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It is unique. If the function f(t) is piecewise continious and square integrable the fourier coffiecients are unique. This is a consequence of Sturm-Liouville theory. Another way to look at things is to consider the fourier series as an expansion in an infinite dimensional Hilbert space with cosnx and sinnx as the orthogonal basis functions.
well above given explanation is a bit more then complex so for beginers so it is enough for u to know that it is "UNIQUE".u can see also any dsp book like www.dspguide.com for simpler explanations.
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