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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?

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  • $\begingroup$ I strongly suspect that he uses the normalised Haar measure on the unit circle, which is Lebesgue measure divided by $2\pi$. $\endgroup$ – Daniel Fischer Jan 14 '16 at 15:30
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    $\begingroup$ A Fourier series on $[-\pi,\pi]$ is $\sum_{n=-\infty}^{\infty}c_n e^{inx}$. No matter how you define things, either by changing inner products or renormalizing the functions, the associated series for $f \in L^2[-\pi,\pi]$ is $\sum_{n=-\infty}^{\infty}c_n e^{inx}$ with $c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi}f(t)e^{-int}dt$. The classical definition is that the $c_n$'s are the Fourier coefficients. $\endgroup$ – DisintegratingByParts Jan 14 '16 at 18:54
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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

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

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  • $\begingroup$ +1. Interesting remark! May I ask you question? But if $\{\phi_n(x)\}$ is orthonormal system how Fourier series $c_n$ defined in this case? $\endgroup$ – ZFR Jan 14 '16 at 14:43
  • $\begingroup$ For example $\phi_n(x)=\frac{e^{inx}}{\sqrt{2\pi}}$ $\endgroup$ – ZFR Jan 14 '16 at 14:43
  • $\begingroup$ As soon as you even mention the word orthonormal it should be (66). The other convention (62) dates back to more innocent times when the integrals were written separately for the sine and the cosine; its only saving grace was easier typography in the pre-LaTeX era. $\endgroup$ – Justpassingby Jan 14 '16 at 14:49
  • $\begingroup$ So Rudin made mistake or not? $\endgroup$ – ZFR Jan 14 '16 at 15:07
  • $\begingroup$ But the only problem that worries me it's that $(62)$ does not coincide with $(66)$ $\endgroup$ – ZFR Jan 14 '16 at 15:08
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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 $\sqrt{2 \pi}$.

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  • $\begingroup$ In (63) he does not consider $\{e^{inx}\}$ as orthonormal system as you said. $\endgroup$ – ZFR Jan 14 '16 at 15:03
  • $\begingroup$ But his (66) does not coincide with (62). That's a problem $\endgroup$ – ZFR Jan 14 '16 at 15:20
  • $\begingroup$ For what it's worth, the discussion around (63) precedes the definition of orthonormal system. However, if you look, for example, at Rudin's Real and Complex Analysis you will see the orthonormal system used to study Fourier series in that book is, in fact, $\{e^{inx}\}$. $\endgroup$ – fourierwho Jan 14 '16 at 15:26

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