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Restriction and extension
Considering a smooth compact hyper-surface $\mathcal{S}$ in $\mathbb{R}^d$ with surface measure $d\sigma$. Given $f \in L^1(\mathbb{R}^d)$, the Fourier transform defined as follow: $$ \begin{equation} \hat{f}(x) = \int_{\mathbb{R}^d}e^{-2\pi i x \xi}f(x)dx \end{equation} $$ which by Riemann-Lebesgue is a bounded, continuous function vanishing at infinity. Since $\hat{f}$ is continuous on $\mathbb{R}^d$, by the Rimann-Lesbegue lemma its restriction to the compact hyper-surface $S \subset \mathbb{R}^d$ is is well-defined pointwise. Specifically, the restriction $\hat{f}\mid_{S}: S \rightarrow \mathbb{C}$ is the continuous function given by $$ \begin{equation} \hat{f}\mid_{S}(\sigma) = \hat{f}(\sigma) = \int_{\mathbb{R}^d}e^{-2\pi i x \xi}f(x)dx \end{equation} $$ for each $\sigma \in S$. This is bounded (as $\hat{f}$ is bounded) and can be integrated against the surface measure $d\sigma$ on $S$.
Proof of Theorem of solution of wave equation in the case $n = 1$
Solution of Brezis Problem 8.24 (1) and (2)
Solution of Evans PDE Problem 13
Collected Lectures on Complex Analysis
📝 Introduction to Complex Analysis - Michael Taylor 📝 An Introduction to Complex Analysis and Geometry - John P. D’Angelo (University of Illinois) 📝 A First Course in Complex Analysis - Matthias Beck, Gerald Marchesi, Dennis Pixton, Lucas Sabalka 📝 A Guide to Complex Variables - Steven G. Krantz 📝 Complex Analysis - Charles Walkden 📝 Complex Analysis - Christian Berg 📝 Complex Variables - R. B. Ash, W.P. Novinger 📝 Complex Analysis - Christer Bennewitz 📝 Complex Analysis - Donald E. Marshall 📝 A Concise Course in Complex Analysis and Riemann Surfaces - Wilhelm Schlag 📝 Complex Analysis - G. Cain (Georgia Tech) 📝 Complex Analysis - Juan Carlos Ponce Campuzano