Tauberian theorem Let fang be a sequence of reals, and suppose that the power series 1 X i F(x) := aix i=0 converges on jxj < 1. en whenever F(x) ! s as x % 1 AND jnanj ≤ C for some constant C, it follows that 1 X ai = s i=0 One of Littlewood’s first major results. In A Mathematical Education he writes (of this period) “ On looking back this time seems to me to mark my arrival at a reasonably assured judgement and taste, the end of my "education". I soon began my 35-year collabora- tion with Hardy.” One of first papers of this collaboration (1914): e Hardy-Littlewood Tauberian theorem P1 i Let fang be a sequence of reals, and suppose that i=0 aix converges for jxj < 1. en whenever 1 X i (1 − x) aix ! s as x % 1 AND an ≥ −C i=0 for some constant C, it follows that n 1 X ai ! s as n ! 1 n i=0 ey later used this result to give a new proof of the prime number theorem: x π(x) ∼ log(x) A quantitative analysis of the Littlewood Tauberian theorem eorem (Finite Littlewood theorem, (unpublished)) Suppose that fang satisfies njanj ≤ C and L is a bound for jF(x)j on (0; 1). en there are constants K1 and K2 such that whenever N1 satisfies 1 X − = − = " a (e ik m − e jk n) ≤ k C=" k=0 4K2 for all (i; m); (j; n) 2 [1; d] × [dN1; dN + g(dN)], where =" l 4LKC m K C N := 2 · N and d := 1 " 1 " then we have P1 a e−k=m − Pn−1 a ≤ " for all m; n 2 [N; N + g(N)]. k=0 k k=0 k Not quite as simple as Tauber’s theorem: Requires in particular quantitative results bounding degree and coeficients of approximating polynomials. Corollary: A quantitative Littlewood’s Tauberian theorem (unpublished) Suppose L is a bound for jF(x)j on (0; 1), and moreover (A) φ :(0;") ! N is a rate of Cauchy convergence for F(x) in the sense that 8" > 0 8x; y 2 [e−1/φ("); 1)(jF(x) − F(y)j ≤ "): (B) C is such that fang satisfies njanj ≤ C en a rate of Cauchy convergence :(0; 1) ! N for fsng i.e. 8" > 0 8n ≥ (")(jsm − snj ≤ ") is given by 1 l DC=" m C;L,φ(") := Lu · φ for u := u " for a suitable constant D. Rates of convergence for Littlewood’s Tauberian theorem have already been studied How does this compare to known remainder theorems? A special case of our analysis Suppose that jF(x) − sj ≤ K(1 − x) for some constant K. en the correponding rate of Cauchy convergence for F(x) ! s is l 2K m φ(") = " Plugging this in to our corollary yields the following rate of Cauchy convergence for the partial sums fsng: C=" (") := LD1 for suitable D1. Rearranging, we can show that C js − sj ≤ 1 N log(N) for suitable C1. So far we have • A quantitative analysis of Abel and Tauber’s theorems (not deep, but maybe useful and a nice simple example of finitization). • A quantitative analysis of Littlewood’s Tauberian theorem (more interesting, work in progress...) Open questions: 1.