Spherical representations and functions are the building blocks for harmonic
analysis on riemannian symmetric spaces. In this paper we consider spherical
functions and spherical representations related to certain infinite dimensional
symmetric spaces Gββ/Kββ=limβGnβ/Knβ. We use the
representation theoretic construction Ο(x)= where e is a
Kββ--fixed unit vector for Ο. Specifically, we look at
representations Οββ=limβΟnβ of Gββ where Οnβ is
Knβ--spherical, so the spherical representations Οnβ and the
corresponding spherical functions Οnβ are related by Οnβ(x)=<enβ,Οnβ(x)enβ> where enβ is a Knβ--fixed unit vector for Οnβ, and we
consider the possibility of constructing a Kββ--spherical function
Οββ=limΟnβ. We settle that matter by proving the equivalence
of condtions (i) {enβ} converges to a nonzero Kββ--fixed vector e,
and (ii) Gββ/Kββ has finite symmetric space rank (equivalently, it
is the Grassmann manifold of p--planes in \F^\infty where p<β and
\F is R, \C or \H). In that finite rank case we also prove the
functional equation Ο(x)Ο(y)=limnββββ«KnββΟ(xky)dk
of Faraut and Olshanskii, which is their definition of spherical functions.Comment: 17 pages. New material added on the finite rank case