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# Copy of Quiz: Convolutions

last edited by 9 years, 11 months ago

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

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

(Key: correct, incorrect, partially correct.)

1. Let and be continuously differentiable -periodic functions. The derivative of the convolution is given by
• CORRECT. This is one of two correct answers.
• INCORRECT. You're thinking of the product rule: . Convolutions behave differently from products.
• INCORRECT. You're thinking of the sum rule: . Convolutions behave differently from sums.
• INCORRECT. You might be thinking of Abel's formula for matrices: . Convolutions behave differently from inverses.
• CORRECT. This is one of two correct answers.
1. In general, there is no simple formula available.
2. Let and be continuously differentiable -periodic functions, and let be an integer. The Fourier coefficient of the convolution is given by
• CORRECT.
1. In general, there is no simple formula available.
3. Let and be continuously differentiable -periodic functions. The average value of is equal to
1. The difference between the average value of and the average value of .
2. The average of the average value of and the average value of .
3. The convolution of the average value of and the average value of .
• PARTIALLY. This is true if one thinks of the average values of and 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 and the average value of .
• CORRECT.
5. The sum of the average value of and the average value of .
6. In general, there is no simple formula available.
4. Let , , be continuous -periodic functions. The expression can also be written as
• CORRECT.
1. None of the above.
5. Let , , be continuous -periodic functions. The expression can also be written as
• CORRECT.
1. None of the above.
6. Let , , be continuous -periodic functions. The expression can also be written as
1. 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 , be -periodic functions. If is continuously differentiable, and is twice continuously differentiable, then the best we can say about is that it is -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 , be -periodic functions. If is continuously differentiable, and is twice continuously differentiable, then the best we can say about is that it is -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 , be -periodic functions. If is continuously differentiable, and is twice continuously differentiable, then the best we can say about is that it is -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 , be -periodic functions. If and are Riemann integrable, then the best we can say about is that it is -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 , be -periodic functions. If and are Riemann integrable, then the best we can say about is that it is -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 be a -periodic function, and let be the constant function . Then is
1. The same function as .
2. The constant function .
3. The constant function with value equal to the mean of .
• CORRECT.
4. The constant function with value equal to .
5. The value of at the point .
6. .
13. Let be a continuous -periodic function, and let be a family of approximations to the identity (a.k.a. good kernels). Which of the following statements is true?
1. For each , converges to as goes to infinity.
• CORRECT.
2. For each , converges to as goes to infinity.
3. For each , converges to as goes to infinity.
4. For each and each , we have .
5. For each , converges to as goes to infinity.
6. The functions converge to zero as goes to infinity.