Fourier Analysis

I have a function $\varphi \in L^2(\mathbb{R})$ such that $\text{supp}(\widehat{\varphi})\subset [0,1]$ and a $2\pi$-periodic Fourier series $R(x):=\sum_{k\in \mathbb{Z}} a_k e^{ikx}$ (of some continu...

Let  $\mathcal{S}(\mathbb{R})$  be the Schwartz-Space. Prove that if  $f \in \mathcal{S}(\mathbb{R})$  and  $\int_{-\infty}^\infty f(t)dt=0$, then there is  $F \in \mathcal{S}(\mathbb{R})$  such that...

We have $M:= \left \{ \left(\begin{array}{c} x \\ y \end{array} \right ) \in \mathbb{R}^2  ; \left | \left | \left(\begin{array}{c} x \\ y \end{array} \right ) - \left(\begin{array}{c} 0 \\ \frac{1}{\...

The whole problem is stated in the attached picture. Please help <3.   Best regards, Robin.

Let $f=\frac{1}{2}-x$ on the interval $[0,1)$, and extend $f$ to be periodic on $\R$. a. Show that $\hat{f}(0)=0$, and $\hat{f}(k)=(2\pi i k)^{-1}$ if $k\neq 0$.b. Prove that $\sum_{k=1}^{\infty}\frac...