Fourier transform on complex L1 classes

Let $n\ge 1$ and $f\in L^1({\mathbb{R}}^n;\mathbb{C})$, with the componentwise integral and quotient conventions of Complex Lp classes and Euclidean test-function conventions, The space $L^p(\mu )$ as the quotient by null functions and The class $L^1(\mu )$ of integrable functions. Define $\widehat{f}(\xi )=\mathcal{F}f(\xi ):={\int }_{{\mathbb{R}}^n}f(x)\exp (-2\pi ix\cdot \xi )dx,\xi \in {\mathbb{R}}^n\mathrm{.}$ Here $x\cdot \xi ={\sum }_{j=1}^n{x}_j{\xi }_j$, Lebesgue measure is used, and the exponential is The complex exponential by its power series. Its unit modulus follows from $\exp (x+iy)=e^x(\cos y+i\sin y)$, $\mathrm{\mid }\exp (x+iy)\mathrm{\mid }=e^x$, and $e^{i\pi }+1=0$. The integral is evaluated using any measurable representative. Absolute convergence and representative independence are the obligations discharged by The integral transform is representative independent ↗. The output is a function defined at every frequency, not merely an almost-everywhere class. At frequency zero the formula reads $\widehat{f}(0)=\int f$. No choice selection of representatives for a family of classes is part of this definition.