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

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

• Sorry, what is a "sinusoid"? Jun 25 '13 at 15:40
• Sorry for a very late reply, but I did not have any access to internet for a couple of days. A Sinusoid is a signal of the following form :- $x(t) = A cos(\omega t + \theta)$ \\ Here, $\omega$ = frequency of the sinusoid (in Hertz) \\ $\phi$ = phase of the sinusoid (in radians). \\ $t$ = time. Sampling theory says that if I have a finite linear combination of such sinusoids, then by performing sampling at twice the highest frequency in the mixture, the signal can be recovered exactly from the samples. By reading the defn. of the theorem in question, you can see its importance. Jul 3 '13 at 12:14