The Fourier transform may be interpreted as an operator on \(L^2(\mathbb{R}^n)\). However, not every \(L^2\) function is integrable and cannot therefore be defined as here . Since the Schwartz functions are dense in \(L^2(\mathbb{R}^n)\) the operator \(\mathcal{F}\colon L^2(\mathbb{R}^n)\to L^2(\mathbb{R}^n)\) can be defined as the continuation of the one defined on Schwartz-spaces [1, Satz V.2.9].
Remark
- Due to Plancherel theorem the operator is bounded . It even implies it is an isometric Isomorphism (see (0x67d92de3) ).
- Using the Hausdorff-Young inequality one can define the Fourier transform on \(L^p(\mathbb{R}^n)\) for \(1\le p\le 2\) (see (0x67470f8d) ).
See also Link to heading
References Link to heading
- D. Werner, Funktionalanalysis. Berlin, Heidelberg: Springer Berlin Heidelberg, 2018. doi:10.1007/978-3-662-55407-4