5.1. Fourier transform, Fourier integral

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Chapter 5. Fourier transform

In this Chapter we consider Fourier transform which is the most useful of all integral transforms.

5.1. Fourier transform, Fourier integral

  1. Heuristics
  2. Definitions and Remarks
  3. $\cos $- and $\sin$-Fourier transform and integral

Heuristics

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}

Definitions and Remarks

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.

  1. Sometimes exponents of $\pm i k x$ is replaced by $\pm 2\pi i k x$ and factor $1/(2\pi)$ dropped.
  2. Sometimes factor $\frac{1}{\sqrt{2\pi}}$ is placed in both Fourier transform and Fourier integral: \begin{align} &\hat{f}( k )= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty f(x)e^{-i k x}\,dx; \tag{FT*}\\ &f(x)= \frac{1}{\sqrt{2\pi}}\int_{-\infty}^\infty \hat{f}( k ) e^{i k x}\,d k \tag{FI*} \end{align} Then FT and IFT differ only by $i$ replaced by $-i$ and Plancherel theorem becomes \begin{equation} \int_{-\infty}^\infty |f(x)|^2\,dx= \int_{-\infty}^\infty |\hat{f}( k )|^2\,d k . \tag{PT*} \end{equation} In this case Fourier transform and inverse Fourier transform differ only by $-i$ instead of $i$ (very symmetric form) and both are unitary operators.

Remark 3. We can consider corresponding operator $LX=-X''$ in the space $L^2(\mathbb{R})$ of the square integrable functions on $\mathbb{R}$.

  1. However, $e^{i k x}$ are no more eigenfunctions since they do not belong to this space. Correspondingly, $k^2$ with $k\in \mathbb{R}$ are not eigenvalues. In advanced Real Analysis such functions and numbers often are referred as generalized eigenfunctions and generalized eigenvalues.
  2. The generalized eigenvalues which are not eigenvalues form continuous spectrum in contrast to point spectrum formed by eigenvalues.
  3. Physicists often omit word "generalized".
  4. More details in Section 13.4.

Remark 4.

  1. For justification see Subsection 5.1.A;
  2. Pointwise convergence of Fourier integrals is discussed in Subsection 5.1.B;
  3. Multidimensional Fourier transform and Fourier integral are discussed in Subsection 5.2.A.
  4. Fourier Transform in the Complex Domain (for those who took "Complex Variables") is discussed in Subsection 5.2.B.
  5. Fourier Series interpreted as Discrete Fourier transform are discussed in Subsection 5.2.C.

$\cos $- and $\sin$-Fourier transform and integral

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

  1. $f(x)$ is even function iff $B( k )=0$;
  2. $f(x)$ is odd function iff $A( k )=0$.

Therefore

  1. Each function on $[0,\infty)$ could be decomposed into $\cos$-Fourier integral \begin{equation} f(x)= \int_0^\infty A( k ) \cos ( k x) \,d k \label{eq-5.1.15} \end{equation} with \begin{equation} A( k )=\frac{2}{\pi}\int_0^\infty f(x)\cos ( k x) \,dx. \label{eq-5.1.16} \end{equation}
  2. Each function on $[0,\infty)$ could be decomposed into $\sin$-Fourier integral \begin{equation} f(x)= \int_0^\infty B( k ) \sin ( k x) \,d k \label{eq-5.1.17} \end{equation} with \begin{equation} B( k )= \frac{2}{\pi}\int_0^\infty f(x)\sin ( k x) \,dx. \label{eq-5.1.18} \end{equation}

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