The Central Idea
Instead of representing a function by values in its original variable, the Fourier transform represents it by the amplitudes and phases of complex exponentials of different frequencies.
Using one common convention, the Fourier transform of a function \(f\) is
\[ \boxed{ \widehat f(\omega) =\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt }. \]The inverse transform reconstructs the original function:
\[ \boxed{ f(t) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat f(\omega)e^{i\omega t}\,d\omega }. \]The variable \(t\) may represent time, position, or another independent variable. The transform variable \(\omega\) represents angular frequency.
From Fourier Series to Fourier Transforms
A Fourier series for a function with period \(T\) uses frequencies
\[ \omega_n=n\omega_0, \qquad \omega_0=\frac{2\pi}{T}. \]These frequencies form a discrete ladder. As the period grows, the spacing between neighboring frequencies becomes smaller:
\[ \Delta\omega=\frac{2\pi}{T}. \]In the limit of an infinitely long period, the discrete frequency ladder becomes a continuous frequency axis. A sum over harmonics is replaced by an integral over \(\omega\). This is the conceptual passage from Fourier series to the Fourier transform.
A Continuous Spectrum
A Fourier series produces coefficients such as
\[ c_n, \qquad n\in\mathbb Z. \]A Fourier transform instead produces a function
\[ \widehat f(\omega), \qquad \omega\in\mathbb R. \]The quantity \(\widehat f(\omega)\) tells us how strongly the frequency \(\omega\) contributes to the original function. Because \(\omega\) varies continuously, the result is called a continuous spectrum.
Complex Exponentials as Frequency Modes
Euler's formula gives
\[ e^{i\omega t} =\cos(\omega t)+i\sin(\omega t). \]Thus, complex exponentials package sine and cosine oscillations into one compact expression. The Fourier transform measures how much of each mode
\[ e^{i\omega t} \]is present in the function.
This complex notation is not a different theory from sine-and-cosine analysis. It is a more efficient representation of the same oscillatory structure.
The Transform Pair
Under the convention used in this lesson, we write
\[ \boxed{ f(t)\ \longleftrightarrow\ \widehat f(\omega) }. \]The forward transform is
\[ \widehat f(\omega) =\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt, \]while the inverse transform is
\[ f(t) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat f(\omega)e^{i\omega t}\,d\omega. \]Different books distribute factors of \(2\pi\) differently. The formulas are equivalent when one convention is used consistently.
Frequency in Cycles per Unit
Some applications use ordinary frequency \(\nu\), measured in cycles per unit, rather than angular frequency \(\omega\). The variables are related by
\[ \boxed{\omega=2\pi\nu}. \]A common alternative convention is therefore
\[ \widehat f(\nu) =\int_{-\infty}^{\infty} f(t)e^{-2\pi i\nu t}\,dt. \]The underlying mathematics is unchanged. Only the scaling of the frequency variable differs.
What the Magnitude Means
The Fourier transform is generally complex-valued. Its magnitude
\[ \boxed{|\widehat f(\omega)|} \]measures the strength of the frequency \(\omega\). A large magnitude indicates that the corresponding oscillation contributes strongly to the function.
Graphs of \(|\widehat f(\omega)|\) are often called magnitude spectra or amplitude spectra.
What the Phase Means
A nonzero complex Fourier transform can be written in polar form as
\[ \widehat f(\omega) =|\widehat f(\omega)|e^{i\phi(\omega)}. \]The function \(\phi(\omega)\) is the phase spectrum. Magnitude describes how much of a frequency is present, while phase describes how that frequency component is positioned relative to the others.
Existence: A Practical Starting Condition
A convenient sufficient condition for the classical Fourier integral is absolute integrability:
\[ \boxed{ \int_{-\infty}^{\infty}|f(t)|\,dt<\infty }. \]Functions satisfying this condition belong to \(L^1(\mathbb R)\), and their Fourier transforms are well-defined as ordinary integrals.
Fourier analysis extends far beyond this setting. Square-integrable functions and generalized functions can also be transformed, but those extensions require a broader functional framework.
A Basic Example: The Rectangular Pulse
Consider
\[ f(t)= \begin{cases} 1, & |t|\le a,\\ 0, & |t|>a. \end{cases} \]Its Fourier transform is
\[ \widehat f(\omega) =\int_{-a}^{a}e^{-i\omega t}\,dt. \]For \(\omega\ne0\), integration gives
\[ \widehat f(\omega) =\frac{e^{-i\omega a}-e^{i\omega a}}{-i\omega} =\frac{2\sin(a\omega)}{\omega}. \]Thus,
\[ \boxed{ \widehat f(\omega) =\frac{2\sin(a\omega)}{\omega} }. \]The Value at Zero Frequency
The formula
\[ \frac{2\sin(a\omega)}{\omega} \]appears undefined at \(\omega=0\), but the limit exists:
\[ \lim_{\omega\to0} \frac{2\sin(a\omega)}{\omega} =2a. \]This agrees directly with the transform definition:
\[ \widehat f(0) =\int_{-a}^{a}1\,dt =2a. \]More generally,
\[ \boxed{ \widehat f(0) =\int_{-\infty}^{\infty}f(t)\,dt }. \]The Sinc Pattern
The rectangular pulse produces a transform proportional to a sinc-shaped function. With the unnormalized definition
\[ \operatorname{sinc}(x)=\frac{\sin x}{x}, \]the transform can be written
\[ \boxed{ \widehat f(\omega) =2a\operatorname{sinc}(a\omega) }. \]A sharply localized interval in the original domain therefore spreads across a broad range of frequencies.
Localization and Frequency Spread
The rectangular-pulse example illustrates a general principle: strong localization in one domain tends to require broad spread in the other.
If a pulse becomes narrower in time, its Fourier transform becomes wider in frequency. If a signal becomes more spread out in time, its frequency content can become more concentrated.
This time-frequency tradeoff is one of the recurring themes of transform analysis.
Linearity
The Fourier transform is linear. If
\[ h(t)=af(t)+bg(t), \]then
\[ \boxed{ \widehat h(\omega) =a\widehat f(\omega)+b\widehat g(\omega) }. \]This follows immediately from the linearity of integration and allows transform pairs to be combined efficiently.
Time Shifting
Suppose a function is shifted by \(t_0\):
\[ g(t)=f(t-t_0). \]Then
\[ \boxed{ \widehat g(\omega) =e^{-i\omega t_0}\widehat f(\omega) }. \]A translation in the original domain changes phase in the frequency domain but does not change the magnitude spectrum:
\[ |\widehat g(\omega)|=|\widehat f(\omega)|. \]Frequency Shifting
Multiplying a function by a complex exponential shifts its spectrum. If
\[ g(t)=e^{i\omega_0 t}f(t), \]then
\[ \boxed{ \widehat g(\omega) =\widehat f(\omega-\omega_0) }. \]This property underlies modulation: multiplication by an oscillation in time moves spectral content to a new frequency range.
Scaling
If
\[ g(t)=f(at), \qquad a\ne0, \]then
\[ \boxed{ \widehat g(\omega) =\frac{1}{|a|} \widehat f\left(\frac{\omega}{a}\right) }. \]Compressing a function in the original variable stretches its spectrum, while stretching the function compresses the spectrum.
Reflection
Replacing \(t\) with \(-t\) reflects the spectrum:
\[ \boxed{ f(-t) \ \longleftrightarrow\ \widehat f(-\omega) }. \]This property helps connect symmetry in the original variable with symmetry in the transform.
Even Functions
If \(f\) is real-valued and even, then
\[ f(-t)=f(t). \]The sine contribution cancels, so the transform becomes real and even:
\[ \boxed{ \widehat f(\omega) =2\int_0^{\infty}f(t)\cos(\omega t)\,dt }. \]Recognizing even symmetry can therefore cut the work in half.
Odd Functions
If \(f\) is real-valued and odd, then
\[ f(-t)=-f(t). \]The cosine contribution cancels, leaving a purely imaginary odd transform:
\[ \boxed{ \widehat f(\omega) =-2i\int_0^{\infty}f(t)\sin(\omega t)\,dt }. \]Conjugate Symmetry for Real Signals
If \(f(t)\) is real-valued, then its Fourier transform satisfies
\[ \boxed{ \widehat f(-\omega) =\overline{\widehat f(\omega)} }. \]Consequently,
\[ |\widehat f(-\omega)|=|\widehat f(\omega)|. \]Thus, the magnitude spectrum of a real signal is even, while its phase has the corresponding antisymmetric structure where defined.
Differentiation in the Original Domain
Under appropriate decay and smoothness conditions,
\[ \boxed{ \mathcal F\{f'(t)\}(\omega) =i\omega\widehat f(\omega) }. \]More generally,
\[ \boxed{ \mathcal F\{f^{(n)}(t)\}(\omega) =(i\omega)^n\widehat f(\omega) }. \]Differentiation in the original variable becomes multiplication by a power of frequency.
Why Differentiation Becomes Multiplication
Begin with
\[ \mathcal F\{f'\}(\omega) =\int_{-\infty}^{\infty}f'(t)e^{-i\omega t}\,dt. \]Integrating by parts gives
\[ \left[f(t)e^{-i\omega t}\right]_{-\infty}^{\infty} +i\omega\int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt. \]If the boundary term vanishes, then
\[ \mathcal F\{f'\} =i\omega\widehat f. \]This is one of the main reasons Fourier transforms simplify differential equations.
Multiplication by the Original Variable
Differentiating the transform with respect to frequency gives
\[ \frac{d}{d\omega}\widehat f(\omega) =\int_{-\infty}^{\infty} (-it)f(t)e^{-i\omega t}\,dt. \]Therefore,
\[ \boxed{ \mathcal F\{t f(t)\}(\omega) =i\frac{d}{d\omega}\widehat f(\omega) }. \]Multiplication by \(t\) in one domain corresponds to differentiation in the other.
Convolution
The convolution of two functions is defined by
\[ \boxed{ (f*g)(t) =\int_{-\infty}^{\infty} f(\tau)g(t-\tau)\,d\tau }. \]Convolution combines the values of two functions across all relative shifts. It appears naturally in averaging, smoothing, linear systems, probability, differential equations, and signal processing.
The Convolution Theorem
The Fourier transform turns convolution into multiplication:
\[ \boxed{ \mathcal F\{f*g\}(\omega) =\widehat f(\omega)\widehat g(\omega) }. \]Under the convention used here, multiplication in the original domain becomes scaled convolution in frequency:
\[ \boxed{ \mathcal F\{fg\}(\omega) =\frac{1}{2\pi} (\widehat f*\widehat g)(\omega) }. \]These two relationships are among the most useful structural identities in transform analysis.
Filtering as Multiplication in Frequency
Suppose an input signal \(x(t)\) passes through a linear time-invariant system with impulse response \(h(t)\). The output is
\[ y(t)=(h*x)(t). \]Taking Fourier transforms gives
\[ \boxed{ \widehat y(\omega) =\widehat h(\omega)\widehat x(\omega) }. \]The transform \(\widehat h(\omega)\) is the system's frequency response. It can amplify, attenuate, or phase-shift different frequency components.
Low-Pass and High-Pass Behavior
A low-pass filter preserves low frequencies and suppresses sufficiently high frequencies. A high-pass filter does the opposite.
In the frequency domain, filtering can therefore be visualized directly through multiplication by a transfer function.
Later lessons on filtering and signal analysis will develop these ideas more systematically.
Energy and Parseval's Identity
Fourier transforms preserve total squared magnitude up to the normalization factor imposed by the transform convention. For square-integrable functions,
\[ \boxed{ \int_{-\infty}^{\infty}|f(t)|^2\,dt =\frac{1}{2\pi} \int_{-\infty}^{\infty} |\widehat f(\omega)|^2\,d\omega }. \]This is the continuous analogue of Parseval's identity for Fourier series.
Plancherel's Theorem
Plancherel's theorem gives the rigorous \(L^2\) extension of the Fourier transform. In practical terms, it says that the transform acts as an energy-preserving change of representation, apart from the chosen normalization.
The theorem allows Fourier analysis to apply to many square-integrable functions even when the defining integral is not absolutely convergent in the ordinary \(L^1\) sense.
The Gaussian Transform
Gaussian functions have an especially important Fourier-transform property. For \(a>0\), consider
\[ f(t)=e^{-at^2}. \]Its transform is another Gaussian:
\[ \boxed{ \widehat f(\omega) =\sqrt{\frac{\pi}{a}} e^{-\omega^2/(4a)} }. \]The Gaussian therefore retains its basic shape when transformed.
Why the Gaussian Matters
Gaussian functions appear throughout probability, heat flow, statistics, optics, quantum mechanics, and signal processing. Their self-similar transform behavior makes them especially important in spectral analysis.
The formula also illustrates the scaling tradeoff: a narrow Gaussian in time or space has a wide Gaussian spectrum, while a broad Gaussian has a narrow spectrum.
Example: Transform of a Two-Sided Exponential
Let
\[ f(t)=e^{-a|t|}, \qquad a>0. \]This function is even, so
\[ \widehat f(\omega) =2\int_0^{\infty}e^{-at}\cos(\omega t)\,dt. \]Using the standard integral
\[ \int_0^{\infty}e^{-at}\cos(\omega t)\,dt =\frac{a}{a^2+\omega^2}, \]we obtain
\[ \boxed{ \widehat f(\omega) =\frac{2a}{a^2+\omega^2} }. \]Example: Using a Time Shift
Suppose
\[ f(t)\ \longleftrightarrow\ \widehat f(\omega). \]Define
\[ g(t)=f(t-3). \]Then the shift property gives immediately
\[ \boxed{ \widehat g(\omega) =e^{-3i\omega}\widehat f(\omega) }. \]No new integral is required.
Example: Using Frequency Shifting
If
\[ f(t)\ \longleftrightarrow\ \widehat f(\omega), \]then
\[ e^{i5t}f(t) \ \longleftrightarrow\ \boxed{\widehat f(\omega-5)}. \]Multiplying by \(e^{i5t}\) shifts the spectrum five units to the right on the angular-frequency axis.
Example: A Derivative Transform
Suppose
\[ f(t)\ \longleftrightarrow\ \widehat f(\omega) \]and the required decay assumptions hold. Then
\[ f''(t) \ \longleftrightarrow\ (i\omega)^2\widehat f(\omega). \]Since \(i^2=-1\),
\[ \boxed{ \mathcal F\{f''\}(\omega) =-\omega^2\widehat f(\omega) }. \]Transforming a Differential Equation
Consider the linear equation
\[ -u''(x)+a^2u(x)=f(x), \qquad a>0. \]Taking Fourier transforms gives
\[ -(-\omega^2)\widehat u(\omega) +a^2\widehat u(\omega) =\widehat f(\omega). \]Therefore,
\[ (\omega^2+a^2)\widehat u(\omega) =\widehat f(\omega), \]so
\[ \boxed{ \widehat u(\omega) =\frac{\widehat f(\omega)}{\omega^2+a^2} }. \]The differential equation has become an algebraic equation in frequency.
Why Transform Methods Help Differential Equations
Differential operators act simply on exponential modes. Since
\[ \frac{d}{dt}e^{i\omega t} =i\omega e^{i\omega t}, \]differentiation becomes multiplication when the function is decomposed into frequency components.
This is a spectral version of diagonalization: a complicated operator can become simple when expressed in a basis or continuum of modes adapted to that operator.
Fourier Transform and Spectral Thinking
In linear algebra, diagonalization replaces a matrix with its action on eigenvectors. In Fourier analysis, exponential functions serve as natural modes for translation-invariant differential operators.
The transform therefore changes the representation of a problem so that its essential components can be treated independently.
This viewpoint prepares the way for eigenfunction expansions and more general spectral decompositions later in this section.
The Inverse Transform as Reconstruction
The forward transform analyzes a function by frequency. The inverse transform synthesizes those frequencies back into the original function:
\[ f(t) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat f(\omega)e^{i\omega t}\,d\omega. \]The expression can be viewed as a continuous superposition of oscillatory modes. Each frequency contributes according to the complex weight \(\widehat f(\omega)\).
When Does Inversion Recover the Function?
Precise inversion theorems depend on the function space and regularity assumptions. Under common sufficient conditions, the inverse transform recovers \(f(t)\) at points of continuity.
At jump discontinuities, Fourier inversion can behave analogously to Fourier series, with midpoint values appearing under appropriate hypotheses.
The key lesson is that transform inversion is a theorem with hypotheses, not merely a formal algebraic reversal.
Riemann-Lebesgue Decay
If \(f\in L^1(\mathbb R)\), then
\[ \boxed{ \widehat f(\omega)\to0 \quad\text{as}\quad |\omega|\to\infty }. \]This is the Riemann-Lebesgue lemma. It says that the Fourier transform of an absolutely integrable function cannot retain a nonzero amplitude indefinitely at arbitrarily high frequencies.
Smoothness and Spectral Decay
Smoothness in the original domain is closely tied to decay in the frequency domain. Because derivatives correspond to multiplication by powers of \(\omega\), sufficiently smooth functions with suitable decay often have Fourier transforms that decrease rapidly at high frequency.
Rough features such as corners, sharp edges, and discontinuities usually require stronger high-frequency content.
Frequency Content and Resolution
Low frequencies describe slowly varying structure. High frequencies describe rapidly varying detail.
In an image, for example, broad smooth regions are associated mainly with lower spatial frequencies, while fine texture and sharp edges require higher spatial frequencies.
This same principle appears in sound, vibration, wave propagation, and numerical approximation.
Fourier Transforms in More Than One Dimension
For a function \(f(x)\) on \(\mathbb R^n\), the transform generalizes to
\[ \boxed{ \widehat f(\xi) =\int_{\mathbb R^n} f(x)e^{-i\xi\cdot x}\,dx }. \]Here \(\xi\) is a frequency vector and \(\xi\cdot x\) is the dot product.
Multidimensional transforms are fundamental in imaging, partial differential equations, wave physics, tomography, optics, and spatial data analysis.
Spatial Frequency
When the independent variable represents position rather than time, the transform variable represents spatial frequency. A large spatial frequency corresponds to rapid variation over short distances.
This interpretation makes Fourier methods useful for analyzing textures, edges, repeating spatial patterns, diffraction, and wave propagation.
A Continuous Transform Is Still an Infinite Expansion
Fourier series use a countable sum of orthogonal modes. Fourier transforms use a continuous superposition of modes. Conceptually,
\[ \sum_n c_n e^{i\omega_n t} \]becomes
\[ \frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat f(\omega)e^{i\omega t}\,d\omega. \]The transform is therefore an extension of orthogonal expansion from discrete coefficients to a continuous spectral density.
The Dirac Delta as an Idealized Frequency Marker
Some important signals, such as pure sinusoids or constants on the whole real line, are not absolutely integrable. Their Fourier transforms are handled using generalized functions such as the Dirac delta.
Formally, for example,
\[ e^{i\omega_0 t} \ \longleftrightarrow\ 2\pi\,\delta(\omega-\omega_0). \]This says that a pure complex sinusoid has all of its spectral content concentrated at one frequency.
Delta distributions are introduced here only as a useful spectral idealization; a rigorous distribution theory requires additional machinery.
Cosines Produce Two Symmetric Frequencies
Since
\[ \cos(\omega_0 t) =\frac{1}{2} \left(e^{i\omega_0 t}+e^{-i\omega_0 t}\right), \]its generalized Fourier transform contains two spectral lines:
\[ \boxed{ \mathcal F\{\cos(\omega_0 t)\} =\pi\left[ \delta(\omega-\omega_0) +\delta(\omega+\omega_0) \right] }. \]Positive and negative frequencies are both needed in the complex exponential representation of a real cosine.
Common Transform Pairs
Under the convention used in this lesson, several useful transform pairs are:
- \(e^{-at^2}\longleftrightarrow \sqrt{\pi/a}\,e^{-\omega^2/(4a)}\), for \(a>0\);
- \(e^{-a|t|}\longleftrightarrow 2a/(a^2+\omega^2)\), for \(a>0\);
- the rectangular pulse on \([-a,a]\) transforms to \(2\sin(a\omega)/\omega\);
- time shifts multiply the transform by a phase factor;
- derivatives multiply the transform by powers of \(i\omega\).
Transform tables become useful because properties can generate many new pairs from a small collection of known ones.
A Reliable Procedure for Direct Fourier Transforms
- Write the transform definition using one consistent convention.
- Check whether symmetry can simplify the integral.
- Identify intervals on which the function has a simpler formula.
- Integrate carefully, keeping the complex exponential intact when useful.
- Simplify the result using Euler identities when appropriate.
- Evaluate removable singularities by limits.
- Check symmetry properties of the result.
- Check the value at \(\omega=0\) against the total integral when applicable.
A Reliable Procedure for Using Transform Properties
- Identify a known transform pair.
- Rewrite the new function as shifts, scalings, modulations, derivatives, or combinations of the known function.
- Apply one transform property at a time.
- Track factors of \(a\), \(i\), \(\omega\), and \(2\pi\) carefully.
- Simplify only after the structural transformation is correct.
- Use symmetry or dimensional reasoning as a check when possible.
A Reliable Procedure for Transforming Differential Equations
- Take the Fourier transform of every term.
- Replace derivatives by powers of \(i\omega\).
- Solve the resulting algebraic equation for the transformed unknown.
- Factor or simplify the frequency-domain expression.
- Recognize a known inverse transform or convolution structure.
- Apply the inverse transform.
- Check that the reconstructed solution satisfies the original equation under the required assumptions.
Common Error: Mixing Transform Conventions
A frequent mistake is to use one definition for the forward transform and an inverse formula from a different convention.
For example, if
\[ \widehat f(\omega) =\int f(t)e^{-i\omega t}\,dt, \]then the corresponding inverse formula used here contains
\[ \frac{1}{2\pi}. \]The safest habit is to write the convention at the beginning of a calculation.
Common Error: Forgetting the Absolute Value in Scaling
If
\[ g(t)=f(at), \]the correct scaling rule is
\[ \boxed{ \widehat g(\omega) =\frac{1}{|a|} \widehat f\left(\frac{\omega}{a}\right) }. \]The factor is \(1/|a|\), not merely \(1/a\), because reversing orientation when \(a<0\) does not create a negative measure scale.
Common Error: Treating Magnitude as the Whole Transform
Two functions can have the same magnitude spectrum but different phase spectra. Therefore,
\[ |\widehat f(\omega)| \]generally does not contain enough information to reconstruct \(f\) uniquely.
Phase is an essential part of the Fourier representation.
Common Error: Assuming Every Function Has an Ordinary Transform Integral
Constants, pure sinusoids, and many other important functions fail the usual absolute-integrability condition on the whole real line.
These cases are not useless or forbidden. They are handled using broader notions such as \(L^2\) transforms or distributions. The important point is to know which framework is being used.
Common Error: Losing the Sign in Differentiation
Because
\[ \mathcal F\{f'\}=i\omega\widehat f, \]a second derivative gives
\[ (i\omega)^2\widehat f =-\omega^2\widehat f. \]The factor \(i^2=-1\) is essential in transformed differential equations.
Common Error: Confusing Fourier Series with Fourier Transforms
Fourier series are naturally associated with periodic functions and a discrete sequence of harmonic frequencies. Fourier transforms are naturally associated with nonperiodic functions on an unbounded domain and a continuous frequency variable.
The two methods are closely related, but they are not interchangeable without accounting for their different domains and spectra.
Checking a Fourier Transform
Several quick checks can catch errors:
- if \(f\) is real and even, \(\widehat f\) should be real and even;
- if \(f\) is real and odd, \(\widehat f\) should be imaginary and odd;
- \(\widehat f(0)\) should equal \(\int f(t)\,dt\) when that integral exists;
- translations should change phase but not transform magnitude;
- narrowing the function should broaden the spectrum;
- a derivative should introduce a factor of \(i\omega\).
Example: Check the Rectangular Pulse
For
\[ f(t)=1\quad\text{on }[-a,a] \]and zero elsewhere, we found
\[ \widehat f(\omega) =\frac{2\sin(a\omega)}{\omega}. \]The original pulse is real and even. The transform is also real and even because both \(\sin(a\omega)\) and \(\omega\) change sign when \(\omega\) changes sign.
Also,
\[ \widehat f(0)=2a, \]matching the area of the pulse.
Transform Methods as a Change of Representation
The Fourier transform does not change the underlying signal or function. It changes the coordinates used to describe it.
In the original domain, localization and pointwise variation are easy to see. In the frequency domain, oscillation, bandwidth, filtering, and differential action are often easier to see.
Good mathematical modeling frequently depends on choosing the representation in which the important structure becomes simplest.
Practice 1: Transform at Zero
Suppose
\[ f(t)= \begin{cases} 2, & -1\le t\le1,\\ 0, & \text{otherwise}. \end{cases} \]Find \(\widehat f(0)\) without computing the entire transform.
Practice 2: Use Even Symmetry
If \(f(t)=e^{-3|t|}\), write its Fourier transform as an integral over \([0,\infty)\) using symmetry.
Practice 3: Time Shift
If
\[ f(t)\ \longleftrightarrow\ \widehat f(\omega), \]find the transform of
\[ g(t)=f(t-4). \]Practice 4: Frequency Shift
If
\[ f(t)\ \longleftrightarrow\ \widehat f(\omega), \]find the transform of
\[ g(t)=e^{i7t}f(t). \]Practice 5: Scaling
If
\[ f(t)\ \longleftrightarrow\ \widehat f(\omega), \]find the transform of \(f(2t)\).
Practice 6: Second Derivative
Express
\[ \mathcal F\{f''(t)\} \]in terms of \(\widehat f(\omega)\).
Practice 7: Convolution
If
\[ h(t)=(f*g)(t), \]express \(\widehat h(\omega)\) in terms of \(\widehat f\) and \(\widehat g\).
Practice 8: Symmetry Check
A real-valued function \(f\) is even. A student obtains
\[ \widehat f(\omega)=\frac{1+i\omega}{1+\omega^2}. \]Explain why this result should be questioned immediately.
Practice 9: Transform a Differential Equation
Suppose
\[ -u''(x)+9u(x)=f(x). \]Assuming the Fourier-transform rules apply, solve algebraically for \(\widehat u(\omega)\) in terms of \(\widehat f(\omega)\).
Practice 10: Frequency Interpretation
In words, explain why a sharply localized pulse generally requires a broad range of Fourier frequencies.
Practice Answers
Practice 1
At zero frequency,
\[ \widehat f(0) =\int_{-\infty}^{\infty}f(t)\,dt. \]The pulse has height \(2\) and width \(2\), so
\[ \boxed{\widehat f(0)=4}. \]Practice 2
Since \(e^{-3|t|}\) is real and even,
\[ \boxed{ \widehat f(\omega) =2\int_0^{\infty}e^{-3t}\cos(\omega t)\,dt }. \]Practice 3
A shift by \(4\) multiplies the transform by a phase factor:
\[ \boxed{ \widehat g(\omega) =e^{-4i\omega}\widehat f(\omega) }. \]Practice 4
Multiplication by \(e^{i7t}\) shifts the spectrum:
\[ \boxed{ \widehat g(\omega) =\widehat f(\omega-7) }. \]Practice 5
Using the scaling rule with \(a=2\),
\[ \boxed{ \mathcal F\{f(2t)\}(\omega) =\frac{1}{2}\widehat f\left(\frac{\omega}{2}\right) }. \]Practice 6
\[ \boxed{ \mathcal F\{f''(t)\}(\omega) =-\omega^2\widehat f(\omega) }. \]Practice 7
Convolution becomes multiplication:
\[ \boxed{ \widehat h(\omega) =\widehat f(\omega)\widehat g(\omega) }. \]Practice 8
The Fourier transform of a real even function should be real and even. The term \(i\omega/(1+\omega^2)\) is imaginary and odd, so the proposed transform has the wrong symmetry structure.
Practice 9
Transforming the equation gives
\[ -(-\omega^2)\widehat u(\omega) +9\widehat u(\omega) =\widehat f(\omega). \]Therefore,
\[ (\omega^2+9)\widehat u(\omega) =\widehat f(\omega), \]so
\[ \boxed{ \widehat u(\omega) =\frac{\widehat f(\omega)}{\omega^2+9} }. \]Practice 10
A sharply localized pulse changes rapidly near its edges. Reconstructing that rapid variation requires high-frequency components in addition to lower frequencies. Narrower localization therefore tends to spread the transform over a broader frequency range.
What to Remember
The Fourier transform converts a function into a continuous frequency representation:
\[ \boxed{ \widehat f(\omega) =\int_{-\infty}^{\infty} f(t)e^{-i\omega t}\,dt }. \]Under the convention used here, reconstruction is given by
\[ \boxed{ f(t) =\frac{1}{2\pi} \int_{-\infty}^{\infty} \widehat f(\omega)e^{i\omega t}\,d\omega }. \]- Fourier series use discrete harmonics; Fourier transforms use a continuous spectrum.
- Transform magnitude measures frequency strength, while phase carries alignment information.
- time shifting becomes multiplication by a phase factor;
- frequency modulation becomes translation in frequency;
- scaling in one domain produces inverse scaling in the other;
- differentiation becomes multiplication by \(i\omega\);
- convolution becomes multiplication;
- Parseval and Plancherel connect energy in the two representations;
- smoothness and localization have direct spectral consequences.
The most important structural idea is that a transform is a change of representation. Problems that are complicated in time or space can become much simpler when expressed by frequency.
Where We Go Next
The Fourier transform developed here uses a continuous independent variable and a continuous frequency variable. Computers, sensors, experimental instruments, and stored datasets usually work with finitely many sampled values instead.
The next lesson, Discrete Fourier Transforms, converts a finite sequence of samples into a finite collection of frequency coefficients. It will connect continuous Fourier ideas to digital computation, frequency bins, sampling, aliasing, and the fast Fourier transform.