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.

  • $\begingroup$ I would just download the pdf files from here and from here and send each through Google translate; it should have no difficulty to translate a German text into your favorite language. $\endgroup$ – Carlo Beenakker Sep 22 at 13:25
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    $\begingroup$ @CarloBeenakker This is indeed useful. Not sure why the article journal web pages did not show up during my google searches. I did try different paragraphs through google translate and while it requires some effort, it is not totally incomprehensible. If you leave an answer, it would be the accepted one at the moment. All the best. $\endgroup$ – A_A Sep 22 at 16:26

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.

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.


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