Comment. Convolutions quiz
Comment. This quiz is designed to test your knowledge of convolutions of 2pi - periodic functions.
Comment. Note: in this entire quiz, the expression f*g denotes convolution of f and g, while the expression fg denotes the pointwise product of f and g.
Comment. The expression hat(f) denotes the Fourier transform of f, thus hat(f)(n) is the n^th Fourier coefficient of f.
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Question. Let f and g be continuously differentiable 2pi-periodic functions. The derivative (f*g)' of the convolution f*g is given by
Correct Answer. (f')*g
Comment. This is one of two correct answers.
Answer. f*(g') + (f')*g
Comment. You're thinking of the product rule: (fg)' = f' g + f g'. Convolutions behave differently from products.
Answer. (f')*(g')
Comment. You're thinking of the sum rule: (f+g)' = f' + g'. Convolutions behave differently from sums.
Answer. (g')*(f')
Comment. You might be thinking of Abel's formula for matrices: (AB)^{-1} = B^{-1} A^{-1}. Convolutions behave differently from inverses.
Correct Answer. f*(g')
Comment. This is one of two correct answers.
Answer. In general, there is no simple formula available.
Question. Let f and g be continuously differentiable 2pi-periodic functions, and let n be an integer. The n^th Fourier coefficient hat(f*g)(n) of the convolution f*g is given by
Answer. hat(f) * g(n)
Correct Answer. hat(f)(n) hat(g)(n)
Answer. hat(f) * hat(g)(n)
Answer. hat(f)(n) + hat(g)(n)
Answer. hat(f)(n) g + f hat(g)(n)
Answer. In general, there is no simple formula available.
Question. Let f and g be continuously differentiable 2pi-periodic functions. The average value of f*g is equal to
Answer. The difference between the average value of f and the average value of g.
Answer. The average of the average value of f and the average value of g.
Partially Correct Answer. The convolution of the average value of f and the average value of g.
Comment. This is true if one thinks of the average values of f and g as constant functions rather than numbers, but this is a rather clumsy way to phrase the answer.
Correct Answer. The product of the average value of f and the average value of g.
Answer. The sum of the average value of f and the average value of g.
Answer. In general, there is no simple formula available.
Question. Let f, g, h be continuous 2pi-periodic functions. The expression f*(g+h) can also be written as
Answer. (f+g)*h
Answer. f*h + g*h
Correct Answer. g*f + f*h
Answer. f*(g*h)
Answer. g*(f+h)
Answer. None of the above.
Question. Let f, g, h be continuous 2pi-periodic functions. The expression (f+3h)*(2g) can also be written as
Correct Answer. 2(f*g) + 6(h*g)
Answer. 6*f*g*h
Answer. 2*f*g + 3*h*g
Answer. (2f)*g + (3h)*g
Answer. 6*h*g + 2*f*g
Answer. None of the above.
Question. Let f, g, h be continuous 2pi-periodic functions. The expression f*(gh) can also be written as
Answer. (fg)*h
Answer. (f*g)(f*h)
Answer. f*g + f*h
Answer. f(g+h)
Answer. f(g*h)
Correct Answer. None of the above.
Comment. In general, there is no useful formula for pulling a product out of a convolution (or a convolution out of a product).
Question. Let f, g be 2pi-periodic functions. If f is continuously differentiable, and g is twice continuously differentiable, then the best we can say about f*g is that it is 2pi-periodic and
Answer. Riemann integrable.
Answer. Piecewise continuous.
Answer. Continuous.
Answer. Continuously differentiable.
Answer. Twice continuously differentiable.
Correct Answer. Three times continuously differentiable.
Comment. Convolving two functions combines their orders of smoothness together.
Answer. Infinitely differentiable.
Question. Let f, g be 2pi-periodic functions. If f is continuously differentiable, and g is twice continuously differentiable, then the best we can say about f+g is that it is 2pi-periodic and
Answer. Riemann integrable.
Answer. Piecewise continuous.
Answer. Continuous.
Correct Answer. Continously differentiable.
Comment. In general, the sum of two functions is only as smooth as the rougher of its two factors.
Answer. Twice continuously differentiable.
Answer. Three times continuously differentiable.
Answer. Infinitely differentiable.
Question. Let f, g be 2pi-periodic functions. If f is continuously differentiable, and g is twice continuously differentiable, then the best we can say about fg is that it is 2pi-periodic and
Answer. Riemann integrable.
Answer. Piecewise continuous.
Answer. Continuous.
Correct Answer. Continously differentiable.
Comment. In general, the product of two functions is only as smooth as the rougher of its two factors.
Answer. Twice continuously differentiable.
Answer. Three times continuously differentiable.
Answer. Infinitely differentiable.
Question. Let f, g be 2pi-periodic functions. If f and g are Riemann integrable, then the best we can say about f*g is that it is 2pi-periodic and
Answer. Bounded.
Partially Correct Answer. Riemann integrable.
Comment. While this true, more can be said.
Answer. Piecewise continuous.
Correct Answer. Continuous.
Answer. Continously differentiable.
Answer. Twice continuously differentiable.
Answer. Infinitely differentiable.
Question. Let f, g be 2pi-periodic functions. If f and g are Riemann integrable, then the best we can say about fg is that it is 2pi-periodic and
Answer. Bounded.
Comment. While this true, more can be said.
Correct Answer. Riemann integrable.
Answer. Piecewise continuous.
Answer. Continuous.
Answer. Continously differentiable.
Answer. Twice continuously differentiable.
Answer. Infinitely differentiable.
Shuffle Answers.
Question. Let f be a 2pi-periodic function, and let 1 be the constant function 1. Then f*1 is
Answer. The same function as f.
Answer. The constant function 1.
Correct Answer. The constant function with value equal to the mean of f.
Answer. The constant function with value equal to f(1).
Answer. The value of f(x) at the point x=0.
Answer. 0.
Question. Let f be a continuous 2pi-periodic function, and let K_n be a family of approximations to the identity (a.k.a. good kernels). Which of the following statements is true?
Correct Answer. For each x, f*K_n(x) converges to f(x) as n goes to infinity.
Answer. For each n, f*K_n(x) converges to f(x) as x goes to infinity.
Answer. For each x, f*K_n(x) converges to 1 as n goes to infinity.
Answer. For each x and each n, we have f*K_n(x) = f(x).
Answer. For each x, K_n(x) converges to f(x) as n goes to infinity.
Answer. The functions f*K_n converge to zero as n goes to infinity.