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\title{ HW V for Math 6210\\ due December 10}
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\noindent 
1) Consider the Hilbert space $L^2(\mathbb R/\mathbb Z)$ of periodic measurable functions. Recall that it has a Hilbert space basis consisting of functions 
$\chi_n(x)=e^{2\pi i n x}$ where $n\in \mathbb Z$. Let $f\in L^2(\mathbb R/\mathbb Z)$ be the periodic function defined by $f(x)= x$, for $x\in [0,1)$. Compute 
$||f||^2$ in two ways, directly, and then using the Parseval's identity, i.e. using the Fourier series expansion of $f$. 

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2)  
Let $\varphi \geq 0$ be a smooth function supported on $[-1,1]$ such that $\int \varphi ~dx=1$. For every positive integer $n$, let $\varphi_n(x)=n^{-1} \varphi(nx)$. 
For every $f\in L^1(\mathbb R)$, let $f_n = f \ast \varphi_n$. Prove that $f_n \rightarrow f$ in the $L^1$-norm.  


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\noindent 
3) Let $f\in L^1(\mathbb R)$ and $\varphi$ a smooth compactly supported function. Prove that $f\ast \varphi$ is a smooth function whose derivative is 
$f\ast \varphi$. There is another notion of derivative:  $h\in L^1(\mathbb R)$ is the $L^1$-derivative of $f$ if
\[ 
\lim_{y \rightarrow 0}  \frac{\tau_y f -f}{y} =h 
\] 
where the limit is with respect to the $L^1$-norm. Prove that the $L^1$-derivative of  $f\ast \varphi$ 
 is $f\ast \varphi'$.  


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4) Let $f\in L^2(\mathbb R)$. Then $f$ is $L^2$-differentiable if and only if $x \widehat{f}(x) \in L^2(\mathbb R)$. Hint: use Fourier transform isometry. 

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5) Let $1 < p < q < \infty$. Then $L^p$ is not contained in $L^q$ iff $X$ contains sets of arbitrarily small positive measure, and $L^q$ is not contained in 
$L^p$ iff $X$ contains sets of arbitrarily large finite measure. This is exercise 5, page 187, see the hints there. 

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6)  Exercise 18 on page 191.  Follow the hints there to prove the Radon-Nikodym Theorem for $(X,\mu)$ of finite measure, using $L^2(X)^*\cong L^2(X)$ which is 
a part of the Hilbert space theory. (As usually, for simplicity assume we deal with real valued functions.)  


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