Lemma 1.5 Chapter 3 Fourier Analysis an Introduction

Question

A space $\mathcal R$ of integrable functions on the circle with inner product:

$$(f,g)=\frac{1}{2\pi}\int_{0}^{2\pi}f(\theta)\overline {g(\theta)}d\theta$$ and the norm $||f||$ defined by

$$\Vert f\Vert ^{2}=(f,f)=\frac{1}{2\pi}\int_{0}^{2\pi}\vert f(\theta)\vert ^{2}d\theta$$

Lemma 1.5:

Suppose $F$ and $G$ are integrable on the circle with $$F \sim \sum a_{n}e^{in\theta}~~and~~G\sim \sum b_{n}e^{in\theta}$$ Then $$\frac{1}{2\pi}\int_{0}^{2\pi}F(\theta)\overline{G(\theta)}d\theta=\sum_{n=-\infty}^{\infty}a_{n}\overline{b_{n}}.$$

Now I am not sure if the orthonormal property of the family $\{e_{n}(\theta)\}_{n\in \mathbb z}$, that is:

$$(e_{n},e_{m})=\lbrace^{1~~~~if~~n=m}_{0~~~~if~~n\neq m.}$$ Can prove the lemma above or not.

The hint states that from the Parseval's identity:

$$\sum_{n=-\infty}^{\infty}\vert a_{n}\vert ^{2}=\Vert f \Vert ^{2}$$

And

$$(F,G)=\frac{1}{4}[\Vert F+G \Vert ^{2}-\Vert F-G\Vert ^{2}+i(\Vert F+iG\Vert^{2}-\Vert F-iG\Vert^{2})]$$ which holds n every hermitian inner product space.

I don't know how to verify this fact.

Answer 1

To verify the identity just expand all the norm squares on RHS. For example $\|F+iG\|^{2}=\langle (F+iG),(F+iG)\rangle>=\|F\|^{2}+v-2\Re i\langle F,G\rangle$ by properties of inner product. If you do this with all the four terms on RHS you will get LHS.

Answer 2

Note that the orthonormal realtion is:

$$ \cfrac{1}{2\pi}\int_0^{2\pi}a_je^{ij\theta}e^{-ik\theta} = a_j \delta_{jk}$$

The hint seems unnecessarily complicated , you can simply plug in:

$$F(\theta) = \sum_ja_je^{ij\theta}, \ \ G(\theta)^*\ =\sum_kb^*_ke^{-ik\theta} $$

into the lemma and use the orthonormal relation to get the required result:

$$ \cfrac{1}{2\pi}\int_0^{2\pi}\sum_j\sum_ka_jb_k^*e^{ij\theta}e^{-ik\theta} = \cfrac{1}{2\pi}\sum_j\sum_k\int_0^{2\pi}a_jb_k^*e^{ij\theta}e^{-ik\theta} $$

$$ \hspace{28mm}= \cfrac{1}{2\pi}\sum_{j=k}{2\pi}a_jb_k^*$$

$$ \hspace{17mm}= \sum_{j}a_jb_k^*$$