Math 461 Bard College

Homework 10

Due Date: Friday, May 6

Let \(C(T)\) be the set of all continuous, periodic functions \(\mathbb{R}\to\mathbb{R}\) with period \(2\pi\). The convolution of functions in \(C(T)\) is defined as in Homework 2.

Recall that a trigonometric polynomial is any polynomial function of \(\cos x\) and \(\sin x\). Equivalently, a trigonometric polynomial is any function that can be written as a finite Fourier sum: \[ f(x) \,=\, a + \sum_{k=1}^n \bigl(a_k \cos kx + b_k \sin kx\bigr). \] The goal of this assignment is to prove the following theorem.

Theorem. Every function in \(C(T)\) is a uniform limit of trigonometric polynomials.

  1. Prove that if \(f\in C(T)\) is a trigonometric polynomial and \(g\in C(T)\), then \(f*g\) is a trigonometric polynomial.
  2. Let \(f \in C(T)\), and let \(\{g_n\}\) be a sequence in \(C(T)\) that satisfies the following conditions:
    • \(g_n \geq 0\) and \(\displaystyle\frac{1}{2\pi}\int_{-\pi}^{\pi} g_n(t)\,dt = 1\) for all \(n\).
    • For each \(\epsilon>0\), the sequence \(\{g_n\}\) converges to \(0\) uniformly on \([-\pi,-\epsilon]\cup [\epsilon,\pi]\).
    Prove that \(f*g_n \to f\) uniformly on \(\mathbb{R}\).
  3. Prove that there exists a constant \(C>0\) so that \[ \int_{-\pi}^{\pi} \biggl(\frac{1 + \cos x}{2}\biggr)^{\!\!n}dx \,>\, \frac{C}{n} \] for all \(n\in\mathbb{N}\).
  4. Use the sequence \[ g_n(x) \,=\, \frac{1}{c_n}\biggl(\frac{1+\cos x}{2}\biggr)^{\!\!n}\quad\text{where}\quad c_n = \frac{1}{2\pi}\int_{-\pi}^{\pi} \biggl(\frac{1+\cos x}{2}\biggr)^{\!\!n} dx \] to prove the given theorem.