Comment. Symmetries of the Fourier transform quiz
Comment. This quiz is designed to test your knowledge of how the Fourier transform behaves under
Comment. various transformations. The quiz is phrased so that it does not matter what your normalization
Comment. conventions are for the Fourier transform. Note: some of these questions lie outside the scope of Math 133.
Comment. All abelian groups are assumed to be amenable (if you don't know what this means, ignore it).
Shuffle Questions.
Shuffle Answers.
Question. If f is a function on an abelian group G, and f is translated by a shift h, then the Fourier transform of f is
Answer. Shifted in the direction h.
Answer. Shifted in the direction -h.
Correct Answer. Modulated by a character with frequency -h or h (depending on conventions)
Answer. Dilated by h.
Comment. This couldn't be right (consider the h=0 case).
Answer. Dilated by 1/h.
Comment. This couldn't be right (consider the h=0 case).
Answer. Convolved by h.
Comment. This doesn't even make sense; h is a group element, not a function.
Answer. Multiplied by h.
Comment. This couldn't be right (consider the h=0 case).
Question. If f is a function on an abelian group G, and f is modulated by a character with frequency xi, then the Fourier transform of f is
Answer. Modulated by the character with frequency xi.
Answer. Modulated by the character conjugate to xi.
Correct Answer. Shifted in the direction xi or -xi (depending on conventions)
Answer. Dilated by xi.
Answer. Dilated by 1/xi.
Answer. Convolved by xi.
Answer. Multiplied by xi.
Question. If f is a function on R^n, and f is dilated by a factor lambda (i.e. f(x) is replaced by f(x/lambda)), then the Fourier transform of f is
Answer. Dilated by a factor lambda.
Answer. Dilated by a factor 1/lambda.
Correct Answer. Dilated by a factor 1/lambda, and multiplied by lambda^n.
Answer. Dilated by a factor 1/lambda, and multiplied by lambda^{-n}.
Answer. Dilated by a factor lambda, and multiplied by lambda^n.
Answer. Dilated by a factor lambda, and multiplied by lambda^{-n}.
Answer. Multiplied by a factor lambda^n.
Question. If f is a function on R^n, and f is rotated by an orthogonal matrix O, then the Fourier transform of f is
Correct Answer. Rotated by O.
Answer. Rotated by O^{-1}.
Answer. Rotated by the transpose O^t.
Answer. Rotated by O, and then conjugated.
Correct Answer. Rotated by the inverse of the transpose of O.
Comment. For orthogonal matrices, the inverse of the transpose O is the matrix O itself.
Answer. Rotated by O, and then reflected around the origin.
Correct Answer. Rotated by the inverse of the transpose of O, and multiplied by |det O|.
Comment. For orthogonal matrices, inverse transpose of O is O itself, and |det O| = 1.
Question. If f is a function on R^n, and f is composed with an invertible linear transformation L (thus f is replaced by f o L), then the Fourier transform of f is
Answer. Composed with L
Answer. Composed with the inverse of L, and multiplied by det L
Answer. Composed with the transpose of L
Correct Answer. Composed with the inverse transpose of L, and multiplied by 1/|det L|
Answer. Composed with the inverse transpose of L, and multiplied by |det L|
Answer. Composed with the transpose of L, and multiplied by 1/det L
Answer. Composed with L, and multiplied by |det L|
Question. If f is a function on an abelian group G, and f is restricted to a subgroup H of G, then the Fourier transform of f is (after identifying the dual group of H in a canonical manner)
Correct Answer. Projected from G^* to G^* / H^perp.
Answer. Restricted from G^* to H^perp.
Answer. Convolved with the indicator function of H^perp.
Answer. Convolved with the normalized indicator function on H^perp.
Answer. Averaged over cosets of H^perp.
Answer. Divided by the index of H in G.
Answer. Divided by the cardinality of H.
Question. If f is a function on an abelian group G, and f is projected onto a quotient G/H of G, then the Fourier transform of f is (after identifying the dual group of G/H in a canonical manner)
Answer. Projected from G^* to H^perp.
Correct Answer. Restricted from G^* to H^perp.
Answer. Convolved with the indicator function of H^perp.
Answer. Convolved with the normalized indicator function on H^perp.
Answer. Averaged over cosets of H^perp.
Answer. Divided by the index of H in G.
Answer. Divided by the cardinality of H.
Question. If f is a function on R^n, and f is complex conjugated, then the Fourier transform of f is
Answer. Complex conjugated.
Correct Answer. Complex conjugated, and reflected around the origin.
Answer. Reflected around the origin.
Answer. Multiplied by i.
Answer. Reflected around the x-axis.
Answer. Reflected around both the x-axis and the origin.
Answer. Complex conjugated, and multiplied by -1.
Question. If f is a function on R^n, and f is reflected around the origin, then the Fourier transform of f is
Answer. Complex conjugated.
Answer. Complex conjugated, and reflected around the origin.
Correct Answer. Reflected around the origin.
Answer. Multiplied by i.
Answer. Reflected around the x-axis.
Answer. Reflected around both the x-axis and the origin.
Answer. Complex conjugated, and multiplied by -1.
Question. If f is a function on R^n, and f is both complex conjugated and reflected around the origin, then the Fourier transform of f is
Correct Answer. Complex conjugated.
Answer. Complex conjugated, and reflected around the origin.
Answer. Reflected around the origin.
Answer. Multiplied by i.
Answer. Reflected around the x-axis.
Answer. Reflected around both the x-axis and the origin.
Answer. Complex conjugated, and multiplied by -1.
Question. If f is a function on R, and f is differentiated, then the Fourier transform of f is
Correct Answer. Multiplied by some multiple of xi.
Answer. Differentiated.
Answer. Integrated.
Answer. Reflected around the origin.
Answer. Divided by some multiple of xi.
Answer. Multiplied by a constant.
Question. If f is a function on R, and f is multiplied by the identity function x, then the Fourier transform of f is
Correct Answer. Differentiated, and multiplied by a constant.
Answer. Unchanged.
Answer. Inverted.
Answer. Reflected around the line x=y.
Answer. Integrated, and multiplied by a constant.
Answer. Multiplied by xi.
Answer. Divided by xi.
Question. If f and g are two functions on an abelian group G, then the Fourier transform of the convolution of f with g is
(up to normalization constants)
Answer. The convolution of the Fourier transforms of f and g.
Correct Answer. The pointwise product of the Fourier transforms of f and g.
Answer. The composition of the Fourier transforms of f and g.
Answer. The inner product of the Fourier transforms of f and g.
Answer. The sum of the Fourier transforms of f and g.
Answer. The tensor product of the Fourier transforms of f and g.
Question. If f and g are two functions on an abelian group G, then the Fourier transform of the pointwise product of f with g is (up to normalization constants)
Correct Answer. The convolution of the Fourier transforms of f and g.
Answer. The pointwise product of the Fourier transforms of f and g.
Answer. The composition of the Fourier transforms of f and g.
Answer. The inner product of the Fourier transforms of f and g.
Answer. The sum of the Fourier transforms of f and g.
Answer. The tensor product of the Fourier transforms of f and g.
Question. If f and g are two functions on two abelian groups G and H, then the Fourier transform of the tensor product of f with g is (up to normalization constants)
Answer. The convolution of the Fourier transforms of f and g.
Answer. The pointwise product of the Fourier transforms of f and g.
Answer. The composition of the Fourier transforms of f and g.
Answer. The inner product of the Fourier transforms of f and g.
Answer. The sum of the Fourier transforms of f and g.
Correct Answer. The tensor product of the Fourier transforms of f and g.