1
Fourier Analysis Midterm
Name:
Problem 1 2 3 4 5 Total:
Max 20 20 20 20 20 100
Scores
Instructions
• Time for test: 75 minutes.
• Do not use any notes or calculators or any textbooks.
2
1. (a) State the definition of the Fourier coefficients and the Fourier se-
ries for a function f defined in the interval [0, 2π].
(b) State the definition of Cesaro summability.
(c) State the definition of a family of good kernels on the circle.
(d) State the definition of Fourier transform on S(R).
3
2. (a) State a criteria for the uniform convergence of a Fourier series.
(b) State the Riemann-Lebesgue Lemma.
(c) State the uniqueness theorem for a Fourier series.
(d) State the isoperimetric inequality.
4
3. State and prove the best approximation lemma.
5
4. Show that the series
sinx+
sin(3x)
3
+
sin(5x)
5
+
sin(7x)
7
+ …
is constant for all x ∈ (0, π).
Compute this constant and write the identity when x = π
2
.
6
5. Suppose that f is a continuous function on R periodic of period 2π and
α/π is irrational. Prove that
lim
N→∞
1
N
N∑
n=1
f(x+ nα) =
1
2π
∫ π
−π
f(t)dt
for every x.
(Hint: Prove this first for trigonometric polynomials.)