Fourier transform
Definition for Lebesgue integrable functions

Until now, we have been dealing with Schwartz functions, which decay rapidly at infinity, with all derivatives. This excludes many functions of practical importance from the definition, such as the rect function. A measurable function f: R → C {\displaystyle f:\mathbb {R} \to \mathbb {C} } is called (Lebesgue) integrable if the Lebesgue integral of its absolute value is finite: ‖ f ‖ 1 = ∫ R | f (x) | d x < ∞. {\displaystyle \|f\|_{1}=\int _{\mathbb {R} }|f(x)|\,dx<\infty.} Two measurable functions are equivalent if they are equal except on a set of measure zero. The set of all equivalence classes of integrable functions is denoted L 1 (R) {\displaystyle L^{1}(\mathbb {R})}. Then:

Definition — The Fourier transform of a Lebesgue integrable function f ∈ L 1 (R) {\displaystyle f\in L^{1}(\mathbb {R})} is defined by the formula Eq.1.

The integral Eq.1 is well-defined for all ξ ∈ R, {\displaystyle \xi \in \mathbb {R},} because of the assumption ‖ f ‖ 1 < ∞ {\displaystyle \|f\|_{1}<\infty }. (It can be shown that the function f ^ ∈ L ∞ ∩ C (R) {\displaystyle {\widehat {f}}\in L^{\infty }\cap C(\mathbb {R})} is bounded and uniformly continuous in the frequency domain, and moreover, by the Riemann–Lebesgue lemma, it is zero at infinity.)

However, the class of Lebesgue integrable functions is not ideal from the point of view of the Fourier transform because there is no easy characterization of the image, and thus no easy characterization of the inverse transform.
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