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\lhead{CS 592--ATK, SPRING 2022}
\rhead{Name: Hemanta K. Maji} %%% <-- REPLACE Hemanta K. Maji WITH YOUR NAME HERE
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\title{Homework 4}
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\begin{enumerate}
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%%%%%%%%%%%% PROBLEM 1 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Fourier Analysis on Larger Domains.} (5+5+5+5 points)
Recall that we apply discrete Fourier Analysis on the Boolean Hypercube to analyze functions with domain $\zo^n$.
We will generalize this analysis to arbitrary domains.
\begin{enumerate}
\item Consider the space of all function $\bbZ_p\to\bbC$, where $p$ is a prime number.
Here $\bbZ_p$ is the set $\{0,1,\dotsc,p-1\}$.
And addition and multiplication of two elements from this set is defined using integer addition and multiplication, respectively, $\mod p$.
The set of complex numbers is represented by \bbC.
Suppose $f,g\colon\bbZ_p\to\bbC$ be two functions.
Recall that the {\em complex conjugate} of a complex number $z=a + \imath b$, represented by $\overline z$, is defined to be $a -\imath b$.
The inner-product of these two functions is defined by
$$\tuple{f,g} \defeq \frac1p \sum_{x\in\bbZ_p}f(x)\overline{g(x)}$$
Let $\omega_p \defeq \exp\left(2\pi\imath/p\right)$ and define $\chi_a(x) \defeq \omega_p^{ax}$, for $a\in\bbZ_p$.
Prove that $\{\chi_a \colon a\in\bbZ_p$ is an orthonormal basis for the space of all function $\bbZ_p\to\bbC$.
\item Consider the space of all functions $\bbZ_p^n \to \bbC$.
Define the inner-product of functions, write the Fourier basis functions, and show their orthonormality.
\item Consider the space of all functions $\bbZ_p\times\bbZ_q\to\bbC$, for primes $p$ and $q$.
The primes $p$ and $q$ need not necessarily be distinct.
Define the inner-product of functions, write the Fourier basis functions, and show their orthonormality.
\item Consider the space of all functions $\bbZ_{p_1}\times \bbZ_{p_2} \times\dotsi\times\bbZ_{p_n} \to \bbC$.
Note that the primes $p_1,\dotsc,p_n$ need not be distinct.
Define the inner-product of functions, write the Fourier basis functions, and show their orthonormality.
\end{enumerate}
\noindent{\bfseries Solution.}\newline
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%%%%%%%%%%%% PROBLEM 2 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Majority Functions.} ( 5 + 15 points)
Let $n$ be an odd number, and $f(x)\colon\zo^n\to\{+1,-1\}$ be the majority function.
That is, if the majority of the bits in $x$ is $0$, then $f(x)=+1$; otherwise $f(x)=-1$.
\begin{enumerate}
\item Compute the Fourier coefficients of $f$ when $n=3$.
\item For $x\in\zo^n$, define $\pred{flip}(x)$ to be the string where we flip every bit of $x$.
For example, we have $\pred{flip}(00101)=11010$.
A function is {\em odd} if $f(\pred{flip}(x)) = -f(x)$, for all $x\in\zo^n$.
Note that the majority function defined above is an odd function.
A set $S\in\zo^n$ is {\em even} if the number of 1s in $S$ is even. For example, when $n=3$, the sets $S=000, 011, 101, 110$ are even sets.
Prove that if $f$ is an odd function then $\hf(S)=0$ for all even $S\in\zo^n$.
\end{enumerate}
\noindent{\bfseries Solution.}\newline
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%%%%%%%%%%%% PROBLEM 3 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries Generalized BLR.} (20 points)
Recall that a function $f\colon\zo^n\to\{+1,-1\}$ is linear if $f(0^n) = +1$ and $f(x+y)=f(x)\cdot f(y)$, for all $x,y\in\zo^n$.
Consider the following generalization of the BLR algorithm to test whether a function $f$ or $-f$ is close to linear.
\begin{boxedalgo}
$\pred{BLR-Gen}^f$:
\begin{enumerate}
\item Let $a,b,c \getsr\zo^n$
\item Let $w=f(a)$, $x=f(b)$, $y=f(c)$, and $z=f(a+b+c)$
\item Return $(q\cdot x\cdot y == z)$
\end{enumerate}
\end{boxedalgo}
State and prove a theorem that intuitively proves that ``the algorithm returns true with high probability'' if and only if ``the function $f$ or $-f$ is close to a linear function.''
\noindent{\bfseries Solution.}\newline
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%%%%%%%%%%%% PROBLEM 4 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries An Alternative Proof.} (5+15 points)
Recall that the convolution of two function $f,g\colon\zo^n\to\bbR$ is defined as follows
$$(f*g)(x) \defeq\frac1N \sum_{y\in\zo^n} f(y)g(x-y)$$
In this problem we shall develop a new technique to prove that $\widehat{(f*g)} = \hf \hg$.
\begin{enumerate}
\item Compute the function $(\chi_S * \chi_T)$
\item Note that the convolution operator is a bilinear operator.
That is, we have $\left((f_1+f_2)*g\right) = (f_1*g) + (f_2*g)$ and $(cf)*g = c(f*g)$ from the definition of convolution.
Similarly, we have $\left(f*(g_1+g_2)\right) = (f*g_1) + (f*g_2)$ and $f*(cg) = c(f*g)$.
Recall that we have $f = \sum_{S\in\zo^n} \hf(S) \chi_S$ and $g=\sum_{S\in\zo^n} \hg(S)\chi_S$.
Prove that $$(f*g) = \sum_{S\in\zo^n} \hf(S)\hg(S) \chi_S$$
\end{enumerate}
\noindent{\bfseries Solution.}\newline
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%%%%%%%%%%%% PROBLEM 5 %%%%%%%%%%%%%%%%%%%%%%%%%%%%
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\item {\bfseries A Few Properties of Fourier Transformation.} (5+5+5+5 points)
Let $f,g\colon\zo^n\to\bbR$ be two functions.
\begin{enumerate}
\item Express $\widehat{(fg)}$ using the functions \hf and \hg.
Here the function $(fg)$ defined as $(fg)(x) = f(x)\cdot g(x)$, for all $x\in\zo^n$.
\item Let $\max\{f,g\}$ is the function that satisfies $\max\{f,g\}(x) = \max\{f(x),g(x)\}$, for all $x\in\zo^n$.
Suppose the range of $f$ and $g$ is $\{+1,-1\}$.
Express $\widehat{\max\{f,g\}}$ is terms of \hf and \hg.
\item Recall that if $f(x) = g(x-c)$ for some $c\in\zo^n$ then we have $\hf = \chi_c\hg$.
Find a function $h\colon\zo^n\to\bbR$ such that $f = (h* g)$.
\item For $1\leq i