Fourier Analysis Second Semester 2009/10 Homework Assignment 3 (Due on 23rd February 2010)

Only the questions marked with an asterisk (*) will count towards the assessment for this course.

Fourier Series.

1. Prove that l^{2}(Z) is complete, that is, every Cauchy sequence in l^{2}(Z) has a limit in l^{2}(Z). *2. Consider the function f: [−π, π] → C defined by f(θ) = |θ|. Use Parseval’s Identity to prove

∑_{k=0}^{∞} \frac{1}{(2k + 1)^{4}} = \frac{π^{4}}{96} and ∑_{n=1}^{∞} \frac{1}{n^{4}} = \frac{π^{4}}{90} .

3. In addition to what we discovered in question 2, it is, in fact, possible to calculate a formula for ∑_{n} 1/n^{k} for every even k. However, nobody has yet managed to do this for odd k, even for k = 3. If you have some spare time, perhaps you might want to try to evaluate ∑_{n} 1/n^{3}. If you succeed you can publish your first paper.

4. Suppose that f is a continuously differentiable 2π-periodic function such that

∫_{−π}^{π} f(θ) dθ = 0.

(a) Use Parseval’s Identity to show that

∫_{−π}^{π} |f(θ)|^{2} dθ ≤ ∫_{−π}^{π} |f′(θ)|^{2} dθ.

(b) Find a function f_{0} which satisfies the same conditions as f but

∫_{−π}^{π} |f_{0}(θ)|^{2} dθ = ∫_{−π}^{π} |f′_{0}(θ)|^{2} dθ.

(c) Find a function f_{1} which satisfies the same conditions as f but

∫_{−π}^{π} |f_{1}(θ)|^{2} dθ < ∫_{−π}^{π} |f′_{1}(θ)|^{2} dθ.

(d) Suppose g is continuously differentiable and f is as above. Prove that

|∫_{−π}^{π} \overline{f(θ)}g(θ) dθ|^{2} ≤ ∫_{−π}^{π} |f(θ)|^{2} dθ ∫_{−π}^{π} |g′(θ)|^{2} dθ.

(e) Suppose h is continuously differentiable on [0, π] and h(0) = h(π) = 0. Prove

∫_{0}^{π} |h(θ)|^{2} dθ ≤ ∫_{0}^{π} |h′(θ)|^{2} dθ.

5. Suppose f is 2π-periodic function which is k-times continuously differentiable. Prove that

f̂(n) = O(1/|n|^{k})

as |n| → ∞.

6. 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}).

The Fourier transform. 7. Prove the following proposition, which we stated in class.

Proposition. If f ∈ S(R) then

(i) f(x + h) → f̂(ξ)e^{2πihξ} whenever h ∈ R, (ii) f(x)e^{−2πixh} → f̂(ξ + h) whenever h ∈ R, (iii) f(δx) → δ^{−1}f̂(δ^{−1}ξ) for δ > 0, (iv) f′(x) → 2πiξf̂(ξ),

(v) −2πixf(x) → (f̂)′(ξ).

8. The aim of this question is to prove that g: R → C defined by

g(x) = e^{−πx^{2}}

for all x ∈ R is equal to its own Fourier transform.

(a) By a simple calculation, check that g′(x) = −2πxg(x).

(b) Set

G(ξ) = ĝ(ξ) = ∫_{−∞}^{∞} g(x)e^{−2πixξ} dx

and use the above fact to prove that G′(ξ) = −2πξG(ξ).

(c) Define H(ξ) = G(ξ)e^{πξ^{2}} and show that H′(ξ) = 0 for all ξ ∈ R.

(d) Conclude that G(ξ) = e^{−πξ^{2}}. You may assume G(0) = 1 without proof.

9. Let f and g be functions defined by

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

and

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

Although f is not continuous, observe the integral defining the Fourier transform of f still makes sense. Show that

f̂(ξ) = \frac{sin(2πξ)}{πξ} and ĝ(x) = (\frac{sin(2πξ)}{πξ})^{2} ,

for ξ ≠ 0 and compute f̂(0) and ĝ(0). *10. 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.

11. Suppose f is a function of moderate decrease (hence, also continuous). (a) Prove that f̂ is continuous and that f̂(ξ) → 0 as |ξ| → ∞ by proving the formula

f̂(ξ) = \frac{1}{2} ∫ [f(x) − f(x − 1/(2ξ)]e^{−2πixξ} dξ.

(b) Show that if f̂(ξ) = 0 for all ξ ∈ R, then f(x) = 0 for all x ∈ R. [Hint: Check the multiplication formula ∫ f̂(y)g(y) dy = ∫ f(y)ĝ(y) dy is valid for such f when g ∈ S(R).]