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Quiz: Convolutions

Page history last edited by RH 11 years, 7 months ago

This quiz is designed to test your knowledge of convolutions of Formula-periodic functions.

 

In this entire quiz, the expression Formula denotes convolution of Formula and Formula, while the expression Formula denotes the pointwise product of Formula and Formula. The expression Formula denotes the Fourier transform of Formula, thus Formula is the Formula Fourier coefficient of Formula.

 

Discuss this quiz 

(Key: correct, incorrect, partially correct.)

 

  1. Let Formula and Formula be continuously differentiable Formula-periodic functions. The derivative Formula of the convolution Formula is given by
    1. Formula
      • CORRECT. This is one of two correct answers.
    2. Formula
      • INCORRECT. You're thinking of the product rule: Formula. Convolutions behave differently from products.
    3. Formula
      • INCORRECT. You're thinking of the sum rule: Formula. Convolutions behave differently from sums.
    4. Formula
      • INCORRECT. You might be thinking of Abel's formula for matrices: Formula. Convolutions behave differently from inverses.
    5. Formula
      • CORRECT. This is one of two correct answers.
    6. In general, there is no simple formula available.
  2. Let Formula and Formula be continuously differentiable Formula-periodic functions, and let Formula be an integer. The Formula Fourier coefficient Formula of the convolution Formula is given by
    1. Formula
    2. Formula
      • CORRECT.
    3. Formula
    4. Formula
    5. Formula
    6. In general, there is no simple formula available.
  3. Let Formula and Formula be continuously differentiable Formula-periodic functions. The average value of Formula is equal to
    1. The difference between the average value of Formula and the average value of Formula.
    2. The average of the average value of Formula and the average value of Formula.
    3. The convolution of the average value of Formula and the average value of Formula.
      • PARTIALLY. This is true if one thinks of the average values of Formula and Formula as constant functions rather than numbers, but this is a rather clumsy way to phrase the answer.
    4. The product of the average value of Formula and the average value of Formula.
      • CORRECT.
    5. The sum of the average value of Formula and the average value of Formula.
    6. In general, there is no simple formula available.
  4. Let Formula, Formula, Formula be continuous Formula-periodic functions. The expression Formula can also be written as
    1. Formula
    2. Formula
    3. Formula
      • CORRECT.
    4. Formula
    5. Formula
    6. None of the above.
  5. Let Formula, Formula, Formula be continuous Formula-periodic functions. The expression Formula can also be written as
    1. Formula
      • CORRECT.
    2. Formula
    3. Formula
    4. Formula
    5. Formula
    6. None of the above.
  6. Let Formula, Formula, Formula be continuous Formula-periodic functions. The expression Formula can also be written as
    1. Formula
    2. Formula
    3. Formula
    4. Formula
    5. Formula
    6. None of the above.
      • CORRECT. In general, there is no useful formula for pulling a product out of a convolution (or a convolution out of a product).
  7. Let Formula, Formula be Formula-periodic functions. If Formula is continuously differentiable, and Formula is twice continuously differentiable, then the best we can say about Formula is that it is Formula-periodic and
    1. Riemann integrable.
    2. Piecewise continuous.
    3. Continuous.
    4. Continuously differentiable.
    5. Twice continuously differentiable.
    6. Three times continuously differentiable.
      • CORRECT. Convolving two functions combines their orders of smoothness together.
    7. Infinitely differentiable.
  8. Let Formula, Formula be Formula-periodic functions. If Formula is continuously differentiable, and Formula is twice continuously differentiable, then the best we can say about Formula is that it is Formula-periodic and
    1. Riemann integrable.
    2. Piecewise continuous.
    3. Continuous.
    4. Continously differentiable.
      • CORRECT. In general, the sum of two functions is only as smooth as the rougher of its two factors.
    5. Twice continuously differentiable.
    6. Three times continuously differentiable.
    7. Infinitely differentiable.
  9. Let Formula, Formula be Formula-periodic functions. If Formula is continuously differentiable, and Formula is twice continuously differentiable, then the best we can say about Formula is that it is Formula-periodic and
    1. Riemann integrable.
    2. Piecewise continuous.
    3. Continuous.
    4. Continously differentiable.
      • CORRECT. In general, the product of two functions is only as smooth as the rougher of its two factors.
    5. Twice continuously differentiable.
    6. Three times continuously differentiable.
    7. Infinitely differentiable.
  10. Let Formula, Formula be Formula-periodic functions. If Formula and Formula are Riemann integrable, then the best we can say about Formula is that it is Formula-periodic and
    1. Bounded.
    2. Riemann integrable.
      • PARTIALLY. While this true, more can be said.
    3. Piecewise continuous.
    4. Continuous.
      • CORRECT.
    5. Continously differentiable.
    6. Twice continuously differentiable.
    7. Infinitely differentiable.
  11. Let Formula, Formula be Formula-periodic functions. If Formula and Formula are Riemann integrable, then the best we can say about Formula is that it is Formula-periodic and
    1. Bounded.
      • INCORRECT. While this true, more can be said.
    2. Riemann integrable.
      • CORRECT.
    3. Piecewise continuous.
    4. Continuous.
    5. Continously differentiable.
    6. Twice continuously differentiable.
    7. Infinitely differentiable.
  12. Let Formula be a Formula-periodic function, and let Formula be the constant function Formula. Then Formula is
    1. The same function as Formula.
    2. The constant function Formula.
    3. The constant function with value equal to the mean of Formula.
      • CORRECT.
    4. The constant function with value equal to Formula.
    5. The value of Formula at the point Formula.
    6. Formula.
  13. Let Formula be a continuous Formula-periodic function, and let Formula be a family of approximations to the identity (a.k.a. good kernels). Which of the following statements is true?
    1. For each Formula, Formula converges to Formula as Formula goes to infinity.
      • CORRECT.
    2. For each Formula, Formula converges to Formula as Formula goes to infinity.
    3. For each Formula, Formula converges to Formula as Formula goes to infinity.
    4. For each Formula and each Formula, we have Formula.
    5. For each Formula, Formula converges to Formula as Formula goes to infinity.
    6. The functions Formula converge to zero as Formula goes to infinity.

 

 

Score:  

Comments (1)

Mohammad said

at 1:40 am on Dec 29, 2010

Have anyone checked the "PDF version"?

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