We consider smeared zeta functions and heat-kernel coefficients on the boun
ded, generalized cone in arbitrary dimensions. The specific case of a ball
is analyzed in detail and used to restrict the form of the heat-kernel coef
ficients A(n) on smooth manifolds with boundary. Supplemented by conformal
transformation techniques, it is used to provide an effective scheme for th
e calculation of the A(n). As an application, the complete A(5/2) coefficie
nt is given. (C) 2001 American Institute of Physics.