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. For a Schwartz function, periodized Fourier series with periods tending to infinity lead to its whole-line Fourier integral.

    Circle one:   True   False

  2. For f in L1 and frequency xi in cycles per unit, which integral defines the forward transform used in this lesson?

    • ∫_R f(x) exp(-2pi i xi x) dx
    • ∫_R f(x) exp(+2pi i xi x) dx
    • ∫ from 0 to 1 of f(x) exp(-2pi i xi x) dx

  3. Under the lesson's convention, what is the Fourier transform of the centered Gaussian exp(-pi a x^2), with a > 0?

    • A nonzero pure sine wave
    • Another centered Gaussian
    • An indicator of an interval

  4. The indicators of (-3,3) and [-3,3] have the same ordinary Fourier transform, because changing their two endpoint values does not change the defining integral.

    Circle one:   True   False

  5. Using the kernel exp(-2pi i xi x), what is the transform of f(x) = exp(-9pi x^2)?

    • 3 exp(-pi xi^2/9)
    • (1/3) exp(-pi xi^2/9)
    • (1/3) exp(-9pi xi^2)
    • 3 exp(-9pi xi^2)

  6. A Schwartz function is periodized with period L = 5. Its transform satisfies fhat(2/5) = 15. What is the complex Fourier coefficient c_2 of the periodized function?

    • 75
    • 15
    • 1/3
    • 3

  7. Assume f is continuously differentiable, f and f' are in L1, and f tends to zero at both infinities. With kernel exp(-2pi i xi x), what is F(f')(xi)?

    • 2pi i xi fhat(xi)
    • -2pi i xi fhat(xi)
    • (2pi i xi)^2 fhat(xi)

  8. Let f be in L1 and have bounded variation near x = 0. Its left limit is 2 and its right limit is 8. What value does symmetric truncated Fourier inversion approach at zero?

    • 5
    • 2
    • 8
    • 10

  9. The ordinary Fourier integrals in this lesson define a transform and recover every function on the line, without any hypotheses.

    Circle one:   True   False

  10. Let f be the indicator of [2,4]. With forward kernel exp(-2pi i xi x), which formula gives fhat(xi) for xi != 0?

    • exp(-2pi i xi) sin(6pi xi)/(pi xi)
    • exp(+6pi i xi) sin(2pi xi)/(pi xi)
    • exp(-6pi i xi) sin(2pi xi)/(pi xi)
    • exp(-6pi i xi) sin(4pi xi)/(pi xi)
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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 · The series samples frequencies n/L with weight 1/L. For Schwartz functions, this weighted sum approaches the inverse Fourier integral.
  2. ∫_R f(x) exp(-2pi i xi x) dx · The stated forward convention integrates over the whole real line and uses a negative exponential sign.
  3. Another centered Gaussian · Its transform is a^(-1/2) exp(-pi xi^2/a), another centered Gaussian.
  4. True · The functions differ only at two points, a null set. Their integrals against the Fourier kernel are equal.
  5. (1/3) exp(-pi xi^2/9) · Set a = 9 in a^(-1/2) exp(-pi xi^2/a). The amplitude is 1/3 and the exponent is -pi xi^2/9.
  6. 3 · Use c_n = fhat(n/L)/L. Thus c_2 = 15/5 = 3.
  7. 2pi i xi fhat(xi) · Integration by parts gives a zero boundary term. Subtracting the kernel derivative leaves the positive factor 2pi i xi.
  8. 5 · The theorem returns the midpoint: (2+8)/2 = 5.
  9. False · For f(x) = 1, the forward integral at zero frequency is ∫_R 1 dx, which diverges.
  10. exp(-6pi i xi) sin(2pi xi)/(pi xi) · The center is 3 and half-width is 1. Multiply the phase exp(-6pi i xi) by sin(2pi xi)/(pi xi).
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