The Fourier Transform on the Line · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

From repeating tones to a continuum

Mathematics · Calculus & Analysis · ages 22-23
Name ______________________   Date ____________

  1. The Fourier transform arises when the period of a Fourier series tends to infinity.

    Circle one:   True   False

  2. A Fourier series on an interval of length L has discrete frequencies spaced 1 over L apart. What happens as L grows without bound?

    • The frequencies spread further apart
    • The frequencies stay exactly fixed
    • The frequencies pack closer and the sum approaches an integral

  3. The Fourier transform of a Gaussian bell curve is:

    • A pure sine wave
    • Another Gaussian
    • Zero at every frequency

  4. Zoe says the Fourier transform arises when the period of a Fourier series tends to infinity. Is Zoe right?

    Circle one:   True   False

  5. The Fourier transform of a Gaussian bell curve is:

    • Another Gaussian
    • An indicator function of an interval
    • A pure sine wave
    • Zero at every frequency

  6. A Fourier series on an interval of length L has discrete frequencies spaced 1/L apart. What happens as L grows without bound?

    • The frequencies pack closer together and the sum approaches an integral over all frequencies
    • The frequencies spread further apart
    • The frequencies stay exactly fixed
    • Only whole number frequencies survive

  7. Why do differential equations love the Fourier transform?

    • Differentiation becomes multiplication by frequency, turning calculus into algebra
    • It removes all constants from the equation
    • It makes every function smooth

  8. What does the Fourier inversion theorem say, roughly?

    • Applying the transform twice, with matching normalisation, recovers the original function
    • Every function equals its own Fourier transform
    • The transform of any signal is zero
    • Fourier series of smooth functions never converge

  9. Fourier inversion holds for every function with no hypotheses at all.

    Circle one:   True   False

  10. The Fourier transform of the indicator function of an interval is a multiple of:

    • sin(x)/x, a sinc function
    • e to the x
    • A step function
    • A polynomial in x
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Answer key

For grown-ups. Fold this page away before handing over the rest.

From repeating tones to a continuum W1-mt_BMrt5FPe3U-s1

  1. True · Stretching the interval packs the tones until the sum becomes an integral.
  2. The frequencies pack closer and the sum approaches an integral · Spacing 1 over L shrinks as L grows, so the ladder becomes a continuum.
  3. Another Gaussian · Gaussians map to Gaussians, trading width between the two domains.
  4. True · Zoe is right. Letting the interval length grow without bound turns the series into the transform integral.
  5. Another Gaussian · The Gaussian is the eigenfunction of the transform: its transform is another Gaussian.
  6. The frequencies pack closer together and the sum approaches an integral over all frequencies · Wider intervals give finer frequency spacing, and in the limit the sum over discrete tones becomes the Fourier integral.
  7. Differentiation becomes multiplication by frequency, turning calculus into algebra · Derivatives turn into algebra, which is far easier to solve before inverting back.
  8. Applying the transform twice, with matching normalisation, recovers the original function · Inversion says the transform is reversible: the inverse transform applied to the transform returns the starting function.
  9. False · Inversion needs decay and smoothness, so wild functions can break it.
  10. sin(x)/x, a sinc function · Integrating the exponential over one finite interval gives a sine divided by the frequency, the sinc shape.
Worksheet · LightMySky