Definition of Fourier coefficients in PMA Rudin

Question

W.Rudin in his book "Principles of MA" defines Fourier coefficients by $(62)$ i.e. $$c_m=\dfrac{1}{2\pi}\int _{-\pi}^{\pi}f(x)e^{-imx}dx \qquad (62)$$ But in 8.10 Definiton for orthonormal system $\{\phi_n(x)\}$ on $[a,b]$ he defines Fourier coefficients by $(66)$ i.e. $$c_n=\int _{a}^{b}f(t)\overline{\phi_n(t)}dt \qquad (66)$$ But we know that system $\left\{\dfrac{e^{inx}}{\sqrt{2\pi}}\right\}_{n\in \mathbb{Z}}$ is orthonormal on $[-\pi,\pi]$ and putting this into $(66)$ we get that $c_n=\dfrac{1}{\sqrt{2\pi}}\int \limits_{-\pi}^{\pi}f(t)e^{-int}dt$ and this does not coincide with $(62)$

Can anyone explain this confusing moment please?

Answer 1

He, too, appears to be confused by the mutually incompatible definitions in the literature. Interestingly, in chapter 7 of his book "Functional Analysis" he solves the problem by redefining Lebesgue measure. The normalized Lebesgue measure on $\mathbb R^n$ is the measure $m_n$ defined by

$$dm_n(x)=(2\pi)^{-n/2}dx.$$

Answer 2

Unfortunately, he's employing two different conventions. Notice that in $(63)$ he writes $$\sum c_{n} e^{inx}$$ so he's thinking of $\{e^{inx}\}_{n \in \mathbb{N}}$ as his orthonormal basis, not $\{\frac{e^{inx}}{\sqrt{2 \pi}}\}_{n \in \mathbb{N}}$. Of course, you could multiply and divide by $\sqrt{2 \pi}$ to get $$\sum a_{n} \frac{e^{inx}}{\sqrt{2 \pi}},$$ where $a_{n}$ satisfies the formula you were expecting $$a_{n} = \frac{1}{\sqrt{2 \pi}} \int_{- \pi}^{\pi} f(t) e^{-int} \, dt.$$

Ultimately, it's a question of what exactly you mean by orthonormality. I don't know if you know what an inner product space is, but I'll put it this way. If you define an orthonormal basis $\{\phi_{n}\}_{n \in \mathbb{N}}$ by requiring $$\int_{a}^{b} \phi_{n}(x) \overline{\phi_{m}(x)} \, dx = \left \{ \begin{array}{c c} 1, & n = m \\ 0, & \text{otherwise} \end{array} \right.$$ then $$\left \{\frac{e^{inx}}{\sqrt{2 \pi}} \right\}_{n \in \mathbb{N}}$$ is the right choice as far as Fourier series are concerned. On the other hand, you can also define an orthornormal basis $\{\phi_{n}\}_{n \in \mathbb{N}}$ using the formula $$\frac{1}{b - a} \int_{a}^{b} \phi_{n}(x) \overline{\phi_{m}(x)} \, dx = \left \{ \begin{array}{c c} 1, & n = m \\ 0, & \text{otherwise} \end{array} \right.$$ There are a number of reasons this is a nice alternative. In our case, it's nice that $\{e^{inx}\}_{n \in \mathbb{N}}$ is now orthornormal so we needn't keep track of factors of $\sqrt{2 \pi}$.