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no begin{enumerate}

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1 parent 4a2368e commit 41c5e6440093969253ee635d8dc98b07b9281daf @jlazarow committed Apr 29, 2012
Showing with 30 additions and 12 deletions.
  1. +30 −12 M365D Final.page
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42 M365D Final.page
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** Due: May 16 **
### Problems ###
-\begin{enumerate}
-\item[1.]
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+1.
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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.
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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.]
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+3.
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Show that the cardinality of a $\sigma$-algebra is either finite or
uncountable.
-\item[4.]
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+4.
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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.
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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.]
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+6.
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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.]
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+7.
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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.]
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+8.
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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.]
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+9.
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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.]
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+10.
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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}
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### Commentary ###
+
1.
2. Naturally related to approximation to the identity using convolution.

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