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In this Chapter we consider Fourier transform which is the most useful of all integral transforms.
In Section 4.5 we wrote Fourier series in the complex form \begin{equation} f(x)= \sum_{n=-\infty}^\infty c_n e^{\frac{i\pi nx}{l}} \label{eq-5.1.1} \end{equation} with \begin{equation} c_n= \frac{1}{2l}\int_{-l}^l f(x)e^{-\frac{i\pi n x}{l}}\,dx \qquad n=\ldots,-2, -1, 0,1,2,\ldots \label{eq-5.1.2} \end{equation} and \begin{equation} 2l\sum_{n=-\infty}^\infty |c_n|^2=\int_{-l}^l|f(x)|^2\,dx. \label{eq-5.1.3} \end{equation}
From this form we formally, without any justification, will deduct Fourier integral.
First we introduce \begin{equation} k _n := \frac{\pi n}{l}\qquad \text{and}\qquad \Delta k _n = k _{n}- k _{n-1}= \frac{\pi}{l} \label{eq-5.1.4} \end{equation} and rewrite (\ref{eq-5.1.1}) as \begin{equation} f(x)= \sum_{n=-\infty}^\infty C( k _n) e^{i k _n x}\Delta k _n \label{eq-5.1.5} \end{equation} with \begin{equation} C( k )= \frac{1}{2\pi}\int_{-l}^l f(x)e^{-i k x}\,dx \label{eq-5.1.6} \end{equation} where we used $C( k _n) := c_n /(\Delta k _n)$; (\ref{eq-5.1.3}) should be rewritten as \begin{equation} \int_{-l}^l|f(x)|^2\,dx= 2\pi\sum_{n=-\infty}^\infty |C( k _n)|^2\Delta k _n. \label{eq-5.1.7} \end{equation} Now we formally set $l\to +\infty$; then integrals from $-l$ to $l$ in the right-hand expression of (\ref{eq-5.1.6}) and the left-hand expression of (\ref{eq-5.1.7}) become integrals from $-\infty$ to $+\infty$.
Meanwhile, $\Delta k _n \to +0$ and Riemannian sums in the right-hand expressions of (\ref{eq-5.1.5}) and (\ref{eq-5.1.7}) become integrals: \begin{equation} f(x)= \int_{-\infty}^\infty C( k ) e^{i k x}\,d k \label{eq-5.1.8} \end{equation} with \begin{equation} C( k )= \frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-i k x}\,dx; \label{eq-5.1.9} \end{equation} (\ref{eq-5.1.3}) becomes \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx= 2\pi\int_{-\infty}^\infty |C( k )|^2\,d k . \label{eq-5.1.10} \end{equation}
Definition 1. Formula (\ref{eq-5.1.9}) gives us a Fourier transform of $f(x)$, it usually is denoted by "hat": \begin{equation} \hat{f}( k )= \frac{1}{2\pi}\int_{-\infty}^\infty f(x)e^{-i k x}\,dx; \tag{FT}\label{FT} \end{equation} sometimes it is denoted by "tilde" ($\tilde{f}$), and seldom just by a corresponding capital letter $F( k )$.
Definition 2. Expression (\ref{eq-5.1.8}) is a Fourier integral a.k.a. inverse Fourier transform: \begin{equation} f(x)= \int_{-\infty}^\infty \hat{f}( k ) e^{i k x}\,d k \tag{FI}\label{FI} \end{equation} a.k.a. \begin{equation} \check{F}(x)= \int_{-\infty}^\infty F( k ) e^{i k x}\,d k. \label{IFT}\tag{IFT} \end{equation}
Remark 1. Formula (\ref{eq-5.1.10}) is known as Plancherel theorem \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx=2\pi\int_{-\infty}^\infty |\hat{f}( k )|^2\,d k . \tag{PT}\label{PT} \end{equation}
Remark 2.
Remark 3. We can consider corresponding operator $LX=-X''$ in the space $L^2(\mathbb{R})$ of the square integrable functions on $\mathbb{R}$.
Remark 4.
Applying the same arguments as in Section 4.5 we can rewrite formulae (\ref{eq-5.1.8})--(\ref{eq-5.1.10}) as \begin{equation} f(x)= \int_0^\infty \bigl( A( k ) \cos( k x) +B( k ) \sin ( k x)\bigr) \,d k \label{eq-5.1.11} \end{equation} with \begin{align} & A( k )= \frac{1}{\pi}\int_{-\infty}^\infty f(x)\cos ( k x) \,dx, \label{eq-5.1.12}\\ & B( k )= \frac{1}{\pi}\int_{-\infty}^\infty f(x)\sin ( k x) \,dx, \label{eq-5.1.13} \end{align} and \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx= \pi\int_0^\infty \bigl( |A( k )|^2+|B( k )|^2\bigr)\,d k . \label{eq-5.1.14} \end{equation}
$A( k )$ and $B( k )$ are $\cos$- and $\sin$- Fourier transforms and
Therefore
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