Reading Rudin, I've got several questions related to the Fourier transform in Schwartz space (some not clear consequences or omitted proofs).
$ $ We define $\mathcal{S}\left(\mathbf{R}^{2}\right)$ the function space $\phi$ : $C^{\infty}$ $\longrightarrow$ $\mathbf{R}^{2}$ - rapidly decreasing (as well as all the derivatives), i.e. verifying $$ \forall \alpha, \beta \in \mathbf{N}^{2}, \quad \sup _{x \in \mathbf{R}^{2}}\left|x^{\alpha} \partial^{\beta} \phi(x)\right|<\infty $$ where $x^{\alpha}=x_{1}^{\alpha_{1}} x_{2}^{\alpha_{2}}$ et $\partial^{\beta} \phi=\partial_{x 1}^{\beta_{1}} \partial_{x_{2}}^{\beta_{2}} \phi$
For $f, g \in L^{2}\left(\mathbf{R}^{2}\right)$, we note $$ \|f\|=\left(\int_{\mathbf{R}^{2}}|f(x)|^{2} d x\right)^{\frac{1}{2}}, \quad(f, g)=\int_{\mathbf{R}^{2}} \bar{f}(x) g(x) d x . $$
For any function $\phi \in \mathcal{S}\left(\mathbf{R}^{2}\right)$, et $\kappa>0$, we consider $\phi^{>\kappa}$ and $\phi^{<\kappa}$ defined by their Fourier transform $$ \mathcal{F}\left(\phi^{>\kappa}\right)(\xi)=\hat{\phi}(\xi) 1_{\{|\xi|>\sqrt{\kappa}\}}(\xi), \quad \mathcal{F}\left(\phi^{<\kappa}\right)(\xi)=\hat{\phi}(\xi) 1_{\{|\xi|<\sqrt{\kappa}\}}(\xi) $$ so that $\phi$ decomposes in the form $$ \phi=\phi^{>\kappa}+\phi^{<\kappa} . $$
- How can I show that $$ \forall \phi \in \mathcal{S}\left(\mathbf{R}^{2}\right), \quad \int_{\mathbf{R}^{2}}|\nabla \phi(x)|^{2} d x=\int_{0}^{+\infty}\left(\int_{\mathbf{R}^{2}}\left|\phi^{>\kappa}(x)\right|^{2} d x\right) d \kappa $$
How can I show that for any finite family of functions $\left\{\phi_{j}\right\}_{j=1}^{N}$ $\in$ $\mathcal{S}\left(\mathbf{R}^{2}\right)$: $$ \forall \kappa>0, \forall x \in \mathbf{R}^{2}, \quad\left(\sum_{j=1}^{N}\left|\phi_{j}^{>\kappa}(x)\right|^{2}\right)^{\frac{1}{2}} \geq\left(\sum_{j=1}^{N}\left|\phi_{j}(x)\right|^{2}\right)^{\frac{1}{2}}-\left(\sum_{j=1}^{N}\left|\phi_{j}^{<\kappa}(x)\right|^{2}\right)^{\frac{1}{2}} $$
Let $\left\{\psi_{j}\right\}_{j=1}^{N}$ be a finite family of functions of $\mathcal{S}\left(\mathbf{R}^{2}\right)$ orthonormal for (, ). How can I show that $$ \forall f \in L^{2}\left(\mathbf{R}^{2}\right), \quad \sum_{j=1}^{N}\left|\int_{\mathbf{R}^{2}} \bar{f} \psi_{j}\right|^{2} \leq\|f\|^{2} $$
Finally, how can I show that $$ \sum_{j=1}^{N} \int_{\mathbf{R}^{2}}\left|\nabla \phi_{j}(x)\right|^{2} d x \geq \frac{2 \pi}{3} \int_{\mathbf{R}^{2}}\left(\sum_{j=1}^{N}\left|\phi_{j}(x)\right|^{2}\right)^{2} d x $$
Thanks in advance for any hints on any of these questions.