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I have the following formula for the Fourier series of a integrable function $g:[0,N] \to \mathbb{C}$

$$ g(x) = \sum_{n=-\infty}^\infty c_n e^{in2\pi x/N}. $$

I was able to derive the formula for the coefficients $c_n$, but what if a want to show that any integrable function can be actually approximated by the formula given above? Is there something like a theorem?

wrong_path
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  • There are some things HERE and HERE each of which is not only "like a theorem," but actually is one. – Mark Viola Dec 18 '16 at 18:10
  • @Dr.MV I wrote "like ..." simply because I do not know if it a theorem or a lemma or something else. Thanks but I am looking for a derivation: from $g(x)$ to the RHS of the given formula. I always see in textbooks "this is the Fourier series, use it". – wrong_path Dec 18 '16 at 18:14
  • For point-wise convergence, you need to show that $\left|g(x)-\sum_{n=-M}^M c_ne^{in2\pi x/N}\right|$ , where $c_n$ are the Fourier coefficients, can be made arbitrarily small by taking $M$ sufficiently large. What do you mean by a derivation from $g$ to the RHS? – Mark Viola Dec 18 '16 at 18:19
  • @Dr.MV I mean the following: how do we know that any integrable function can be actually approximated by that sum? I do not think this is simply a definition that we have to take as it is. For example, the formula for $c_n$ can be derived: is there something similar for the Fourier series of $g(x)$? Thank you. – wrong_path Dec 18 '16 at 18:22
  • Have you read the articles in the embedded links? – Mark Viola Dec 18 '16 at 18:23
  • @Dr.MV Yes, but I am not able to understand if there is the answer to my question in there. – wrong_path Dec 18 '16 at 18:25

1 Answers1

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The statement as stated in the title of the question is false. An arbitrary integrable function $g$ on $[0,N]$ does not necessarily have a Fourier series that converges pointwise to $g$. Precisely how badly this can fail was a major research topic of harmonic analysis well into the later 20th century, and the situation is largely summarized in the links Dr. MV has helpfully provided.

In simpler terms than the Wikipedia article, here is a summary of the situation:

  1. Arbitrary integrable functions on $[0,N]$ are not necessarily pointwise well-approximated by their Fourier series. In fact, the situation is as bad as it could possibly get: Kolmogorov showed that there is a continuous function whose Fourier series diverges on a set of full measure (and in particular, diverges on a dense set).

  2. With an alternative integrability condition, things get better. The Carleson-Hunt theorem states that if $1<p<\infty$ and $g$ is $p$-integrable, i.e. if $|g|^p$ is integrable, then its Fourier series converges to $g$ almost everywhere (i.e. approximates $g$ well, except on a negligible set of points).

  3. If $g$ is piecewise continuous and has a jump discontinuity at $x_0$, then the Fourier series approximation tends to overshoot the actual value of the function from both directions by about $9\%$. This is known as the Gibbs phenomenon, and is of importance in signal processing. In particular, this overshoot will happen at the endpoints $0$ and $N$ unless $g(0) = g(N)$.

  4. Better convergence results can be obtained by prescribing better regularity than merely integrability and continuity. For example, if $g$ is integrable and Holder continuous of order $\alpha\in(frac{1}{2},1]$ (basically slightly better than mere continuity), then the Fourier series of $g$ will converge uniformly to $g$. In particular, if $g$ is continuously differentiable on $[0,N]$, or even merely differentiable with bounded derivative, then the Fourier series of $g$ will converge uniformly to $g$.

  5. If one replaces the usual notion of convergence with an alternative convergence criterion, then one can obtain much better results. For example, it is well-known that if $g$ is square-integrable, then its Fourier series converges to $g$ in mean-square sense. If $g$ is integrable, then its Fourier series is Cesaro summable and Abel summable almost everywhere and in mean to $g$, and the convergence is uniform if $g$ is continuous. However, these notions of convergence do not imply pointwise convergence, and in the case of Cesaro and Abel summability are slightly weaker criteria.

Gyu Eun Lee
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