Exercise 10.23

Question

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

Suppose that $f$ is a function mapping $F$ to $\C$. Define $\hat{f}(s) = (1/q) \sum_t f(t) \overline{\psi(st)}$ and prove that $f(t) = \sum_s \hat{f}(s) \psi(st)$. The last sum is called the finite Fourier series expansion of $f$.

Richard Ganaye · Accepted Answer

\def\imp{\Rightarrow}
\def\gcro{\mbox{[\hspace{-.15em}[}}
\def\dcro{\mbox{]\hspace{-.15em}]}}

ewcommand{\D}{\mathrm{d}}

ewcommand{\Q}{\mathbb{Q}}

ewcommand{\Z}{\mathbb{Z}}

ewcommand{\N}{\mathbb{N}}

ewcommand{\R}{\mathbb{R}}

ewcommand{\C}{\mathbb{C}}

ewcommand{\F}{\mathbb{F}}

ewcommand{e}{\,\mathrm{Re}\,}

ewcommand{\ord}{\mathrm{ord}}

ewcommand{
}{\mathrm{N}}

ewcommand{\legendre}[2]{\genfrac{(}{)}{}{}{#1}{#2}}

\begin{proof} Using the proposition 10.3.3, we obtain, for all $t \in \F_q$,
\begin{align*}
 \sum_{s \in \F_q} \hat{f}(s) \psi(st) &=  \frac{1}{q} \sum_{s \in \F_q} \left(\sum_{u \in \F_q} f(u) \overline{\psi(su)} ight ) \psi(st)\
 &= \frac{1}{q}  \sum_{u \in \F_q} f(u)  \sum_{s \in \F_q} \psi(s(t-u))\
 &= \frac{1}{q}  \sum_{u \in \F_q} f(u)  q \delta(t,u)\
 &= f(t).
\end{align*}

\end{proof}

Exercise 10.23

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Exercise 10.23

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