list 35 publications in the bibliography appended to his obituary, and also a search performed on the "Jahrbuch über die Fortschritte der Mathematik" database results in a list 35 mathematical works authored by him, spanning a period of time from 1891 to 1940. However, cites two papers on actuarial mathematics which do not appear in these two bibliographical lists and Binder's bibliography of Tauber's works, while listing 71 entries including the ones in the bibliography of and the two cited by Hlawka, does not includes the short note so the exact number of his works is not known. According to, his scientific research can be divided into three areas: the first one comprises his work on the theory of functions of a complex variable and on potential theory, the second one includes works on linear differential equations and on the Gamma function, while the last one includes his contributions to actuarial science. give a more detailed list of research topics Tauber worked on, though it is restricted to mathematical analysis and geometric topics: some of them are infinite series, Fourier series, spherical harmonics, the theory of quaternions, analytic and descriptive geometry. Tauber's most important scientific contributions belong to the first of his research areas, even if his work on potential theory has been overshadowed by the one of Aleksandr Lyapunov.
Tauberian theorems
His most important article is. In this paper, he succeeded in proving a converse to Abel's theorem for the first time: this result was the starting point of numerous investigations, 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 is summable according to a given summability method and satisfies an additional condition, called "Tauberian condition", then it is a convergent series. Starting from 1913 onward, G. H. Hardy and J. E. Littlewood used the term Tauberian to identify this class of theorems. Describing with a little more detail [|Tauber's 1897 work], it can be said that his main achievements are the following two theorems: This theorem is, according to, the forerunner of all Tauberian theory: the condition is the first Tauberian condition, which later had many profound generalizations. In the remaining part of his paper, by using the theorem above, Tauber proved the following, more general result:
is Abel summable and
.
This result is not a trivial consequence of. 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 jointly with Tauberian condition on the other. claims that this latter result must have appeared to Tauber much more complete and satisfying respect to the 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, though it has its rightful place in all detailed developments of summability of series.
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. Precisely, considers the real part and imaginary part of a power series, where
Under the hypothesis that is less than the convergence radius of the power series, Tauber proves that and satisfy the two following equations: Assuming then, he is also able to prove that the above equations still hold if and are only absolutely integrable: 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 and are equivalent to the following pair of Hilbert transforms: Finally, it is perhaps worth pointing out an application of the results of, given by Tauber himself in the short research announce :
