Why is the complex fourier series defined this way?

Question

The definition of complex fourier series of a function is always given as the limit of symmetric partial sums $S_N(x)=\sum_{-N}^N c_n\exp(2\pi i nx)$, provided that series is convergent.

1. Why do we consider the symmetric partial sums rather than defining the fourier series as the sum of the two series $\sum_{-\infty}^0c_n\exp(2\pi i nx)+\sum_{1}^\infty c_k\exp(2\pi i kx)$? (If I'm not mistaken the fourier transform is not defined as the limit of a symmetric integral over the real line).

I believe that a possible answer to this question is "because by taking the symmetric partial sums we can combine the $n$ and $-n$ terms to get the classical sine and cosine terms..." which brings me to the second question:

2. If we had chosen the "non-symmetric" definition, would we get a different theory, or maybe put in a different way, how would the theory of those "non-symmetric" series be different from the usual complex fourier series?

3. Is there a simple example of a sequence $\{c_n\}_{-\infty}^{\infty}$ for which $$\lim_{N\to \infty} \sum_{-N}^N c_n\exp(2\pi inx)\ne \sum_{-\infty}^0c_n\exp(2\pi i nx)+\sum_{1}^\infty c_k\exp(2\pi i kx)?$$

I apologize if these are too naive questions, but since I could not come up with good answers, decided to ask. References are also welcome!

Answer 1

The definition of complex Fourier series of a function is always given as the limit of symmetric partial sums

This is not so, for two reasons.

1. Some textbook authors explicitly define the Fourier series as the sum of two infinite series, over $n\ge 0$ and $n<0$ (e.g., Strauss, PDE, page 116).
2. The authors of more advanced textbooks separate the process of defining a series and studying its convergence. The reason for doing this is that convergence of a Fourier series is treated in multiple ways: the same series may converge in one sense and diverge in another sense. One should not have to immediately deal with all of this just to write down the series.

Following approach 2, one says that the Fourier series is
$$ \sum_{n\in\mathbb{Z}} c_ne^{2\pi i nx} $$ where $c_n$ is defined as $\int_0^1 f(x)e^{-2\pi i nx}\,dx$, and the symbol $\sum$ is just that — a symbol that I put at the beginning of that line because I felt like it. I don't have to explain in what order I'm adding the terms, because I'm not actually adding them: the notation expresses the vague idea that I might decide to add them some day. No claim is made about the convergence of the series in any sense.

When it comes to proving theorems about convergence, one has to specify exactly what kind of convergence is proved. It can be convergence of symmetric partial sums, or of Cesàro means, or unconditional convergence (independent of order) in the norm of some function space, like $L^2(\mathbb{R})$ or $C_b(\mathbb{R})$, or something else.

For some convergence theorems the order is immaterial: e.g., if one shows $(c_n)\in \ell^2$, then the series of partial sums converges in $L^2$ no matter how the terms are ordered. Similarly, if one proves $(c_n)\in \ell^1$, then the series converges uniformly, regardless of order.

When proving results about pointwise convergence it is convenient to focus on symmetric sums, because they can be expected to converge to $f(x)$. We don't want just to prove that the series converges; we want it to converge to $f(x)$. And to do that, we need some partial sums that get closer to $f(x)$. The symmetric partial sums often do that; the one-sided partial sums do not, simply because they don't include half of the series.

Some results about symmetric partial sums (e.g., pointwise convergence for Hölder functions and Carleson's theorem about a.e. convergence for $L^2$ functions) imply corresponding statements for one-sided sums. This is because the Hilbert transform $H$ flips the sign of the Fourier coefficients with negative indices, while preserving the Hölder spaces (of fractional order) and the $L^2$ space. Adding the symmetric partial sums of $f$ and $Hf$ yields a one-sided partial sum for $2f$, and conclusion follows.

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Summary:

• the definition of Fourier series is agnostic of the methods of summation
• symmetric partial sums are convenient because they can be expected to converge to $f$, and also because they can be written as the convolution with a real-valued kernel (Dirichlet's kernel)