Superstability of functional equations related to spherical functions

Abstract In this paper we prove stability-type theorems for functional equations related to spherical functions. Our proofs are based on superstability-type methods and on the method of invariant means.


Introduction
In this paper C denotes the set of complex numbers. We suppose that G is a topological group and K is a compact topological group of continuous automorphisms of G. Hence, as a group K is a subgroup of the group of Aut .G/ of all continuous automorphisms of G. We also assume that the mapping k 7 ! k.x/ from K into G is continuous for each x in G. The normed Haar measure on K is denoted by m K . Hence m K is right and left invariant and m K .K/ D 1. We shall consider the functional equation Z K f .xk.y//d m K .k/ D g.x/h.y/ C p.y/ ; (1) where f; g; h; p W G ! C are continuous functions, and f is non-identically zero. Important special cases are Z K f .xk.y//d m K .k/ D f .x/ C f .y/ (2) corresponding to the case h D 1, f D g D p, and Z K f .xk.y//d m K .k/ D f .x/f .y/ (3) corresponding to the case p D 0, f D g D h. Nonzero solutions f of the latter equation are called generalized K-spherical functions. We note that if f is a bounded solution of (3), then we call it a K-spherical function. For K-spherical functions see [1]. Functional equations related to spherical functions have been studied in [2][3][4]. In the case G is a discrete group and K D fid G g then (1) reduces to f .xy/ D g.x/h.y/ C p.y/ which is a Levi-Civitá-type functional equation and its stability was studied on hypergroups in [5]. The stability of sine and cosine functional equations was investigated in [6].
In this paper we study stability properties of functional equations of type (1). The ideas are similar to those in [5][6][7].

Superstability of the functional equation (1) when p D 0
The following lemma is crucial.
for each x; y; z in G.
Proof. We apply Fubini's Theorem and the invariance of m K to get The next theorem is about the superstability of the functional equation of type (1) where f and g are equal and p D 0.
Theorem 2. Suppose that f; g W G ! C are continuous functions such that the function is bounded on G for each y in G. Then either f is bounded or g is a generalized K-spherical function.
Proof. We let for each x; y in G. Then F W G G ! C is continuous and it satisfies jF .x; y/j Ä A.y/ with some function A W G ! C for each x; y in G.
Substituting yl.z/ for y and using the fact that l 7 ! F .x; yl.z// is continuous, hence integrable on K, we have, by Lemma On the other hand, substituting xk.y/ for x and z for y we obtain Moreover, we have which implies, together with (5) Now, from (4) and (6) we derive for each x; y; z in G. Obviously, the right hand side, as a function of x, is bounded on G. Hence, if f is unbounded, then we must have Z K g.yl.z//d m K .l/ D g.y/g.z/ for each y; z in G, which was to be proved.
As a consequence we obtain the superstability of the functional equation of K-spherical functions.
Corollary 3. Suppose that f W G ! C is a continuous function such that the function Now we are in the position to prove the general superstability-type result for equation (1) in the case p D 0. it follows that f is bounded, which implies immediately that the function x 7 ! g.x/h.y/ is bounded for each y, too. As h ¤ 0 we infer that g is bounded. This means that we may assume that h.e/ ¤ 0. In this case, obviously, we may replace h by h= h.e/, that is we assume h.e/ D 1. We use similar ideas like above. We introduce the function 3 Stability of the functional equation (1) If in the functional equation (1) we have p ¤ 0, then the equation has some "additive character", too, as it includes equation (2) if h D 1. Hence we cannot expect a purely superstability result, which is a common feature of multiplicative-type equations. On the other hand, in the case of additive-type equations our experience shows that invariant means can be utilized. This is illustrated in the following general result. for each x; y in G.
Proof. As f is unbounded, hence g is unbounded, too, and the function  From these two inequalities, by (10) and the property of !, we infer j!.z/ Z K g.xl.y//d m K .l/ g.x/!.y/ '.y/ Á j Ä 2L for each x; y; z in G. As ! D h is unbounded it follows that we have that is, g D ' C , where W G ! C satisfies (11) for each x; y in G. The theorem is proved.
We note that the above results can be generalized to some extent. In fact, we haven't used inverses neither in G, nor in K. It follows that similar results can be obtained if we suppose that G and K are just some types of topological semigroups satisfying reasonable conditions so that the existence of an invariant integral on K and -in the case of the general equation (2) -an invariant mean on G is guaranteed.