Radon problems for hyperboloids

Abstract

We offer a variant of Radon transforms for a pair X and Y of hyperboloids in R^3 defined by [x,x] = 1 and [y,y] = -1, y_1 ≥ 1, respectively, here [x,y] = -x_1 y_1+x_2 y_2+x_3 y_3. For a kernel of these transforms we take δ([x,y]), δ(t) being the Dirac delta function. We obtain two Radon transforms D(X) →C^∞ (Y) and D(Y) →C^∞ (X). We describe kernels and images of these transforms. For that we decompose a sesqui-linear form with the kernel δ([x,y]) into inner products of Fourier components.