My notes when reading Fourier Analysis An Introduction by Stein and Shakarchi. This is for chapter 2 Basic Properties of Fourier Series, section 3 Convolutions.

3 Convolutions

Given two $$2\pi$$-periodic integrable functions $$f$$ and $$g$$ on $$\mathbb{R}$$, we define their convolution $$f*g$$ on $$[-\pi,\pi]$$ by

$$ (f*g)(x)=\frac{1}{2\pi}\int_{-\pi}^\pi f(y)g(x-y)dy\mathrm{.} $$

Proposition 3.1 Suppose that $$f$$, $$g$$, and $$h$$ are $$2\pi$$-periodic integrable functions. Then:

1. $$f*(g+h)=f*g+f*h$$.
2. $$(cf)*g=c(f*g)=f*(cg)$$ for any $$c\in\mathbb{C}$$.
3. $$f*g=g*f$$.
4. $$(f*g)*h=f*(g*h)$$.
5. $$f*g$$ is continuous.
6. $$\hat{f*g}(n)=\hat{f}(n)\hat{g}(n)$$¹.

Lemmar 3.2 Suppose $$f$$ is integrable on the circle and bounded by $$B$$. Then there exists a sequence $$\{f_k\}^\infty_{k=1}$$ of continuous functions on the circle so that

$$ \sup_{x\in[-\pi,\pi]}\lvert f_k (x)\rvert\leq B\quad\mathrm{for\:all}\:k=1,2,\dots\mathrm{,} $$

and

$$ \int_{-\pi}^\pi \lvert f(x)-f_k (x)\rvert dx\to 0\quad as\: k\to\infty\mathrm{.} $$

Footnotes