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.