Fourier Analysis

Prove that $\hat{\overline{f}}(n) = \overline{\hat{f}(-n)}$ for all $f \in L^{1}(\partial \mathbf{D})$ and all $n \in \mathbf{Z}$.

Suppose $1 \leq p \leq \infty$ and $n \in \mathbf{Z}$.

Show that if $0 \leq r < 1$ and $t \in \mathbb {R}$, then

Suppose $f \in L^{1}(\partial \mathbf{D}), z \in \partial \mathbf{D}$, and $f$ is continuous at $z$. Prove that

Suppose $a, b \in \mathbb {C}, f \in L^{1}(\partial \mathbf{D}), z \in \partial \mathbf{D}, \lim_{t \downarrow 0} f\left(e^{i t} z\right) = a$, and $\lim_{t \uparrow 0} f\left(e^{i t} z\right) = b$. Prove that

Prove that for each $p \in [1, \infty )$, there exists $f \in L^{1}(\partial \mathbf{D})$ such that

Suppose $\zeta \in \partial \mathbf{D}$. Show that the function

is harmonic on $\mathbb {C} \backslash \left\{ \zeta \right\}$ by finding an analytic function on $\mathbb {C} \backslash \left\{ \zeta \right\}$ whose real part is the function above.

Suppose $f: \partial \mathbf{D} \rightarrow \mathbb {R}$ is the function defined by

for $(x, y) \in \mathbb {R}^{2}$ with $x^{2}+y^{2} = 1$. Find a polynomial $u$ of two variables $x, y$ such that $u$ is harmonic on $\mathbb {R}^{2}$ and $\left.u\right|_{\partial \mathbf{D}} = f$.

Find a formula (in closed form, not as an infinite sum) for $\mathcal{P}_{r} f$, where $f$ is the function in the second bullet point of Example 11.8.

Suppose $f: \partial \mathbf{D} \rightarrow \mathbb {C}$ is three times continuously differentiable. Prove that

for all $z \in \partial \mathbf{D}$.

Let $C(\partial \mathbf{D})$ denote the Banach space of continuous function from $\partial \mathbf{D}$ to $\mathbb {C}$, with the supremum norm. For $M \in \mathbb {N}$, define a linear functional $\varphi_{M}: C(\partial \mathbf{D}) \rightarrow \mathbb {C}$ by

Thus $\varphi_{M}(f)$ is a partial sum of the Fourier series $\sum_{n = -\infty }^{\infty } \widehat{f}(n) z^{n}$, evaluated at $z = 1$.

Define $f: \partial \mathbf{D} \rightarrow \mathbb {R}$ by

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 $a, b$.

Suppose $f \in L^{1}(\mathbb {R})$. Prove that $\left\| \widehat{f}\right\|_{\infty } = \left\| f\right\|_{1}$ if and only if there exists $\zeta \in \partial \mathbf{D}$ and $t \in \mathbb {R}$ such that $\zeta f(x) e^{-i t x} \geq 0$ for almost every $x \in \mathbb {R}$.

Suppose $f(x) = x e^{-\pi x^{2}}$ for all $x \in \mathbb {R}$. Show that $\widehat{f} = -i f$.

Suppose $f(x) = 4 \pi x^{2} e^{-\pi x^{2}}-e^{-\pi x^{2}}$ for all $x \in \mathbb {R}$. Show that $\widehat{f} = -f$.

Find $f \in L^{1}(\mathbb {R})$ such that $f \neq 0$ and $\widehat{f} = i f$.

Prove that if $p$ is a polynomial on $\mathbb {R}$ with complex coefficients and $f: \mathbb {R} \rightarrow \mathbb {C}$ is defined by $f(x) = p(x) e^{-\pi x^{2}}$, then there exists a polynomial $q$ on $\mathbb {R}$ with complex coefficients such that $\operatorname {deg} q = \operatorname {deg} p$ and $\widehat{f}(t) = q(t) e^{-\pi t^{2}}$ for all $t \in \mathbb {R}$.

Suppose

Show that $\widehat{f}(t) = \frac{1}{4 \pi^{2}(1+i t)^{2}}$ for all $t \in \mathbb {R}$.

Prove the formulas in 11.55 for the Fourier transforms of translations, rotations, and dilations.

Suppose $f \in L^{1}(\mathbb {R})$ and $n \in \mathbb {N}$. Define $g: \mathbb {R} \rightarrow \mathbb {C}$ by $g(x) = x^{n} f(x)$. Prove that if $g \in L^{1}(\mathbb {R})$, then $\widehat{f}$ is $n$ times continuously differentiable on $\mathbb {R}$ and

for all $t \in \mathbb {R}$.

Suppose $n \in \mathbb {N}$and $f \in L^{1}(\mathbb {R})$ is $n$ times continuously differentiable and $f^{(k)} \in L^{1}(\mathbb {R})$ for $k = 1, \ldots , n$. Prove that if $t \in \mathbb {R}$, then

Suppose $1 \leq p \leq \infty , f \in L^{p}(\mathbb {R})$, and $g \in L^{p^{\prime }}(\mathbb {R})$. Prove that $f * g$ is a uniformly continuous function on $\mathbb {R}$.

Suppose $f \in L^{\infty }(\mathbb {R}), x \in \mathbb {R}$, and $f$ is continuous at $x$. Prove that

Suppose $p \in [1, \infty ]$ and $f \in L^{p}(\mathbb {R})$. Prove that $\mathcal{P}_{y}\left(\mathcal{P}_{y^{\prime }} f\right) = \mathcal{P}_{y+y^{\prime }} f$ for all $y, y^{\prime } > 0$.

Suppose $p \in [1, \infty ]$ and $f \in L^{p}(\mathbb {R})$. Prove that if $0 < y < y^{\prime }$, then

Suppose $f \in L^{1}(\mathbb {R})$.

Define $f \in L^{1}(\mathbb {R})$ by $f(x) = e^{-x^{4}} \chi_{[0, \infty )}(x)$. Show that $\widehat{f} \notin L^{1}(\mathbb {R})$.

Suppose $f \in L^{1}(\mathbb {R})$ and $\widehat{f} \in L^{1}(\mathbb {R})$. Prove that $f \in L^{2}(\mathbb {R})$ and $\widehat{f} \in L^{2}(\mathbb {R})$.

Prove there exists a continuous function $g: \mathbb {R} \rightarrow \mathbb {R}$ such that $\lim_{t \rightarrow \pm \infty } g(t) = 0$ and $g \notin \left\{ \widehat{f}: f \in L^{1}(\mathbb {R})\right\}$.

Prove that if $f \in L^{1}(\mathbb {R})$, then $\left\| \widehat{f}\right\|_{2} = \left\| f\right\|_{2}$.

Suppose $y > 0$. Define on operator $T$ on $L^{2}(\mathbb {R})$ by $T f = f * P_{y}$.

Prove that if $f \in L^{1}(\mathbb {R})$ and $g \in L^{2}(\mathbb {R})$, then $\mathcal{F}(f * g) = \widehat{f} \mathcal{F} g$.

Prove that if $f, g \in L^{2}(\mathbb {R})$, then $\widehat{f g} = (\mathcal{F} f) *(\mathcal{F} g)$.