2015年丘赛分析组个人赛试题 - Eufisky - The lost book
Problems of the Miklós Schweitzer Memorial Competition

2015年丘赛分析组个人赛试题

Eufisky posted @ 2015年8月10日 14:55 in 竞赛 with tags 丘赛 , 1699 阅读
  1. Let $f_n\in L^2(R)$ be a sequence of measurable functions over the line, $f_n\rightarrow f$ almost everywhere. Let $||f_n||_{L^2}\rightarrow||f||_{L^2}$, prove that $||f_n-f||_{L^2}\rightarrow 0$.

    enlightenedProof.(Weingarten) $L^2(\mathbf R)$ is a Hilbert space, and $\|f_n\|_{L^2}\to\|f\|_{L^2}$ as $n\to\infty$ , so we only need to prove $f_n\in f$ weakly in $L^2(\mathbf R)$, that is, for each $g\in L^2(\mathbf R)$, there holds

    \[\lim_{n\to\infty}\int_{\mathbf R}f_ng=\int_{\mathbf R} fg.\]

    For each $\epsilon>0$, there exists $R>0$, such that \[\int_{|x|>R}|g|^2<\epsilon^2,\]

    and by the absolute continuity of integration of $g$, there exists a positive $\delta$, such that: for any Lebesgue measurable subset $E$ of $\mathbf R$ with $m(E)<\delta$, there holds \[\int_E|g|^2<\epsilon^2.\]

    By the Egoroff's thoerem, there exists a subset $E_\delta$ of $(-R,R)$ with $m((-R,R)\setminus E_\delta)<\delta$, such that the convergence $\lim\limits_{n\to\infty}f_n=f$ is uniform on the $E_\delta$, so there exists $N\in\mathbf Z_+$, such that \[(\forall n>N,x\in E_\delta),(|f_n-f|<\epsilon/\sqrt{2R}).\]

    Assume $M=\|g\|_{L^2}+\|f\|_{L^2}+\sup\limits_n\|f_n\|_{L^2}$, hence $\forall n>N$, we have the following estimations

    \begin{align*}\int_{\mathbf R}|f_n-f|\cdot|g|&=\left(\int_{E_\delta}+\int_{(-R,R)\setminus E_\delta}+\int_{|x|>R}\right)|f_n-f|\cdot|g|\\&\leq\sqrt{\int_{E_\delta}|f_n-f|^2\cdot\int_{E_\delta}|g|^2}+\sqrt{\int_{(-R,R)\setminus E_\delta}|f_n-f|^2\cdot\int_{(-R,R)\setminus E_\delta}|g|^2}\\&+\sqrt{\int_{|x|>R}|f_n-f|^2\cdot\int_{|x|>R}|g|^2}\\&\leq M\epsilon+2\sqrt{2}M\epsilon.\end{align*}

    the proof is finished.


     

  2. Let $f$ be a continuous function on $[a,b]$, define $M_n=\int ^b_a f(x)x^n{\rm d}x$. Suppose that $M_n=0$ for all $n$, show that $f(x)=0$ for all $x$.
  3. Determine all entire functions $f$ that satisfying the inequality $$|f(z)|\leq|z|^2|{\rm Im}(z)|^2$$ for $z$ sufficlently large.
  4. Describe all holomorphic functions over the unit disk $D=\{z||z|\leq 1\}$ which maps the boundary of the disk into the boundary of the disk.
  5. Let $T:H_1\rightarrow H_2, Q:H_2\rightarrow H_1$ be bounded linear operators of Hilbert spaces $H_1,\ H_2$. Let $QT={\rm Id}-S_1,TQ={\rm Id}-S_2$ where $S_1$ and $S_2$ are compact operators. Prove ${\rm Ker}T=\{v\in H_1,Tv=0\},{\rm Coker}T=H_2/\overline{{\rm Im}T}$, where ${\rm Im}T=\{Tv\in H_2,v\in H_1\}$ are finite dimensional and ${\rm Im}(T)$ is closed in $H_2$.
  6. Let $H_1$ be the Sobolev space on the unit interval $[0,1]$, i.e. the Hilbert space consisting of functions $f\in L^2([0,1])$ such that $$||f||_1^2=\sum^\infty_{n=-\infty}(1+n^2)|\hat f(n)|^2<\infty;$$ where $$\hat f(n)=\frac 1 {2\pi}\int_0^1f(x)e^{-2\pi inx}{\rm d}x$$ are Fourier coefficients of $f$. Show that there exists constant $C>0$ such that $$||f||_{L^\infty}\leq C||f||_1$$ for all $f\in H_1$, where $||\cdot||_{L^\infty}$ stands for the usual supremum norm. (Hint: Use Fourier series.)
Avatar_small
bdpscresult2018.com 说:
2023年4月17日 01:46

Education Board DPE has conducted the class 8th grade of Junior School Certificate Exam and Junior Dakhil Certificate Exam on 1st to 15th November at all centers in division wise under Ministry of Primary and Mass Education (MOPME), and the class 8th grade terminal examination tests are successfully conducted for all eligible JSC/JDC students for the academic year The students who have qualified in this bdjscresult2018.com JSC/JDC exams they will get stipend from the government of to continue their higher class easily and this very helpful to every student, there are 9 education boards are working under Primary and Maas education ministry, according to the information 18,778 school students are participating in the exam from all division education boards in the country.


登录 *


loading captcha image...
(输入验证码)
or Ctrl+Enter