Proof of Caratheodory's theorem about the unique determination of a linear combination of sinusoids

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Following is a statement of Caratheodory's Theorem about a positivelinear combination of sinusoids :-

Any positive linear combination of k sinusoids is uniquely determined by its value at time t = 0 and any other 2k instants of time

How can we prove this statement ? Can someone please enlighten. The mathematical paper in which this is presented a 1907 paper listed below. Unfortunately I am unable to access its english version.

Carathéodory, C. (1907), "Über den Variabilitatsbereich der Koeffizienten von Potenzreflien, die gegebene Werte nicht annehmen", Math. Ann. 64: 95–115