Properties of Fourier coefficients, Fourier transform on L²(-D,D), Fourier integral theorem - One Line Questions
1.
What is the Fourier transform of the function f(x) = 1/x for x ≠ 0 and 0 at x=0? —
-iπ sgn(ω)
2.
If f(x) is defined on [-D, D], the Fourier transform on L²(-D, D) is defined using integrals over what interval? —
(-D, D)
3.
The Fourier transform of the function f(x) = x² e^(-x²) is proportional to: —
(1 - ω²) e^(-ω²/4)
4.
If F(ω) is the Fourier transform of f(x), what is the Fourier transform of f(ax) for a > 0? —
(1/a) F(ω/a)
5.
What is the Fourier transform of the function f(x) = sech(ax)? —
(π/a) sech(πω/(2a))
6.
What is the Fourier transform of the function f(x) = e^(-ax²) for a > 0? —
√(2π/a) e^(-ω²/4a)
7.
Parseval's identity for Fourier transforms states that the integral of |f(x)|² over the real line is related to the integral of |F(ω)|². What is the relationship? —
∫₋∞^∞ |f(x)|² dx = (1/2π) ∫₋∞^∞ |F(ω)|² dω
8.
The Fourier transform of the function f(x) = e^(-ax)u(x) for a > 0, where u(x) is the Heaviside step function, is: —
1 / (a - iω)
9.
Consider the Fourier transform of f(x) = e^(-|x|). What is the form of its Fourier transform F(ω)? —
2 / (1 + ω²)
10.
What is the Fourier transform of a rectangular pulse function, f(x) = 1 for |x| ≤ D and 0 otherwise? —
2D sinc(ωD/2)
11.
What is the Fourier transform of the function f(x) = 1 for -D ≤ x ≤ D and 0 otherwise, in the limit as D → ∞? —
δ(ω)
12.
For a function f(x) on L²(-D, D), the Fourier transform is defined as an integral over the finite interval [-D, D]. This is essentially: —
A truncated Fourier transform.
13.
If f(x) is a function such that f(x) → 0 as |x| → ∞, then its Fourier transform F(ω) is guaranteed to be: —
Bounded.
14.
The Fourier transform of a constant function f(x) = c is: —
2πc δ(ω)
15.
If the Fourier coefficients of a function f(x) on [-L, L] are denoted by c_n, and the function is odd, what can be said about c_n for n ≠ 0? —
c_n = -c_{-n}
16.
If the Fourier coefficients of a function f(x) on [-L, L] are denoted by c_n, and the function is even, what can be said about c_n for n ≠ 0? —
c_n = c_{-n}
17.
If f(x) is a function such that ∫₋∞^∞ |f(x)| dx < ∞, its Fourier transform F(ω) is guaranteed to be: —
Continuous and bounded.
18.
The Fourier transform of f(x) = x is proportional to: —
d/dω δ(ω)
19.
The Fourier integral theorem is valid for functions f(x) that satisfy: —
Dirichlet conditions.
20.
Which property of Fourier transforms relates the transform of f(x) to the transform of its inverse Fourier transform? —
Duality property
21.
If F(ω) is the Fourier transform of f(x), what is the Fourier transform of f(x-x₀)? —
e^(-iωx₀) F(ω)
22.
The Fourier transform of a time-shifted signal f(t-t₀) is related to the Fourier transform of f(t) by a factor of: —
e^(-jωt₀)
23.
If f(x) is a real-valued function, what is the relationship between F(ω) and F(-ω)? —
F(-ω) = F(ω)* (complex conjugate)
24.
If f(x) has a jump discontinuity at x=c, the Fourier integral theorem converges to: —
(f(c+) + f(c-))/2
25.
The Fourier Integral Theorem states that a function f(x) can be represented as an integral of its Fourier transform components. What is the general form of this representation? —
f(x) = (1/2π) ∫₋∞^∞ F(α)e^(iαx) dα
26.
What is the condition for a function f(x) to be absolutely integrable over the real line, which is a prerequisite for its Fourier transform to exist? —
The integral of |f(x)| from -infinity to +infinity must be finite.
27.
What is the condition for the convergence of the Fourier integral representation of f(x)? —
f(x) must be absolutely integrable.
28.
The property that the Fourier transform of a derivative of f(x) is (iω) times the Fourier transform of f(x) holds under which condition? —
f(x) must be identically zero at infinity.
29.
If F(ω) is the Fourier transform of f(x), what is the Fourier transform of e^(iω₀x) f(x)? —
F(ω - ω₀)
30.
The Fourier transform of the integral of a function f(x), i.e., F{∫f(t)dt}, is related to F(ω) by: —
F(ω) / (iω) + πF(0)δ(ω)
31.
If f(x) is an odd function, how does its Fourier transform F(ω) behave? —
F(ω) is an odd function.
32.
If f(x) is an even function, how does its Fourier transform F(ω) behave? —
33.
The convolution of two functions f(x) and g(x) is defined as (f * g)(x) = ∫₋∞^∞ f(τ)g(x-τ) dτ. What is the Fourier transform of the convolution? —
F(ω)G(ω)
34.
What is the definition of the Fourier cosine transform of a function f(x) for x ≥ 0? —
Fc(ω) = √(2/π) ∫₀^∞ f(x) cos(ωx) dx
35.
What is the definition of the Fourier sine transform of a function f(x) for x ≥ 0? —
Fs(ω) = √(2/π) ∫₀^∞ f(x) sin(ωx) dx
36.
What is the Fourier transform of the function f(x) = x for -1 ≤ x ≤ 1 and 0 otherwise? —
i (sin(ω) - ωcos(ω)) / ω²
37.
What is the property of Fourier coefficients related to the symmetry of the function? —
If f(x) is odd, all cosine coefficients are zero.
38.
What is the Fourier transform of the function f(x) = sin(ω₀x)? —
iπ [δ(ω - ω₀) - δ(ω + ω₀)]
39.
The Fourier transform of f(x) = x e^(-x²) is: —
-iω √(π/2) e^(-ω²/4)
40.
The Fourier transform of a derivative of a function, i.e., F{f'(x)}, is related to the Fourier transform of f(x), F(ω), by which formula? —
iω F(ω)
41.
The Fourier integral theorem can be viewed as an extension of which concept to non-periodic functions? —
Fourier Series
42.
The Fourier integral theorem represents a function as a superposition of: —
Sinusoids of continuous frequencies.
43.
Which property of Fourier coefficients states that if f(x) is periodic with period 2L, then the coefficients for f(-x) are related to the original coefficients? —
Symmetry property
44.
For a function f(x) defined on [-D, D], what is the Fourier transform of f(x) on L²(-D, D) related to? —
The Fourier series coefficients of f(x).
45.
What is the domain of the Fourier transform of a function f(x) in L²(-∞, ∞)? —
The set of all complex numbers.
46.
If f(x) is an even function, what can be said about its Fourier cosine coefficients? —
They are non-zero.
47.
If f(x) is an odd function, what can be said about its Fourier sine coefficients? —
They are non-zero.
48.
The Fourier transform of the Dirac delta function δ(x) is: —
1
49.
What is the Fourier transform of the function f(x) = cos(ω₀x)? —
π [δ(ω - ω₀) + δ(ω + ω₀)]