# Fourier Series (in reverse)

In electronics (and in other fields of engineering), we study that every periodic signal (whatever its shape) may be decomposed in a series of sine waves with frequency $f, 2f, 3f, \ldots, nf$ with decreasing amplitudes. This is the basics of Fourier Series.

I've carried with me one question that I never have seen any answer: can we prove the opposite of this?

Let me explain in more detail: if I have a sine wave of amplitude $A$ and frequency $f$, can I decompose the sine wave in a series of square waves (or triangular, or whatever) with frequencies $f, 2f, 3f, \ldots, nf$ with decreasing amplitudes?

If the answeer is yes, please indicate me bibliography where can I find such math demonstration.

Thank you

Carlos Lisbon, Portugal

• Where do you see a requirement that the amplitudes are decreasing? Assuming of course that the signal is square-integrable over one period, the requirement is that the sum of the squares of the amplitudes is finite. – Robert Israel Aug 17 '12 at 17:49

• Not actually periods $f, 2f, 3f, \dots$ as in the question, though. – GEdgar Aug 17 '12 at 14:53
• Or you could try the Walsh functions, or any other orthonormal basis of $L^2$ of an interval. – Robert Israel Aug 17 '12 at 19:56
• @carlos: The physicist's answer to what families can express all these functions is any family that can represent a $\delta$ function. – Ross Millikan Aug 18 '12 at 16:33