# jlazarow/wiki

no begin{enumerate}

 @@ -1,21 +1,27 @@ ** Due: May 16 ** ### Problems ### -\begin{enumerate} -\item[1.] + +1. + Let $\{a_n\}$ be a sequence of real numbers such that $|\sum a_n| < \infty$ but that $\sum |a_n| = \infty$. Can it be that $\sum a_n^3 = \infty$. Explain. -\item[2.] +2. + Let $\phi(x) = \frac{1}{\pi} \frac{\sin{x}}{1 + x^2}$ over $\mathbb{R}$. For $f \in L^2(\mathbb{R})$, determine (with proof) the limit in $L^2(\mathbb{R})$ of $\phi_s * f$ as $\delta \to 0$ with $\phi_s - \delta^{-1} \phi(\delta x)$. -\item[3.] + +3. + Show that the cardinality of a $\sigma$-algebra is either finite or uncountable. -\item[4.] + +4. + Show by the most elementary'' proof that: $\sum_{n=1}^\infty \frac{1}{\ln{1 + n!}} = \infty$ @@ -29,12 +35,15 @@ Use the gamma function and dominated converge theorem to show Stirling's formula. -\item[5.] +5. + Let $\{ f_n \}$ be a sequence of Lebesgue measurable functions on $[0,1]$ with $|f_n(x)| < \infty$ a.e. Show that there exists a sequence of positive real numbers $\{c_n\}$ such that $\frac{f_n(x)}{c_n} \to 0$ a.e. by using Borel-Cantelli. -\item[6.] + +6. + Let $\{ a_n\}$ be a sequence of real numbers. Suppose $f(t)$ is a positive integrable function: $f(t) > 0$ and $\int_{-\infty}^\infty f < \infty$. @@ -44,11 +53,15 @@ Then suppose that for some sequence of reals $\{ r_n \}$ that $\sum_{n=1}^\infty r_n |\cos{nt + a_n}| < \infty$. Then show that $\sum_{n=1}^\infty r_n < \infty$. -\item[7.] + +7. + Let $f$ be measurable on a space $M$. Suppose that $|f(x) - f(y)|$ is integral over $M \times M$. Show that $f(x)$ is integrable over $M$. (using Fubini's theorem). -\item[8.] + +8. + Let $F$ be a closed set in $\mathbb{R}$ whose complement has finite measure. Let $\delta(x)$ denote the distance of $x$ to $F$. i.e. $\delta(x) = d(x,F) = \inf \{ |x-y| : y \in F\}$. Let $I(x) = @@ -59,21 +72,26 @@ Show that$I(x) < \infty$a.e. for$x \in F$, by looking at$\int_F I(x) dx$for the case of the real line, why would it suffice to consider$F^C$as an open interval. -\item[9.] + +9. + A sequence$\{ f_n \}$in$L^p$converges weakly to$f \in L^p$for$1 \leq p < \infty$if$\int f_n g d\mu \to \int fg d\mu$for every$g \in (L^p)^*$If$||f_n - f||_{L^p} \to 0$, show that$f_n \to f$weakly in$L^p$. -\item[10.] + +10. + Let$\lambda > 0$. Find$\lim_{A \to \infty} \int_{-A}^A \frac{\sin{\lambda t}{t} e^{i tx} dt$. The Dirichlet integral$\int_0^\infty \frac{\sin{t}}{t} = \frac{\pi}{2}\$ might be useful. -\end{enumerate} + ### Commentary ### + 1. 2. Naturally related to approximation to the identity using convolution.