Translated version of a Caratheodory article This 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.
 A: Google translate output of the first page of the 1907 paper, 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.

