Inverse Fourier Transforms

Inverse Fourier Transform-1:  Remaining Part of Inverse Fourier Transform are given as:

In the second integrals;

If K ≥ 1,

Since are both convergent integrals,

For the integrals over [0, k],

Since f is a piecewise smooth, f' (x ) exists for all x ∈ R and Therefore g is bounded on [0,k] and hence g ∈ L1 (R). By the Riemann - Lebessgue lemma, g^ exists, is continuous, g^(±a) → 0 as a → \infty  and therefore;


For K ≥ 1. A virtually identical arguments works for the first integrals.