2009/11/24 by E K Narayanan, Narayanan, E K, Rakesh
Mathematics · #35L05 #35L15 #35R30 #44A05 #44A12 #92C55 #Analysis of PDEs (math.AP) #FOS: Mathematics #math.AP #msc:35L05 #msc:35L15 #msc:35R30 #msc:44A05 #msc:44A12 #msc:92C55
paper · pdf · doi:10.48550/arxiv.0911.4582
Revised version. Corrected some typing errors and added a figure
arxiv created 2010/02/01 · arxiv updated 2010/02/08
Given a real valued function on Rn we study the problem of recovering the function from its spherical means over spheres centered on a hyperplane. An old paper of Bukhgeim and Kardakov derived an inversion formula for the odd n case with great simplicity and economy. We apply their method to derive an inversion formula for the even n case. A feature of our inversion formula, for the even n case, is that it does not require the Fourier transform of the mean values or the use of the Hilbert transform, unlike the previously known inversion formulas for the even n case. Along the way, we extend the isometry identity of Bukhgeim and Kardakov for odd n, for solutions of the wave equation, to the even n case.