A Cardinal Function Method of Solution of the Equation Aw = u — u

The steady-state form of the Klein-Gordon equation is given by (•) Au = u-u3, u = u(X), X&R3. 3 For solutions which are spherically symmetric, (*) takes the form ü + lujr = u — u , 3 u = u(r), where r is the distance from the origin in R . The function y = ru satisfies (* *) j' = y y ¡r . It is known that (•*) has solutions {yn}n=Q, where yn has exactly n zeros in (0, °°), and where y(0) = y(°°) = 0. In this paper, an approximation is obtained for the solution yQ by minimizing a certain functional over a class of functions of the form £ a*s,nc ——• k=-m |_ hm J It is shown that the norm of the error is 0(m ' exp(—am 2)) as m —► °°, where a is positive.