The Fourier transform of a complex-valued (Lebesgue) integrable function $${\displaystyle f(x)}$$ on the real line, is the complex valued function 展开

The frequency variable must have inverse units to the units of the original function's domain (typically named $${\displaystyle t}$$ or $${\displaystyle x}$$). For example, if $${\displaystyle t}$$ is measured in seconds, $${\displaystyle \xi }$$ should … 展开

The integral for the Fourier transform $${\displaystyle {\hat {f}}(\xi )=\int _{-\infty }^{\infty }e^{-i2\pi \xi t}f(t)\,dt}$$ can be studied for complex values of its argument ξ. Depending on the properties of f, this might not converge off the real axis at all, or it might converge to a 展开

The definition of the Fourier transform naturally extends from $${\displaystyle L^{1}(\mathbb {R} )}$$ to $${\displaystyle L^{1}(\mathbb {R} ^{n})}$$. … 展开

History
In 1822, Fourier claimed (see Joseph Fourier § The Analytic Theory of Heat) that any function, whether continuous or discontinuous, can be … 展开

Let $${\displaystyle f(x)}$$ and $${\displaystyle h(x)}$$ represent integrable functions Lebesgue-measurable on the real line satisfying: 展开

The Fourier transform can be defined in any arbitrary number of dimensions n. As with the one-dimensional case, there are many conventions. For an … 展开