程序代写代做代考 1

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.)