Fourier Series and $L^{p}$ of Unit Circle

Show that the family $\left\{ e_{k}\right\}_{k \in \mathbf{Z}}$ of trigonometric functions defined by 11.1 is an orthonormal basis of $L^{2}((-\pi , \pi ])$.

Use the result of Exercise 12(a) in Section 11A to show that

Use techniques similar to Example 11.32 to evaluate $\sum_{n = 1}^{\infty } \frac{1}{n^{4}}$.

Suppose $f, g: \partial \mathbf{D} \rightarrow \mathbb {C}$ are measurable functions. Prove that the function $(w, z) \mapsto f(w) g(z \overline{w})$ is a measurable function from $\partial \mathbf{D} \times \partial \mathbf{D}$ to $\mathbb {C}$.

Where does the proof of 11.42 fail when $p = \infty$?

Suppose $f \in L^{1}(\partial \mathbf{D})$. Prove that $f$ is real valued (almost everywhere) if and only if $\widehat{f}(-n) = \widehat{f}(n)$ for every $n \in \mathbf{Z}$.

Suppose $f \in L^{1}(\partial \mathbf{D})$. Show that $f \in L^{2}(\partial \mathbf{D})$ if and only if $\sum_{n = -\infty }^{\infty }\left|\widehat{f}(n)\right|^{2} < \infty$.

Suppose $f \in L^{2}(\partial \mathbf{D})$. Prove that $\left|f(z)\right| = 1$ for almost every $z \in \partial \mathbf{D}$ if and only if

for all $n \in \mathbf{Z}$.

For this exercise, for each $r \in [0,1)$ think of $\mathcal{P}_{r}$ as an operator on $L^{2}(\partial \mathbf{D})$.

Suppose $f \in L^{1}(\partial \mathbf{D})$. Define $T: L^{2}(\partial \mathbf{D}) \rightarrow L^{2}(\partial \mathbf{D})$ by $T g = f * g$.

Show that if $f, g \in L^{1}(\partial \mathbf{D})$ then

for those $t \in \mathbb {R}$ such that $(f * g)\left(e^{i t}\right)$ makes sense; here $(f * g)^{\sim }, \widetilde{f}$, and $\widetilde{g}$ denote the transfers to the real line as defined in 11.24.

Suppose $1 \leq p \leq \infty$. Prove that if $f \in L^{p}(\partial \mathbf{D})$ and $g \in L^{p^{\prime }}(\partial \mathbf{D})$, then $f * g$ is a continuous function on $\partial \mathbf{D}$.

Suppose $g \in L^{1}(\partial \mathbf{D})$ is such that $\widehat{g}(n) \neq 0$ for infinitely many $n \in \mathbf{Z}$. Prove that if $f \in L^{1}(\partial \mathbf{D})$, then $f * g \neq g$.

Show that there exists a two-sided sequence $\ldots , b_{-2}, b_{-1}, b_{0}, b_{1}, b_{2}, \ldots$ such that $\lim_{n \rightarrow \pm \infty } b_{n} = 0$ but there does not exist $f \in L^{1}(\partial \mathbf{D})$ with $\widehat{f}(n) = b_{n}$ for all $n \in \mathbf{Z}$.

Prove that if $f, g \in L^{2}(\partial \mathbf{D})$, then

for every $n \in \mathbf{Z}$.

Suppose $f \in L^{1}(\partial \mathbf{D})$. Prove that $\mathcal{P}_{r}\left(\mathcal{P}_{s} f\right) = \mathcal{P}_{r s} f$ for all $r, s \in [0,1)$.

Suppose $p \in [1, \infty ]$ and $f \in L^{p}(\partial \mathbf{D})$. Prove that if $0 \leq r < s < 1$, then

Prove Wirtinger's inequality: If $f: \mathbb {R} \rightarrow \mathbb {R}$ is a continuously differentiable $2 \pi$-periodic function and $\int_{-\pi }^{\pi } f(t) d t = 0$, then

with equality if and only if $f(t) = a \sin (t)+b \cos (t)$ for some constants $a, b$.