Convolution, Plancherel and Transforming a Derivative · seed 1 · A4, ink-friendly. The answer key prints on its own page for grown-ups.

Three moves that make the transform useful

Mathematics · Calculus & Analysis · ages 23-24
Name ______________________   Date ____________

  1. Mina says Plancherel means the transform is unitary: it preserves inner products and distances. Is Mina right?

    Circle one:   True   False

  2. What does the convolution theorem state?

    • The transform of a convolution is the product of the two transforms
    • The transform of a convolution is the sum of the two transforms
    • Convolution becomes addition under the transform
    • The transform of a convolution is another convolution

  3. The transform of a convolution of f and g equals what?

    • The product of the two transforms
    • The sum of the two transforms
    • Another convolution of the transforms

  4. The transform of f' equals i times the frequency times the transform of f.

    Circle one:   True   False

  5. Why do we call the transform unitary?

    • It makes every function smooth
    • It removes all negative frequencies
    • It preserves lengths and angles like a rotation

  6. With zero initial data, what does the Laplace transform turn y double prime + 3 y prime + 2 y = f into?

    • (s^2 + 3s + 2) Y(s) = F(s), so divide by the polynomial and invert
    • The second derivative of F(s)
    • s^2 Y(s) = F(s), ignoring the lower terms
    • The sum of the three separate solutions

  7. The integral over the whole line of a convolution f * g looks hard. What is the cheap route via the convolution theorem?

    • Transform each factor, multiply, and evaluate the product at zero
    • Differentiate under the integral sign twice
    • Expand the convolution as a power series
    • Square the integrand first

  8. An integral over the whole line of a convolution looks hard. What is the cheap route?

    • Differentiate under the integral sign
    • Transform both parts and multiply, then read off the value
    • Split the integral into a thousand slices

  9. Two students solve one equation. One divides by a multiplier that is zero at one frequency and keeps going. Who is right?

    • The one who kept going, since one point never matters
    • The one who stopped, since division by zero is not allowed
    • Both, since the transform forgives zero division

  10. The transform of f is the constant 5 and the frequency is 2. The second derivative multiplies by (i times frequency) squared. Type the transform of the second derivative.

    Answer: ______________

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Answer key

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

Three moves that make the transform useful W1-mt_zSwMJrQIMt-s1

  1. True · Preserving norms gives preservation of inner products by polarisation, hence distances too.
  2. The transform of a convolution is the product of the two transforms · Convolution in the original domain becomes plain multiplication in the transform domain.
  3. The product of the two transforms · Convolution in the original domain becomes plain multiplication.
  4. True · That multiplier is exactly what differentiating becomes.
  5. It preserves lengths and angles like a rotation · Plancherel keeps sizes fixed, the way rotations do.
  6. (s^2 + 3s + 2) Y(s) = F(s), so divide by the polynomial and invert · Each derivative becomes multiplication by s, giving an algebraic equation for Y(s).
  7. Transform each factor, multiply, and evaluate the product at zero · Integrating a function is evaluating its transform at zero, so the integral of f * g is F(0) times G(0).
  8. Transform both parts and multiply, then read off the value · The convolution theorem trades the hard integral for a product.
  9. The one who stopped, since division by zero is not allowed · A zero multiplier blocks division, so stopping is correct.
  10. -20 · Squaring 2i gives -4, and -4 times 5 is -20.
Worksheet · LightMySky