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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

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    $\begingroup$ 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. $\endgroup$ – Robert Israel Aug 17 '12 at 17:49
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The paper DN Green, SC Bass, Signal representation with triangular basis functions, IEE Journal on Electronic Circuits and Systems, vol. 3, Mar. 1979, p. 58-68 states the following result: A signal possesses a trigonometric series representation if and only if it has a "triangular series" representation.

However, as Robert points out in the comment, there is no reason to believe that the amplitudes of the triangular wave components are decreasing.

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I think that you will find that the Haar functions are what you are looking for. You can find a plethora of references by googling.

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    $\begingroup$ Not actually periods $f, 2f, 3f, \dots$ as in the question, though. $\endgroup$ – GEdgar Aug 17 '12 at 14:53
  • $\begingroup$ @Gedgar: Yes, you're right. In some depending on the shape of the wave, we may have only the even or the odd harmonics. $\endgroup$ – Carlos Aug 17 '12 at 14:58
  • $\begingroup$ Or you could try the Walsh functions, or any other orthonormal basis of $L^2$ of an interval. $\endgroup$ – Robert Israel Aug 17 '12 at 19:56
  • $\begingroup$ @carlos: The physicist's answer to what families can express all these functions is any family that can represent a $\delta$ function. $\endgroup$ – Ross Millikan Aug 18 '12 at 16:33

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