Math 461 Bard College

# Homework 2

Due Date: Friday, February 12

The goal of this assignment is to prove the following theorem regarding the convergence of Fourier series.

Theorem. Let $$f$$ be a $$C^2$$ function that is periodic with period $$2\pi$$, and let $$a$$, $$b_n$$, and $$c_n$$ denote the Fourier coefficients for $$f$$. Then the series $a + \sum_{n=1}^\infty \bigl(b_n \cos nx + c_n \sin nx\bigr)$ converges pointwise to $$f$$.

Note: Recall here that a function $$f$$ is $$C^n$$ if it is $$n$$-times differentiable and $$f^{(n)}$$ is continuous.

1. If $$f$$ and $$g$$ are continuous periodic functions with period $$2\pi$$, define the convolution $$f*g$$ by the formula $(f*g)(x) \,=\, \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x-t) g(t)\,dt.$ Prove that $$f* g = g * f$$.
2. For each $$n\in\mathbb{N}$$, let $D_n(x) \,=\, 1 + 2\sum_{k=1}^n \cos kx.$ If $$f$$ is a continuous periodic function with period $$2\pi$$, prove that $(D_n*f)(x) \,=\, a + \sum_{k=1}^n \bigl( b_k \cos kx + c_k\sin kx\bigr)$ for all $$n\in\mathbb{N}$$, where $$a$$, $$b_n$$, and $$c_n$$ are the Fourier coefficients of $$f$$.
3. Prove that $D_n(x) = \frac{\sin \bigl((n+\frac{1}{2})x\bigr)}{\sin(x/2)}$ for all $$n\in\mathbb{N}$$ and all values of $$x$$ for which $$\sin(x/2) \ne 0$$.
4. Let $$f$$ be a $$C^2$$ function, let $$a\in\mathbb{R}$$, and let $$g\colon\mathbb{R}\to\mathbb{R}$$ be the function defined by $g(a) = f'(a)\qquad\text{and}\qquad g(x) \,=\, \frac{f(x) - f(a)}{x-a}\text{ for }x\ne a.$ Prove that $$g$$ is $$C^1$$.
5. If $$f$$ is $$C^2$$ and periodic with period $$2\pi$$, prove that $$f * D_n \to f$$ pointwise.
Hint: Start by rewriting $$f(x)$$ as $$\displaystyle \frac{1}{2\pi}\int_{-\pi}^{\pi} f(x)D_n(t)\,dt$$ to combine it with $$(f*D_n)(x)$$, and then use integration by parts on the resulting combined integral.