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I have a lack of mathematical knowledge, and notably in mathematical vocabulary, so maybe a similar question exists but with a different wording.

What I want to know, is actually how to know if a function, given its properties (for example, a polynomial with natural-number coefficients) can be converted to a finite sum of sinuoids, or in other words if the Fourier transform of the function will give a finite sum of frequencies (because, as I understand the FT, which decomposes a signal into sinusoids, an infinite sum of frequencies means that the function is not sinusoidal in its nature and thus can only be approximated to a sinuoid, through an infinite sum of sinuoids).

Is there a theorem or a principle that can told me if a function, as long as it does not involve some specific operations, stay perfectly convertable (= no approximation) to a sinusoid ?

Please avoid complex mathematical notations as possible (unless each indeterminate or non-trivial greek letter is explained), I better understand natural phrasing and analogies, I'm only a beginner in signal processing, which I investigate for the purpose of advancing knowledge in image/graphics processing.

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    $\begingroup$ Here are some necessary (but not sufficient) criteria for a function to be decomposable into a finite number of sinusoidal basis functions: The function is smooth. The function or its analytic continuation is finitely bounded on R. If the function is defined on all of R, then its L2 norm is not finite, otherwise the L2 norm of its analytic continuation is not finite. The inner product of the function with a sinusoid vanishes for almost all frequencies. The function retains all these properties when convolved with any distribution. $\endgroup$
    – Jazzmaniac
    Commented Feb 5 at 14:29

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I think a good rule of thumb is this: "If it isn't already written as a finite sum of sinusoids, then it probably can't be written as a finite sum of sinusoids."

Most functions are not a finite sum of sinusoids. Polynomials certainly are not, and neither are square waves, triangle waves, or sawtooth waves. It seems like being a finite sum of sinusoids is a property that rarely happens by accident, so to speak.

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  • $\begingroup$ Related: How did Leibniz prove that sin(x) rarely happens by accident, so to speak? $\endgroup$
    – Stef
    Commented Feb 5 at 22:07
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    $\begingroup$ Thank you. I would like to find the parallel between your statement (it must be a sum of sinuoids) and @Hilmar statement (that the periodic signal must be bandlimited). My current understanding is that "bandlimited" is synonymous to "is a finite sum of sinuoids". May I get a validation or explanation on this ? $\endgroup$
    – endyx
    Commented Feb 6 at 16:47
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There are actually 4 different types or Fourier Transform. Which one to use depends on the signal properties: specifically whether a signal as periodic vs aperiodic and whether it is continuous vs discrete. These properties are related: a signal that's periodic in one domain is discrete in the other. For more details, see https://www.le.ac.uk/users/dsgp1/LODZLECT/Lodz5.pdf

One of the reasons why we have different transforms is practical: if a signal is discrete you use sums and if signals are continuous you use integrals.

enter image description here

can be converted to a finite sum of sinusoids,

Any signal that's periodic and bandlimited will do that. Periodicity implies that the signal is discrete in the frequency domain. And bandlimited implies that the sum is finite since the content above a certain frequency must be 0.

There are other signals that will meet your ask as well, but the only test I can think of is to actually calculate the Fourier Transform and look at it.

An example of a signal that contains only two frequencies but is aperiodic would be

$$x(t) = \sin(\omega t) + \sin(\sqrt{2} \omega t) \tag{1}$$

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  • $\begingroup$ Thank you for your answer. I think I miss something somewhere, especially after : "Any signal that's periodic and bandlimited will do that". The square wave for example is a periodic function. If you look at its fourier series (en.wikipedia.org/wiki/Square_wave#/media/…) you can see that increasing the number of terms up to infinity won't give us the exact square wave. Thereby no finite sum of sinusoids can represent it. Maybe that's because it's not bandlimited in terms of sinusoids ? Then, how to know if a function is "sinusoids-bandlimited" ? $\endgroup$
    – endyx
    Commented Feb 4 at 14:14
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    $\begingroup$ A triangle wave is NOT bandlimited. Hence the number of sinusoids is infinite. $\endgroup$
    – Hilmar
    Commented Feb 4 at 15:52
  • $\begingroup$ Okay, then my question should have been worded, is there a way to know (formally, not through computing) if a function (by its properties) is bandlimited or not ? $\endgroup$
    – endyx
    Commented Feb 4 at 16:31
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    $\begingroup$ Do you have a function defined via some mathematical expression (e.g. y=log(x)) or are you looking at data? If the former, then you could try to compute its Fourier transform analytically and see if the resulting coefficient was zero for large enough frequency. But my guess is that you are looking at data. $\endgroup$
    – Tunneller
    Commented Feb 4 at 20:31
  • $\begingroup$ @endyx: Being bandlimited IS a property of the signal. What class or type of properties do you have in mind? $\endgroup$
    – Hilmar
    Commented Feb 4 at 21:12

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