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\lhead{CS 590--MTK, SPRING 2019}               
\rhead{Name: Hemanta K. Maji} %%% <-- REPLACE Hemanta K. Maji WITH YOUR NAME HERE

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\title{Homework 6}

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\item {\bfseries  Fourier Transformation Matrix.} (20 points)
  We shall provide an alternate mechanism to construct the Fourier transformation matrix. 
  Recall that, for functions $\zo^n\to\bbR$, we defined the basis functions as follows. 
  For all $S,x\in\zo^n$, we defined
    $$ \chi_S(x) \defeq (-1)^{S_1\cdot x_1 + S_2\cdot x_2 + \dotsi + S_n\cdot x_n}$$
  Given this definition of the Fourier basis functions, the definition of the Fourier transformation matrix $\cF_n \in \frac1N \{+1,-1\}^{N\times N}$, where $N=2^n$, is as follows.
  We shall use row indices $i\in\{0,1,\dotsc,N-1\}$ and $j\in\{0,1,\dotsc,N-1\}$ and define 
    $$ \left(\cF\right)_{i,j} \defeq \frac1N\chi_j(i)$$
    
  Now, we begin the new definition using {\em matrix tensor product}. 
  Let $A\in\bbR^{a\times b}$ and $B\in \bbR^{a'\times b'}$ be two matrices. 
  We define the {\em block matrix} $C=A\otimes B$ as follows.
  For $i\in\{1,\dotsc,a\}$ and $b\in\{1,\dotsc,b\}$
    $$ C_{i,j} \defeq a_{i,j}B$$
    
  {\bfseries Base case.} Define
    $$\cG_1 \defeq \frac12\left[\begin{matrix} 1 &1 \\1&-1\end{matrix}\right]$$
  {\bfseries Recursive construction.} Define, for $n>1$, $\cG_n \defeq \cG_{1}\otimes\cG_{n-1}$.  
  
  Prove, by induction, that $\cF_n = \cG_n$.

\noindent{\bfseries Solution.}\newline 
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\item {\bfseries Smoothed Function Property.} (20 points) 
  Let $f\colon\zo^n\to\bbR$ be a function. 
  Let $L_p(f)$ be the norm defined as follows
    $$ L_p(f) \defeq \left(\frac1N \sum_{x\in\zo^n} \abs{f(x)}^p\right)^{1/p}$$
  For any $\rho\in[0,1]$, prove that $L_p(T_\rho(f)) \leq L_p(f)$. 
  Equality holds if and only if $f$ is a constant function, or $\rho=1$. 

\noindent{\bfseries Solution.}\newline 
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\item {\bfseries Most Random functions are Small Biased.} (20 points) 
   Let $f\colon\zo^n\to\{+1,-1\}$ be a boolean function.
   Suppose we consider a {\em random} boolean function such that, for every $x\in\zo^n$, we assign $f(x)$ independently and uniformly at random from the set $\{+1,-1\}$. 
   Recall that a function $f$ is small biased if $\abs{\bias_f(S)}\leq\eps$ for all $0\neq S\in\zo^n$. 
   
   Formally state and prove a concentration result that proves: ``a random boolean function is small-biased with very high probability.'' 

 


\noindent{\bfseries Solution.}\newline 
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\item {\bfseries Differential Operator.} (20 points) 
  We shall consider functions $\zo^n\to\bbR$. 
  Let us introduce a notation. 
  Given $x\in\zo^n$, we represent $x|_{i=1}$ as the bit-string identical to $x$ except that its $i$-th coordinate is fixed to 1.   
  Similarly, $x|_{i=0}$ is the bit-string that is identical to $x$ except that its $i$-th coordinate is fixed to $0$. 
  
  Let $D_i(f)$ be the function $\zo^n\to\bbR$ defined as follows
    $$ D_i(f)(x) = f(x|_{i=1})-f(x|_{i=0})$$
    
  Express $\widehat{D_i(f)}$ as a function of $\hf$. 

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\item {\bfseries Flats are Small-biased Distribution.} (20 points) 
  We shall consider function $\bbZ_p\to\bbC$ in this problem. 
  Define $\omega = \exp(2\pi \imath /p)$. 
  Recall that we defined, for $S\in\bbZ_p$, as follows 
    $$ \bias_f(S) = \sum_{x\in\bbZ_p} f(x)\omega^{S\cdot x}$$
 
 Let $\X$ be a uniform distribution over the set $\{0,1,\dotsc,t-1\}$, for some integer $t<p$.
 Prove that
   $$ \bias_\X{(1)} \leq \frac{\sinc(\pi t/p)}{\sinc(\pi /p)}, $$
 where $\sinc(x)\defeq \sin(x)/x$

\noindent{\bfseries Solution.}\newline 
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%%%%%%%%%%%% PLEASE LIST COLLABORATORS BELOW  %%%%%
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{\bfseries Collaborators :} \newline 
% ENTER THEIR NAMES HERE  

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