# Proof of the Hardy-Littlewood Tauberian theorem

Can someone point me to a proof of the Hardy-Littlewood Tauberian theorem, that is suitable enough to be shown to high school students? (with knowledge of calculus, sequences and series of course)

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Why should high-school students bother about Tauberian theorems? Ask 100 professional mathematicians what Tauberian theorems are all about, and you will get $\leq1$ usable response. –  Christian Blatter Sep 4 '12 at 14:45

## 2 Answers

Have you looked at the presentation in Titchmarsh's Theory of Functions (Section 7.5)? The only non-elementary part of the argument is Weierstrass's approximation theorem, which you can probably assume as a fact. The preliminary material given also include an "easy" special case where the exposition certainly can be understood by someone with knowledge of calculus, sequences, and series.

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Unfortunately, I've been unable to get a hold of a copy of Titchmarsh's book so far. –  Ishihara Sep 4 '12 at 11:55
It is available on the Internet Archive. –  Willie Wong Sep 4 '12 at 12:05
Found it, thank you. –  Ishihara Oct 1 '12 at 15:27

Take a look at Chapter 1, sections 11 and 12 of Tauberian Theory: A Century of Development by Jacob Korevaar. He also has many references.

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