臺灣大學數學系

九十學年度第一學期碩博士班資格考試試題

分析

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1.

[a] Show that the function $h$ defined by $h(x)=c^{-{1\over {x^2}}}$, $x>0$; $h(x)=0$, $x\leq 0$ is in $C^\infty$.

[b] Show that the function $g(x)=h(x-a)h(b-x)$, $a<b$ is $C^\infty$ and with support
in $[a, b]$.

[c] Construct a function in $C^\infty_0(R^n)$ whose support is a ball.

2.

Let $\phi(x)$ be a bounded positive measurable function such that $\phi(x)=0$ outside $[-1, 1]$ and $\int \phi=1$. For $\varepsilon>0$ let $\phi_{\varepsilon}(x)=$ $\varepsilon^{-1}\phi({x\over \varepsilon})$. Show that $\lim \limits_{\varepsilon \rightarrow 0}$ $(f*\phi_{\varepsilon})(x)=f(x)$ in the Lebesgue set of $f$. ($*$ denote convolution)

3.

[a] Show that in $L^2[0, 1]$, the parallelogram law holds, ie.

$\ \ \ \ \vert\vert f+g\vert\vert^2+\vert\vert f-g\vert\vert^2=2\vert\vert f\vert\vert^2+2\vert\vert g\vert\vert^2$.

[b] Is it true for $L^P$, $p\ne 2$?

4.

Prove that together with $f_n(x)\geq 0$ and $If_n\rightarrow 0$, imply $f_n\rightarrow 0$ in measure, but (in general) not $f_n\rightarrow 0$ almost everywhere. Also, the condition $f_n(x)\geq 0$ cannot be dropped.

5.

[a] $1<p<q<\infty$, show that $L^P[0, 1]\supset L^q[0, 1]$ and that

$\ \ \ \ \vert\vert f\vert\vert _p\leq \vert\vert f\vert\vert _q$ for $f\in L^q$.

[b] $\lim \limits_{p\rightarrow \infty}\vert\vert f\vert\vert _p=\vert\vert f\vert\vert _\infty$ for $f\in L^{\infty}[0, 1]$.

6.

[a] Let $f$ be analytic in a region $D$ in upper half plane, the boundary $\partial D$ intersect the real line in a interval $[a, b]$ (Fig. 1). $f$ is continuous on $D\cup [a, b]$ and takes real values on $[a, b]$. Show that (the Schwarz reflection principle) $f$ can be continued analytically into $D^*$ (reflection of $D$).

[b] What can you conclude about the case in Fig. 2, where $f(z)=z^{1\over 2}$ is

$\ \ \$ defined on $D$?

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