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 for \(f\).
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: Prove that \[ f(x) - (f*D_n)(x) \,=\, \frac{1}{2\pi}\int_{-\pi}^{\pi} g(t)\, \sin\bigl((n+\tfrac{1}{2})t\bigr)\,dt \] for some \(C^1\) function \(g\) and then use integration by parts.