Fourier Transforms

fx fx −2 12−f̂k e −ikx dk ,analogous to the Dirichlet Conditions in Fourier Series. Usually we employ (3.2.9) but weknow that the above form is available if fx is discontinuous. This clearly requires f k tobe absolutely integrable as well.Tables of FTs and their inverses exist but beyond these standard forms, it becomesnecessary to evaluate (3.2.9) for the particular f̂k and this usually involves contourintegration. The integral in (3.2.9) is the same as the model form of Jordan’s Lemmapresented in (2.4.4) and the methods employed therein can be employed.All of this theory assumes that the integrals exist. With the stated conditions for fx, weknow that f̂k exists. The Riemann-Lebesque Lemma states if fx is also continuous thenf̂k → 0ask →, so this provides some theoretical support but not sufficient to ensureexistence in all cases.Combining (3.2.1) and (3.2.9) givesfx 12 fx 12−−f̂k e −ikx dk 12−−−fu e iku due −ikx dkfu e iku−x du dk (3.2.10)and this is the general Fourier Theorem that underpins the analysis.Return to (3.2.9) and replace x by −x to give2 f−x −f̂k e ikx dk ,and interchange x and k,2 f−k −f̂x e ikx dx .Thus the FT of f̂x is 2f−k and this identity can be useful in the manipulation anddetermination of transforms.9