Fourier Analysis Second Semester 2008/9 Homework Assignment 4 (Due on 6th March 2009)

Only the questions marked with an asterisk (*) will count towards the assessment for this course. Most of these exercises are taken from Stein and Shakarchi.

1. 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).

*2. 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. *3. We have proved the following theorem in class.

Theorem. If a continuous function f on [−π, π] (such that f(−π) = f(π)) has a Fourier series which is absolutely convergent, then the Fourier series converges to the function. That is,

lim_{N→∞} ∑_{n=−N}^{N} a_{n}e^{inx} = f(x)

for all x ∈ [−π, π], where

a_{n} = \frac{1}{2π} ∫_{−π}^{π} f(x)e^{−inx} dx.

Suppose now that f: R → C is zero outside an interval [−M, M] and its Fourier transform f̂ is of moderate decrease. The aim of this question is to use the above theorem to find a simpler proof of the Fourier inversion formula for such functions f. Throughout this question f̂ denotes the Fourier transform of f not the Fourier coefficients (the formal difference between the two is only the value of certain constants).

(a) Fix L such that L > 2M and show that f(x) = ∑_{n} a_{n}(L)e^{2πinx/L}, where

a_{n}(L) = \frac{1}{L} ∫_{−L/2}^{L/2} f(x)e^{−2πinx/L} dx = \frac{f̂(n/L)}{L}.

This can be thought of as the theorem above rewritten for a function on [−L/2, L/2] (Observe that a_{n} = a_{n}(2π)). Show this can also be stated as

f(x) = δ ∑_{n=−∞}^{∞} f̂(nδ)e^{2πinx/L}

with δ = 1/L.

(b) Prove that if F is continuous and of moderate decrease, then

∫_{−∞}^{∞} F (ξ) dξ = lim_{δ↘0} δ ∑_{n=−∞}^{∞} F (δn).

[Hint: Approximate the integral by Riemann sums.]

(c) Conclude that f(x) = ∫_{−∞}^{∞} f̂(ξ)e^{2πinξ} dξ.

4. 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). *5. 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.

6. 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).]