Foundations of the Fourier Transform

1Introduction

This lecture explains how the Fourier transform represents a function by the complex amplitudes of its frequency components. The transform provides an invertible correspondence, not merely an informal decomposition, and converts differentiation into multiplication by frequency.

2Schwartz Functions and the Transform Convention

To state convergence results without ambiguity, first work in the Schwartz space $\text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$\mathcal S(\mathbb R). A smooth function $f$f belongs to this space when, for every pair of nonnegative integers $m,n$m,n,

Thus $f$f and all its derivatives decay faster than every reciprocal polynomial. For $f,g\in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$f,g\in\mathcal S(\mathbb R), define

Here $\omega$\omega is angular frequency in radians per unit time. Ordinary frequency $\nu$\nu is measured in cycles per unit time, or Hz when time is in seconds, and satisfies $\omega =2\pi \nu$\omega=2\pi\nu. Communications texts often use

so $X\left(\nu \right)=X_{\omega }\left(2\pi \nu \right)$X(\nu)=X_\omega(2\pi\nu). The sign and placement of $2\pi$2\pi vary among conventions; the forward and inverse definitions must always be specified together.

3Fourier Inversion

If $f\in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$f\in\mathcal S(\mathbb R), then $\hat{f}\in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$\hat f\in\mathcal S(\mathbb R) and

A standard proof regularizes the inverse integral with a Gaussian factor. Interchanging the absolutely convergent integrals produces the convolution of $f$f with a Gaussian approximate identity. As the Gaussian width tends to zero, this convolution converges to $f$f. Rapid decay justifies the exchanges of limits and integrals. This theorem is the basis for reconstructing a function from its spectrum.

4An Acoustic Interpretation

A sound waveform records air-pressure variation in time, while pitch perception distinguishes its high- and low-frequency components. The Fourier transform provides the mathematical complex amplitude and phase of these components.

5Discrete Orthogonality and the Continuous Transform

For integers $m,n\in \mathbb{Z}$m,n\in\mathbb Z, the complex exponentials on $[-\pi ,\pi ]$[-\pi,\pi] satisfy

Fourier series use this orthogonality to extract coefficients at discrete frequencies. The Fourier transform extends the construction to a continuum of angular frequencies: $\hat{f}\left(\omega \right)$\hat f(\omega) is a density of complex amplitude and therefore records both amplitude and phase. Fourier inversion establishes the reciprocal correspondence between $f$f and $\hat{f}$\hat f.

6Differentiation Rule

For $f\in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$f\in\mathcal S(\mathbb R), integration by parts gives

The boundary term vanishes by rapid decay. Hence differentiation in the original variable becomes multiplication by $i\omega$i\omega in the frequency domain. This identity is a principal reason that Fourier methods simplify constant-coefficient differential equations.

7Gaussian Example

For $f\left(x\right)=e^{-x^2}$f(x)=e^{-x^2},

To derive the formula, put

Differentiation under the integral and the identity $x{e}^{-x^2}=-\frac{1}{2}(d/dx)e^{-x^2}$xe^{-x^2}=-\frac12(d/dx)e^{-x^2} yield, after integration by parts,

Together with the Gaussian integral $I\left(0\right)=\sqrt{\pi }$I(0)=\sqrt\pi, this differential equation gives $I\left(\omega \right)=\sqrt{\pi }{e}^{-{\omega }^2/4}$I(\omega)=\sqrt\pi e^{-\omega^2/4}. Thus a smooth localized Gaussian transforms to another Gaussian.

8Functions Outside the Schwartz Space

The Dirac delta $\delta$\delta is the distribution defined by the extraction property $⟨\delta (\omega -a),\phi ⟩=\phi \left(a\right)$\langle\delta(\omega-a),\varphi\rangle=\varphi(a) for every test function $\phi \in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$\varphi\in\mathcal S(\mathbb R). A tempered distribution is a continuous linear functional on the Schwartz space, and the Fourier transform extends to this space.

The function $\cos \left(ax\right)$\cos(ax) does not decay at infinity, so its Fourier integral does not converge in the preceding sense. Under the present convention, its transform is the tempered-distribution identity

This identity means that both sides give the same value when applied to every Schwartz test function; it is not an equality of ordinary convergent integrals.

If $f\in L^1\left(\mathbb{R}\right)$f\in L^1(\mathbb R), the forward integral exists for every $\omega$\omega, but inversion requires additional hypotheses. For a general $L^2\left(\mathbb{R}\right)$L^2(\mathbb R) function, the integral need not exist pointwise. The transform is instead defined as the unique $L^2$L^2 extension supplied by Plancherel's theorem.

9Plancherel's Theorem

For $f\in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$f\in\mathcal S(\mathbb R),

The proof first establishes preservation of the inner product, with the normalization factor dictated by the transform convention, by substituting the inversion formula and justifying interchange of integrals. Since $\text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$\mathcal S(\mathbb R) is dense in $L^2\left(\mathbb{R}\right)$L^2(\mathbb R), the scaled isometry extends uniquely to all of $L^2$L^2. Consequently, time-domain and frequency-domain squared norms contain the same information up to the stated factor.

10Mathematical Interpretations

10.1Analytic Interpretation

Whereas calculus describes pointwise values and local variation, the Fourier transform represents a function as a superposition of frequency components. This representation is particularly appropriate for waves and signals.

10.2Linear-Algebra Interpretation

In finite-dimensional discrete settings, the discrete Fourier transform is multiplication by a matrix whose columns are orthogonal discrete exponentials. The continuous transform analogously supplies continuous frequency coordinates through inversion and Plancherel's identity. However, $e^{i\omega x}$e^{i\omega x} is not in $L^2\left(\mathbb{R}\right)$L^2(\mathbb R), so these functions do not form an ordinary finite- or countable-dimensional basis.

10.3Operator Interpretation

The identity

makes $e^{i\omega x}$e^{i\omega x} a generalized eigenfunction of the differentiation operator with generalized eigenvalue $i\omega$i\omega. It is not an ordinary $L^2\left(\mathbb{R}\right)$L^2(\mathbb R) eigenfunction because it is not square-integrable. Fourier transformation diagonalizes differentiation in this generalized sense, converting differential equations into algebraic equations indexed by $\omega$\omega.

10.4Appropriate Applications

• analyzing waves, oscillations, and periodicity by frequency components;
• converting differential equations into algebraic equations indexed by frequency;
• analyzing global oscillatory structure rather than local information.

11Scope and Transform Contract

Fourier series are the natural elementary model for periodic functions, whereas the Fourier transform is the corresponding model on the entire real line. Function-space and distributional extensions refine this distinction.

Transform contract for this lecture. For $f\in \text{[PARSE ERROR: Undefined("Command("mathcal")")]}S\left(\mathbb{R}\right)$f\in\mathcal S(\mathbb R),

$$\hat{f}\left(\omega \right)={\int }_{\mathbb{R}}f\left(x\right)e^{-i\omega x}dx,f\left(x\right)=\frac{1}{2\pi }{\int }_{\mathbb{R}}\hat{f}\left(\omega \right)e^{i\omega x}d\omega ,$$\hat f(\omega)=\int_{\mathbb R}f(x)e^{-i\omega x}\,dx, \qquad f(x)=\frac1{2\pi}\int_{\mathbb R}\hat f(\omega)e^{i\omega x}\,d\omega,

$$\widehat{f^{\prime }}\left(\omega \right)=i\omega \hat{f}\left(\omega \right),{\int }_{\mathbb{R}}|f{|}^2=\frac{1}{2\pi }{\int }_{\mathbb{R}}|\hat{f}{|}^{2.}$$\widehat{f'}(\omega)=i\omega\hat f(\omega), \qquad \int_{\mathbb R}|f|^2 =\frac1{2\pi}\int_{\mathbb R}|\hat f|^2.

The formula $X\left(\nu \right)=X_{\omega }\left(2\pi \nu \right)$X(\nu)=X_\omega(2\pi\nu) translates this angular-frequency convention to the Hz convention used in communications engineering.

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