臺灣大學數學系

 八十八學年度第二學期碩博士班資格考試試題

 分析(加考)

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1. Let Z be a subset of $\mathbb{R}^1$ with measure zero. Show that the set $\{ x^2 : x \in {\bf Z}\}$ also has measure zero.

2. If $\lambda_1 < \lambda_2 <\cdots < \lambda_m$ is a finite sequence and $-\infty < s < +\infty$, write $\sum\limits_k a_k e^{-s\lambda_k}$ as a Riemann-Stieltjes integral. [Take $f(x) = e^{-sx}$, $\phi$ to be an appropriate step function, and $[a,b]$ to contain all the $\lambda_k$ inits interior.]

3. Let $f(x,y)$, $0 \le x,y \le 1$, satisfy the following conditions: for each $x$, $f(x,y)$ is an integrable function of $y$, and $(\partial f(x,y)/\partial x)$ is a bounded function of $f(x,y)$. Show that
   
   $$\frac{d}{dx} \int_0^1 f(x,y)\ dy = \int_0^1 \frac{\partial}{\partial x} f(x,y)\ dy.$$
   
   

4. (a)

   Let $\{f_k\}$ be a sequence of measurable functions on $E$. Show that $\sum f_k$ converges absolutely a.e. in $E$ if $\sum \int_E \vert f_k \vert < +\infty$. [Use theorem (5.16) and (5.22).]

   (b)

   If $\{ r_k \}$ denotes the rational numbers in $[0,1]$ and $\{a_k\}$ satisfies $\sum \vert a_k\vert < +\infty$, show that $\sum a_k \vert x-r_k\vert^{-1/2}$ converges absolutely a.e. in $[0,1]$.

5. Let $E$ be a measurable subset of $\mathbb{R}^2$ such that for almost every $x \in \mathbb{R}^1$, $\{ y : (x,y) \in E \}$ has $\mathbb{R}^1$-measure zero. Show that $E$ has measure zero, and that for almost every $y \in \mathbb{R}^1$, $\{ x : (x,y) \in E \}$ has measure zero.

6. Show that if $\alpha > 0$, $x^{\alpha}$ is absolutely continuous on every bounded subinterval of $(0,\infty)$.

7. Let $f,\{f_k\} \in L^p$. Show that if $\Vert f-f_k \Vert _p \rightarrow 0$, then $\Vert f_k \Vert _p \rightarrow \Vert f \Vert _p$. Conversely, if $f_k \rightarrow f$ a.e. and $\Vert f_k \Vert _p \rightarrow \Vert f \Vert _p, 1 \le p \le \infty$, show that $\Vert f-f_k \Vert _p \rightarrow 0$.
8. Prove the following generalization of H$\ddot{o}$lder's inequality. If $\sum\limits_{i=1}^k 1/p_i = 1/r$, $p_i,r \ge 1$, then
   
   $$\Vert f_1 \cdots f_k\Vert _r \le \Vert f_1\Vert _{p_1} \cdots \Vert f_k\Vert _{p_k}.$$
   
   
9. For $f \in L(\mathbb{R}^1)$, define the Fourier transform $\hat{f}$ of $f$ by
   
   $$\hat{f}(x) = \int_{-\infty}^{\infty} f(t) e^{-ixt}\ dt\ \ \ \ \ \ (x \in \mathbb{R}^1).$$
   
   (For a complex-valued function $F=F_0+iF_1$ whose real and imaginary parts $F_0$ and $F_1$ are integrable, we define $\int F = \int F_0+i \int F_1$.) Show that if $f$ and $g$ belong to $L(\mathbb{R}^1)$, then
   
   $$\widehat{(f \ast g)} (x) = \hat{f}(x) \hat{g}(x).$$
   　
10. If $p >0$ and $\int_E \vert f-f_k \vert^p \rightarrow 0$ as $k \rightarrow \infty$, show that $f_k \buildrel m \over \longrightarrow f$ on $E$ (and thus that there is a subsequence $f_{k_j} \rightarrow f$ a.e. in $E$).

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