The reproducing kernel of H^2 and radial eigenfunctions of the hyperbolic Laplacian
In the paper we characterize the reproducing kernel K_n,h for the Hardy space H^2 of hyperbolic harmonic functions on the unit ball B in R^n. Specifically we prove that \[ K_n,h(x,y) = _α =0^∞ S_n,α (x vert )S_n,α (y vert ) Z_α (x,y), \] where the series converges absolutely and uniformly on K× B for every compact subset K of B. In the above, S_n,α is a hypergeometric function and Z_α is the reproducing kernel of the space of spherical harmonics of degree α. In the paper we prove that \[ 0K_n,h(x,y) C_n(1-2x,y+ x vert^2 y vert^2)^n-1 , \] where C_n is a constant depending only on n. It is known that the diagonal function K_n,h(x,x) is a radial eigenfunction of the hyperbolic Laplacian _h on B with eigenvalue λ _2 = 8(n-1)^2. The result for n=4 provides motivation that leads to an explicit characterization of all radial eigenfunctions of _h on B . Specifically, if g is a radial eigenfunction of _h with eigenvalue λ _α = 4(n-1)^2α (α -1), then \[ g(r) = g(0) p_n,α (r^2)(1-r^2)^(α -1)(n-1) , \] where p_n,α is again a hypergeometric function. If α is an integer, then p_n,α (r^2) is a polynomial of degree 2(α -1)(n-1). , number=1, journal=MATHEMATICA SCANDINAVICA, author=Stoll, Manfred, year=2019, month=Jan., pages=81–101