臺灣大學數學系

 八十七學年度第二學期碩博士班資格考試試題

分析

[回上頁]

A. Choose 4 from the following 5 problems

1. Let $f_n$ and $f$ be measurable functions on $[0,1]$. Assume $\lim_{n\to\infty}\int_0^1\vert f_n(x)-f(x)\vert^p\,dx=0$ for some $p>0$. Show that $f_n$ converges in measure to $f$ on $[0,1]$.
2. Let $Z$ be a measurable set in $[0,\infty)$ and $E=\{\sqrt{x}(1-x):x\in Z\}$.
   (a) Show that $E$ is measurable.
   (b) Show that $E$ has measure zero if $Z$ has measure zero.
3. Let $\{r_1,r_2,...,r_k,...\}$ be all of the rational numbers in $[0,1]$. Let $f_n(x)=\sum_{k=1}^n \frac1{k^3\vert x-r_k\vert^{1-\frac1{k}}}$.
   (a) Show that $\lim_{n\to\infty} f_n(x)$ exists and $<\infty$ a.e. on $[0,1]$. (b) Show that for a.e. $y\in [0,1]$, we have $\lim_{n\to\infty} [f_n(x)f_n(x+y)]$ exists and $<\infty$ for a.e. $x\in [0,1]$.
4. Let $f$ be a measurable function on $[a,b]$. Determine which of the following conditions implies that
   
   $$f(x)=\int_a^x f'(t)\,dt + f(a) \hbox{ for all }x\in [a, b].$$
   (1) $\vert f(x)- f(y)\vert\le L\vert x-y\vert$ for all $x, y\in [a, b]$. (2) $\vert f(x)- f(y)\vert\le L\sqrt{\vert x-y\vert}$ for all $x, y\in [a, b]$.
5. Show that every nonempty, closed, convex set in a Hilbert space contains a unique element of smallest norm.

B. Choose 1 from the following 2 problems

1. Suppose $f$ is an entire function and $n$ is a nonnegative integer. Show that if
   
   $$f(z)\le a+ b\vert z\vert^n$$
   
   for some positive constants $a$ and $b$, then $f$ is a polynomial of degree at most $n$.
2. Evaluate the intrgral
   
   $$\int_0^{\infty}\frac {xsinx\,dx}{9+x^2}.$$
   
   

[回上頁]