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I read that

Any "well-behaved" function of period $2\pi$ can be expressed as a Fourier series.

What qualifies as "well-behaved"? Any examples of functions that cannot be expressed as a Fourier series?

Thanks.

  • 1
    Depends how you mean "expressed as a Fourier series". Convergence in which mode? In $L^2$? Pointwise (almost everywhere)?2011-10-20
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    @JonasTeuwen: So there are different degrees...? I would be grateful if you could elaborate on the differences, that would be a good ans to this (vaguish) question. Thanks.2011-10-20
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    If the function has an infinite number of jump discontinuities, then that's a problem.2011-10-20
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    @Huh: much ink has been used in discussing the different modes of convergence. For an introductory (but by no means exhaustive) account, you can look at chapters 2 and 3 or Stein & Shakarchi, [Fourier Analysis](http://press.princeton.edu/titles/7562.html).2011-10-20

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