[This](http://www.ecs.umass.edu/~mduarte/images/IntroCS.pdf) excellent introduction to Compressive Sensing cites a couple of (seemingly) interesting Caratheodory papers from 1907-1911.


These are:


* [46] C. Caratheodory. Uber den Variabilitätsbereich der Koeffizienten von Potenzreihen, die gegebene Werte nicht annehmen. Math. Ann., 64:95–115, 1907.
* [47] C. Caratheodory. Uber den Variabilitätsbereich der Fourierschen Konstanten von positiven harmonischen Funktionen. Rend. Circ. Mat. Palermo, 32:193–217, 1911.


These papers keep getting cited in various other articles on the subject as well but to date, I have not been able to find them.


Would anyone know where I could find a preferably English translation? I am not too bothered if it could be found in a printed book of translated works either. Alternatively, it could be a different paper in English that covers the subject sufficiently.


The papers show that if you have a (positive) sum of $k$ sinusoids, you can recover the mix completely by knowing the value of the sum at $t=0$ and **any** $2k$ time points. Even as a subset of Compressed Sensing problems, this is a really interesting proposition and I would like to have a closer look.



Translated version of a Caratheodory article
Google translate output of the first page of [the 1907 paper](https://link.springer.com/content/pdf/10.1007/BF01449883.pdf), no postprocessing (other than LaTeXing the formulas).



> 
> **Over the range of variability of the coefficients of power series that
>  do not assume given values.**
> 
> 
> **Introduction.**  
> 
>  Given an analytic function $ y $ of the complex variable
>  z, which assumes the value $ y = A\_0 $ for $ z = 0 $ and is regular in
>  the neighborhood of this point, there are certain restrictions inside
>  the circle $ z <\rho $ is subject to the question arises whether not
>  at the same time for the coefficients of the functional element
>  $$y=A\_0+\sum\_{k=1}^\infty A\_k z^k,$$
>  which is the function of creating restrictions that can be determined.
> 
> 
> A special case of this kind of question occurs in the well-known
>  generalization that E. Landau has given for Picard's theorem on whole
>  transcendental functions.\*)
> 
> 
> This sentence can be expressed as follows: If the function $ y $ for $ z =
> 0 $ assumes the value $ y = A\_0 $, inside the unit circle is regular
>  and leaves the values ​​zero and one, and if the real and imaginary
>  part of the coefficient $ A\_1 $ as the coordinates of a point of the
>  planes, this point must lie inside a circle whose radius you can
>  specify.
> 
> 
> \*) During the printing of this work, Mr. Landau has published a detailed account of the questions in question (quarterly of the
>  Naturforschenden Gesellschaft in Zurich vol. 51, p. 252). In
>  particular, in par. 15 of this work is an analogous problem to an
>  algebraic problem recurred procedure.
> 
> 
> 


