Math Problems Vault

Problem 3.7.14 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.7 "Fourier Transforms"

Suppose $F$ is an entire function in $\mathbb {C}^{n}$ , and suppose that to each $\varepsilon > 0$ there correspond an integer $N(\varepsilon )$ and a constant $\gamma (\varepsilon ) < \infty$ such that

\left|F(z)\right| \leq \gamma (\varepsilon )\left(1+\left|z\right|\right)^{N(\varepsilon )} e^{\varepsilon \left|\mathrm{Im}\left(z\right)\right|} \quad \left(z \in \mathbb {C}^{n}\right)

Prove that $F$ is a polynomial.

?

Problem 3.8.1 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.8 "Applications to Differential Equations"

(a)(b)(c)

The following simple properties of holomorphic functions of several variables were tacitly used in this chapter. Prove them.

?

(a)

If $f$ is entire in $\mathbb {C}^{n}$ , if $w \in \mathbb {C}^{n}$ , and if $\phi (\lambda ) = f(\lambda w)$ , then $\phi$ is an entire function of one complex variable.

(b)

If $P$ is a polynomial in $\mathbb {C}^{n}$ and if

\int _{T^{n}}\left|P\right| d \sigma _{n} = 0

then $P$ is identically 0. Hint: Compute $\int_{T^{n}}\left|P\right|^{2} d \sigma_{n}$ .

(c)

If $P$ is a polynomial (not identically 0) and $g$ is an entire function in $\mathbb {C}^{n}$ , then there is at most one entire function $f$ that satisfies $P f = g$ . Find generalizations of these three properties.

Problem 3.11.4 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.11 "Commutative Banach Algebras"

Prove that the polynomials are dense in the polydisc algebra $A\left(U^{n}\right)$ . (See Theorem 11.7). Suggestion: If $f \in A\left(U^{n}\right), 0 < r < 1$ , and $f_{r}$ is defined by $f_{r}(z) = f(r z)$ , then $f_{r}$ is the sum of an absolutely (hence uniformly) convergent multiple power series on $\bar{U}^{n}$ .

?

Problem 3.11.21 from Rudin W. "Functional Analysis" [Rud91]More problems in Chapter 3.11 "Commutative Banach Algebras"

In $\mathbb {C}^{2}$ , let $K_{1}$ consist of all points $\left(e^{i \theta }, e^{-i \theta }\right)$ , and let $K_{2}$ consist of the points $\left(e^{i \theta }, e^{i \theta }\right), 0 \leq \theta \leq 2 \pi$ . Of these circles, show that $K_{1}$ is polynomially convex but $K_{2}$ is not. How about $K_{3} = \left\{ (\cos \theta , \sin \theta ): 0 \leq \theta \leq \pi \right\}$ ?

?

Several complex variables

References