Holomorphic Fourier Transforms

Suppose $f$ is an entire function of exponential type and

Prove that either $\varphi (y) = \infty$ for all real $y$ or $\varphi (y) < \infty$ for all real $y$. Prove that $f = 0$ if $\varphi$ is a bounded function.

Suppose $f$ is an entire function of exponential type such that the restriction of $f$ to two nonparallel lines belongs to $L^{2}$. Prove that $f = 0$.

Suppose $f$ is an entire function of exponential type whose restriction to two nonparallel lines is bounded. Prove that $f$ is constant. (Apply Exercise 9 of Chap. 12).

Suppose $f$ is entire, $\left|f(z)\right| < C \exp (A\left|z\right|)$, and $f(z) = \Sigma a_{n} z^{n}$. Put

Prove that the series converges if $\left|w\right| > A$, that

if $\Gamma (t) = (A+\epsilon ) e^{i t}, 0 \leq t \leq 2 \pi$, and that $\Phi$ is the function which occurred in the proof of Theorem 19.3. (See also Sec. 19.4).

Suppose $f$ satisfies the hypothesis of Theorem 19.2. Prove that the Cauchy formula

holds; here $z = x+i y$. Prove that

exists for almost all $x$. What is the relation between $f^{*}$ and the function $F$ which occurs in Theorem 19.2? Is $({ }^{*})$ true with $\epsilon = 0$ and with $f^{*}$ in place of $f$ in the integrand?

Suppose $\varphi \in L^{2}(-\infty , \infty )$ and $\varphi > 0$. Prove that there exists an $f$ with $\left|f\right| = \varphi$ such that the Fourier transform of $f$ vanishes on a half line if and only if

Suggestion: Consider $f^{*}$, as in Exercise 5, where $f = \exp (u+i v)$ and

Let $f$ be a complex function on a closed set $E$ in the plane. Prove that the following two conditions on $f$ are equivalent:

Prove that $C\left\{ n!\right\} = C\left\{ n^{n}\right\}$.

Prove that there are quasi-analytic classes which are larger than $C\left\{ n!\right\}$.

Put $\lambda_{n} = M_{n-1} / M_{n}$, as in the proof of Theorem 19.11. Pick $g_{0} \in C_{c}\left(\mathbb {R}^{1}\right)$, and define

Prove directly (without using Fourier transforms or holomorphic functions) that $g = \lim g_{n}$ is a function which demonstrates that (e) implies (a) in Theorem 19.11. (You may choose any $g_{0}$ that is convenient).

Find an explicit formula for a function $\varphi \in C^{\infty }$, with support in [-2,2], such that $\varphi (x) = 1$ if $-1 \leq x \leq 1$.

Prove that to every sequence $\left\{ \alpha_{n}\right\}$ of complex numbers there corresponds a function $f \in C^{\infty }$ such that $\left(D^{n} f\right)(0) = \alpha_{n}$ for $n = 0,1,2, \ldots$ Suggestion: If $\varphi$ is as in Exercise 11, if $\beta_{n} = \alpha_{n} / n!$, if $g_{n}(x) = \beta_{n} x^{n} \varphi (x)$, and if

then $\left\| D^{k} f_{n}\right\|_{\infty } < 2^{-n}$ for $k = 0, \ldots , n-1$, provided that $\lambda_{n}$ is large enough. Take $f = \Sigma f_{n}$.

Construct a function $f \in C^{\infty }$ such that the power series

has radius of convergence 0 for every $a \in \mathbb {R}^{1}$. Suggestion: Put

where $\left\{ c_{k}\right\}$ and $\left\{ \lambda_{k}\right\}$ are sequences of positive numbers, chosen so that $\Sigma c_{k} \lambda_{k}^{n} < \infty$ for $n = 0,1,2, \ldots$ and so that $c_{n} \lambda_{n}^{n}$ increases very rapidly and is much larger than the sum of all the other terms in the series $\Sigma c_{k} \lambda_{k}^{n}$.

For instance, put $c_{k} = \lambda_{k}^{1-k}$, and choose $\left\{ \lambda_{k}\right\}$ so that

Suppose $C\left\{ M_{n}\right\}$ is quasi-analytic, $f \in C\left\{ M_{n}\right\}$, and $f(x) = 0$ for infinitely many $x \in [0,1]$. What follows?

Let $X$ be the vector space of all entire functions $f$ that satisfy $\left|f(z)\right| \leq C e^{\pi \left|z\right|}$ for some $C < \infty$, and whose restriction to the real axis is in $L^{2}$. Associate with each $f \in X$ its restriction to the integers. Prove that $f \rightarrow \left\{ f(n)\right\}$ is a linear one-to-one mapping of $X$ onto $\ell^{2}$.

Assume $f$ is a measurable function on $(-\infty , \infty )$ such that $\left|f(x)\right| < e^{-\left|x\right|}$ for all $x$. Prove that its Fourier transform $\hat{f}$ cannot have compact support, unless $f(x) = 0$ a.e.