Editing Legendre function (section)

==Legendre function as characters==
The real integral representation of <math>P_s</math> are very useful in the study of harmonic analysis on <math>L^1(G//K)</math> where <math>G//K</math> is the [[Homogeneous space|double coset space]] of <math>SL(2,\R)</math> (see [[Zonal spherical function]]). Actually the Fourier transform on <math>L^1(G//K)</math> is given by
<math display="block">L^1(G//K)\ni f\mapsto \hat{f}</math>
where
<math display="block">\hat{f}(s)=\int_1^\infty f(x)P_s(x)dx,\qquad -1\leq\Re(s)\leq 0 </math>