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Copy of Quiz: Convolutions
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last edited
by Mohammad 13 years, 3 months ago
- Let and be continuously differentiable -periodic functions. The derivative of the convolution is given by
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- CORRECT. This is one of two correct answers.
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- INCORRECT. You're thinking of the product rule: . Convolutions behave differently from products.
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- INCORRECT. You're thinking of the sum rule: . Convolutions behave differently from sums.
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- INCORRECT. You might be thinking of Abel's formula for matrices: . Convolutions behave differently from inverses.
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- CORRECT. This is one of two correct answers.
- In general, there is no simple formula available.
- Let and be continuously differentiable -periodic functions, and let be an integer. The Fourier coefficient of the convolution is given by
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- In general, there is no simple formula available.
- Let and be continuously differentiable -periodic functions. The average value of is equal to
- The difference between the average value of and the average value of .
- The average of the average value of and the average value of .
- 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.
- The product of the average value of and the average value of .
- The sum of the average value of and the average value of .
- In general, there is no simple formula available.
- Let , , be continuous -periodic functions. The expression can also be written as
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- None of the above.
- Let , , be continuous -periodic functions. The expression can also be written as
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- None of the above.
- Let , , be continuous -periodic functions. The expression can also be written as
- 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).
- 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
- Riemann integrable.
- Piecewise continuous.
- Continuous.
- Continuously differentiable.
- Twice continuously differentiable.
- Three times continuously differentiable.
- CORRECT. Convolving two functions combines their orders of smoothness together.
- Infinitely differentiable.
- 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
- Riemann integrable.
- Piecewise continuous.
- Continuous.
- Continously differentiable.
- CORRECT. In general, the sum of two functions is only as smooth as the rougher of its two factors.
- Twice continuously differentiable.
- Three times continuously differentiable.
- Infinitely differentiable.
- 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
- Riemann integrable.
- Piecewise continuous.
- Continuous.
- Continously differentiable.
- CORRECT. In general, the product of two functions is only as smooth as the rougher of its two factors.
- Twice continuously differentiable.
- Three times continuously differentiable.
- Infinitely differentiable.
- Let , be -periodic functions. If and are Riemann integrable, then the best we can say about is that it is -periodic and
- Bounded.
- Riemann integrable.
- PARTIALLY. While this true, more can be said.
- Piecewise continuous.
- Continuous.
- Continously differentiable.
- Twice continuously differentiable.
- Infinitely differentiable.
- Let , be -periodic functions. If and are Riemann integrable, then the best we can say about is that it is -periodic and
- Bounded.
- INCORRECT. While this true, more can be said.
- Riemann integrable.
- Piecewise continuous.
- Continuous.
- Continously differentiable.
- Twice continuously differentiable.
- Infinitely differentiable.
- Let be a -periodic function, and let be the constant function . Then is
- The same function as .
- The constant function .
- The constant function with value equal to the mean of .
- The constant function with value equal to .
- The value of at the point .
- .
- 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?
- For each , converges to as goes to infinity.
- For each , converges to as goes to infinity.
- For each , converges to as goes to infinity.
- For each and each , we have .
- For each , converges to as goes to infinity.
- The functions converge to zero as goes to infinity.
Score:
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Copy of Quiz: Convolutions
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