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.