臺灣大學數學系

 八十六學年度第二學期碩博士班資格考試試題

分析

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There are problems A to F. You have to do Problems B, E, F, and 2 Problems out of A, C, and D.

A.

Let $f:[0, 1] \times [0,1] \to \mathbb{R}$, $f=f(x, y)$. If $f$ is measurable with respect to $x$ for each fixed $y$, and $f$ is continuous in $y$ for almost everywhere fixed $x$, prove that $f$ is measurable. Is the conclusion true if we only assume that $f$ is measurable in $x$ and $y$ separartely?

B.

Let $E \subseteq \mathbb{R}^n$ be measurable, and $f_m(m=1, 2, \cdots)$ be a sequence of functions in $L^p(E)$ with $1 \le p \le \infty$. For an $f \in L^p(E)$, we have the following 4 possible ways of convergence:

(a)

$f_m \to f$ a.e. ;

(b)

$f_m \to f$ in measure ;

(c)

$f_m \to f$ in $L^p$, i.e. , $\lim\limits_{m \to \infty} \int_E \vert f_m(x) - f(x)\vert^p dx =0;$

(d)

$f_m \to f$ weakly, i.e. for all $g \in L^q (E), \lim\limits_{m \to \infty}\int_E f_m(x)g(x)dx =\int_E f(x)g(x)dx$,

where ${1 \over p} + {1 \over q}=1$.

(1)

Prove that $(a)\Rightarrow (b)$, $(c) \Rightarrow (b)$, and $(c) \Rightarrow (d)$. In each case, show by example that the converse implication is false.

(2)

If $\lim_{m \to \infty} \int_E \vert f_m(x)\vert^p dx =\int_E \vert f(x)\vert^p dx$. Prove that $(b) \Leftrightarrow (c)$ and $(a) \Leftrightarrow (d)$.

C.

Determine which of the following conditions implies that

$$f(x)=\int_a^x f'(t) dt + f(a), \, \, \hbox{ for all }\, \, x \in [a, b]$$

where $f:[a,b]\to \mathbb{R}$ is a function of bounded variation.

(1)

$\vert f(x)- f(y)\vert \le L \sqrt{\vert x -y\vert}$ for all $x, y \in [a,b]$.

(2)

$f$ is differentiable at every point of $(a, b)$, and $\vert f'(x)\vert \le L$ for all $a < x < b$.

(3)

$f$ is differentiable at every point of $[a,b)$, $f'$ is bounded in $[a, b-\epsilon]$ for all small $\epsilon > 0$, and the improper Riemann integral of $f'$ on $[a,b]$ exists.

Here $L$ is a positive constant.

D.

Let $f \in L^p(\mathbb{R}^n)$ with $1 \le p \le \infty$. Define

$$u(x,y) ={1 \over \pi}\int_{-\infty}^{\infty}{y \over {y^2+(x-t)^2}} f(t) dt, \quad y > 0.$$

Prove that $u(x,y)$ is $C^{\infty}$ in the upper half plane $y > 0$, $u_{xx} + u_{yy}=0$ in $y > 0$, and $\lim_{y \to 0+} u(x,y)=f(x)$ for a.e. $x \in \mathbb{R}^n$.

E.

Let $E \subset \mathbb{R}$ be a compact set with $\vert E\vert >0$. Define

$$f(z) = \int_E {1 \over {t-z}} dt, \quad z \in \Omega$$

where Ω is the complement of $E$ in the complex plane.

(1)

Prove that $f$ is analytic in Ω.

(2)

Is $z = \infty$ a regular point, or a pole, or an essential singularity of $f$?

(3)

Compute $\int_{\Gamma} f(z) dz$, where Γ is a positvely oriented simple closed curve in the plane which contains $E$ in its interior.

F.

Determine the range of $t > 0$ such that the improper integral $\int_0^{\infty} {{\log (1+x^2)} \over {x^t}} dx$ exists. If it exists, find its value.

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