Fourier Analysis Second Semester 2008/9 Mock Examination

The following questions are of the style you can expect in the final examination.

1. Let f be the characteristic function of the interval [a, b] ⊂ [−π, π].

(a) State Jordan’s criterion for the convergence of Fourier series.

(b) Show that the Fourier series of f is

\frac{b − a}{2π} + ∑_{n≠0} \frac{e^{−ina} − e^{inb}}{2πin} e^{inx}.

(c) Prove that the above Fourier series converges for each value of x and determine the

function to which it converges.

2. Suppose f is 2π-periodic and integrable on [−π, π].

(a) Show that

f̂(n) = \frac{1}{4π} ∫_{−π}^{π}[f(x) − f(x + π/n)]e^{−inx} dx.

(b) Assume that f is Hölder continuous of order α, that is

|f(x) − f(y)| ≤ C|x − y|^{α}

for some C > 0 and all x and y. Use part (a) to prove

f̂(n) = O(1/|n|^{α}).

3. (a) Define the convolution f ∗ g of two functions f and g defined on R.

(b) Prove that (f ∗ g) (ξ) = f̂(ξ)ĝ(ξ).

(c) Define

f(x) = \{ \begin{matrix}1, & if |x| ≤ 1, \\ 0, & if |x| > 1, \end{matrix}

and show by direct calculation that f̂(ξ) = sin(2πξ)/(πξ). Use the above to find a function h for which

ĥ(ξ) = (\frac{sin(2πξ)}{πξ})^{2} .

4. Let f be a function of moderate decrease. This question illustrates the principle that decay in the Fourier transform f̂ is related to the continuity of the function f. Suppose that f̂ is continuous and satisfies

|f̂(ξ)| ≤ \frac{C}{|ξ|^{1+α}}

for some C > 0 and 0 < α < 1.

(a) Derive an expression for f(x + h) − f(x) using the inversion formula for the Fourier transform. This expression should be an integral of some kind.

(b) Split the integral in two, integrating over |ξ| ≤ 1/|h| and |ξ| > 1/|h|. Estimate each integral separately to conclude

|f(x + h) − f(x)| ≤ A|h|^{α}

for some A > 0.