Fourier Analysis
Centre for Mathematical Sciences
Mathematics, Faculty of Science
Review questions for Fourier Analysis
1. Define scalar product in a complex linear space. Give examples.
2. What is a norm on a linear space? Give examples.
3. What does it mean that a sequence in a normed linear space is convergent?
4. For normed linear spaces, what are Cauchy sequences and what does completeness mean? What
are Banach spaces? Give examples.
5. Prove Bessel’s inequality. If one has equality, what does this mean?
6. Explain what it means that an orthonormal sequence is complete.
7. What are Hilbert spaces? Give examples.
8. Define the Fourier coefficients of a function u ∈ L1 (−π, π). Show, under appropriate conditions, how one from the Fourier coefficients of u(x) can find the Fourier coefficients of the
functions u(x + a) and u0 (x).
9. State and prove the Riemann–Lebesgue lemma.
10. Prove Dini’s theorem about pointwise convergence for Fourier series. Give examples of functions
that satisfy the conditions in Dini’s theorem.
11. Show that if u ∈ L1 (−π, π) and ûn = 0 for all integers n, then u = 0 almost everywhere.
12. Show that einx , n = 0, ±1, ±2, . . ., is a complete orthonormal sequence in L2 (−π, π) and
that Parseval’s formula holds.
13. Show that if u is a continuously differentiable function that is periodic with period 2π, then the
Fourier series of u converges to u uniformly on the whole real axis.
14. From the usual Fourier series of a function on the interval (−π, π), derive the function’s sine
and cosine series on the interval (0, π). Do the same for a general interval of the form (0, T ).
15. Describe the method of separation of variables for solving the wave equation. In what way do
the boundary conditions influence the form of the solution? How can one show that a solution
is uniquely determined by its initial conditions?
16. Define the Fourier transform û of a function u ∈ L1 (R) and explain why û(ξ) turns out to be
a continuous, bounded function that converges to 0 as ξ → ±∞.
Please, turn over!
17. Define the Fourier transform û of a function u ∈ L2 (R). Show Parseval’s formula kûk22 =
2πkuk22 for u ∈ L2 (R).
ˆ
18. Show the Fourier inversion formula û(−x)
= 2πu(x) for u ∈ L2 (R).
19. Define the convolution of two functions u, v ∈ L2 (R). Show how one can determine the
Fourier transform of a convolution and the convolution of two Fourier transforms.
20. Show, under appropriate conditions, how one from the Fourier transform of u(x) can find the
Fourier transform of the functions xu(x) and u0 (x).
21. Find the Fourier transform of u(x) = e−x
2 /2
.
22. Describe how one can solve the heat equation ut = uxx , when −∞ < x < ∞ and t > 0.