Tauberian theorems

His most important article is (Tauber 1897).^[3] In this paper, he succeeded in proving a converse to Abel's theorem for the first time:^[12] this result was the starting point of numerous investigations,^[3] leading to the proof and to applications of several theorems of such kind for various summability methods. The statement of these theorems has a standard structure: if a series ∑ a_n is summable according to a given summability method and satisfies an additional condition, called "Tauberian condition",^[13] then it is a convergent series.^[14] Starting from 1913 onward, G. H. Hardy and J. E. Littlewood used the term Tauberian to identify this class of theorems.^[15] Describing with a little more detail Tauber's 1897 work, it can be said that his main achievements are the following two theorems:^[16][17]

Tauber's first theorem.^[18] If the series ∑ a_n is Abel summable to sum s, i.e. lim_{x→ 1⁻}  ∑+∞
n=0 a_n x ⁿ =  s, and if a_n = ο(n⁻¹), then ∑ a_k converges to s.

This theorem is, according to Korevaar (2004, p. 10),^[19] the forerunner of all Tauberian theory: the condition a_n = ο(n⁻¹) is the first Tauberian condition, which later had many profound generalizations.^[20] In the remaining part of his paper, by using the theorem above,^[21] Tauber proved the following, more general result:^[22]

Tauber's second theorem.^[23] The series ∑ a_n converges to sum s if and only if the two following conditions are satisfied:

1. ∑ a_n is Abel summable and
2. ∑n
   k=1 k a_k = ο(n).

This result is not a trivial consequence of Tauber's first theorem.^[24] The greater generality of this result with respect to the former one is due to the fact it proves the exact equivalence between ordinary convergence on one side and Abel summability (condition 1) jointly with Tauberian condition (condition 2) on the other. Chatterji (1984, pp. 169–170) claims that this latter result must have appeared to Tauber much more complete and satisfying respect to the former one as it states a necessary and sufficient condition for the convergence of a series while the former one was simply a stepping stone to it: the only reason why Tauber's second theorem is not mentioned very often seems to be that it has no profound generalization as the first one has,^[25] though it has its rightful place in all detailed developments of summability of series.^[23][25]

Contributions to the theory of Hilbert transform

Frederick W. King (2009, p. 3) writes that Tauber contributed at an early stage to theory of the now called "Hilbert transform", anticipating with his contribution the works of Hilbert and Hardy in such a way that the transform should perhaps bear their three names.^[26] Precisely, Tauber (1891) considers the real part φ and imaginary part ψ of a power series f,^[27][28]

${\displaystyle f(z)=\sum _{k=1}^{+\infty }c_{k}z^{k}=\varphi (\theta )+\mathrm {i} \psi (\theta )}$

where

• z = re ^iθ with r = | z | being the absolute value of the given complex variable,
• c_k r ^k = a_k + ib_k for every natural number k,^[29]
• φ(θ) = ∑+∞
  k=1 a_kcos(kθ) − b_ksin(kθ) and ψ(θ) = ∑+∞
  k=1 a_ksin(kθ) + b_kcos(kθ) are trigonometric series and therefore periodic functions, expressing the real and imaginary part of the given power series.

Under the hypothesis that r is less than the convergence radius R_f of the power series f, Tauber proves that φ and ψ satisfy the two following equations:

(1)     ${\displaystyle \varphi (\theta )={\frac {1}{2\pi }}\int _{0}^{\pi }\left\{\psi (\theta +\phi )-\psi (\theta -\phi )\right\}\cot \left({\frac {\phi }{2}}\right)\,\mathrm {d} \phi }$

(2)     ${\displaystyle \psi (\theta )=-{\frac {1}{2\pi }}\int _{0}^{\pi }\left\{\varphi (\theta +\phi )-\varphi (\theta -\phi )\right\}\cot \left({\frac {\phi }{2}}\right)\mathrm {d} \phi }$

Assuming then r = R_f, he is also able to prove that the above equations still hold if φ and ψ are only absolutely integrable:^[30] this result is equivalent to defining the Hilbert transform on the circle since, after some calculations exploiting the periodicity of the functions involved, it can be proved that (1) and (2) are equivalent to the following pair of Hilbert transforms:^[31]

${\displaystyle \varphi (\theta )={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\psi (\phi )\cot \left({\frac {\theta -\phi }{2}}\right)\mathrm {d} \phi \qquad \psi (\theta )={\frac {1}{2\pi }}\int _{-\pi }^{\pi }\varphi (\phi )\cot \left({\frac {\theta -\phi }{2}}\right)\mathrm {d} \phi }$

Finally, it is perhaps worth pointing out an application of the results of (Tauber 1891), given (without proof) by Tauber himself in the short research announce (Tauber 1895):

the complex valued continuous function φ(θ) + iψ(θ) defined on a given circle is the boundary value of a holomorphic function defined in its open disk if and only if the two following conditions are satisfied

1. the function [φ(θ − α) − φ(θ + α)]/α is uniformly integrable in every neighborhood of the point α = 0, and
2. the function ψ(θ) satisfies (2).