Tauber’s theorem and Karamata’s proof of the Hardy-Littlewood tauberian theorem
The following lemma is attributed to Kronecker by Knopp.11 1 Konrad Knopp, Theory and Application of Infinite Series, p. 129, Theorem 3.
Lemma 1 (Kronecker’s lemma).
If
Proof.
Suppose that
∎
We now use the above lemma to prove Tauber’s theorem.22 2 cf. E. C. Titchmarsh, The Theory of Functions, second ed., p. 10, §1.23.
Theorem 2 (Tauber’s theorem).
If
Proof.
Let
Next,
because
Take
Also, using
we have
Now,
and then
proving the claim. ∎
Lemma 3.
Let
For
and
Proof.
There is some
further, take
Take
For
Define
Because
and
On the other hand, take
One checks that for
Define
which is continuous and satisfies
Because
and
∎
The following is the Hardy-Littlewood tauberian theorem.33 3 E. C. Titchmarsh, The Theory of Functions, second ed., p. 227, §7.53, attributed to Karamata.
Theorem 4 (Hardy-Littlewood tauberian theorem).
If
then
Proof.
For any
as
(1) |
Define
Let
and
Because the coefficients
Taking lower limits and then using (1) we obtain
The above two inequalities do not depend on the polynomials
and
Thus
(2) |