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Why can we express any function in terms of cosine (Fourier) and polynomial (Taylor)? what is special about cosine and polynomial?

I mean can we write any function in terms of exponential (e^x) instead of cosine and polynomial for example?

Elie Makdissi
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  • We can not express every function using cosine and polynomials. Take a step function for instance. –  Nov 19 '21 at 11:26
  • Even for smooth functions the Taylor series does not have to be the the original function, see e.g. https://math.stackexchange.com/questions/3435144/doubt-about-taylor-series-do-successive-derivatives-on-a-point-determine-the-wh/3435157#3435157 – humanStampedist Nov 19 '21 at 11:27
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    As you can "express any function in terms of a polynomial", then you could "express the cosine in terms of a polynomial" and the cosine has nothing special... –  Nov 19 '21 at 11:28
  • But you can interpolate any finite dataset by means of linearly independent functions (such as polynomials or exponentials). Whether this extends to infinite datasets is another matter. –  Nov 19 '21 at 11:31
  • Also learn about the orthogonal functions. https://en.wikipedia.org/wiki/Orthogonal_functions –  Nov 19 '21 at 11:33
  • I don't want to dig deep, just be simple with me please : Taylor formula express a function say x^3 in terms of polynomial, and fourier serie also do the same by expressing in terms of cosine, are there any other series/formulas like the type of fourier and Taylor? – Elie Makdissi Nov 19 '21 at 11:45

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The premise of your question is flawed: You can't express ANY function in terms of cosine / polynomials. Not even in terms of converging infinite series of such terms.

I suggest you read the inluminating https://en.wikipedia.org/wiki/Convergence_of_Fourier_series for further details about what is required for such series to converge.

G. Fougeron
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