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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Gianazza, Ugo / Vespri, Vincenzo

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Volume 15, Issue 1

Issues

Superstability of functional equations related to spherical functions

László Székelyhidi
  • Corresponding author
  • Institute of Mathematics, University of Debrecen, Egyetem tér 1., P.O. Box 12, H 4010 Debrecen, Hungary
  • Department of Mathematics, University of Botswana, Botswana, Southern Africa
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Published Online: 2017-04-17 | DOI: https://doi.org/10.1515/math-2017-0038

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.

Keywords: Spherical function; Stability

MSC 2010: 39B82; 39B52; 43A90

1 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 kk(x) from K into G is continuous for each x in G. The normed Haar measure on K is denoted by mK. Hence mK is right and left invariant and mK(K) = 1. We shall consider the functional equation Kf(xk(y))dmK(k)=g(x)h(y)+p(y),(1) where f, g, h, p : G → ℂ are continuous functions, and f is non-identically zero. Important special cases are Kf(xk(y))dmK(k)=f(x)+f(y)(2) corresponding to the case h = 1, f = g = p, and Kf(xk(y))dmK(k)=f(x)f(y)(3) corresponding to the case p = 0, f = g = 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 [24]. In the case G is a discrete group and K = {idG} then (1) reduces to f(xy)=g(x)h(y)+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 [57].

2 Superstability of the functional equation (1) when p = 0

The following lemma is crucial.

Lemma 1

Let f : G → ℂ be continuous, then we have KKf(xk(y)l(z))dmK(k)dmK(l)=KKf(xk(yl(z)))dmK(k)dmK(l) for each x, y, z in G.

Proof

We apply Fubini’s Theorem and the invariance of mK to get KKf(xk(y)l(z))dmK(k)dmK(l)=KKf(xk(y)l(z))dmK(l)dmK(k)=KKf(xk(y)(kl)(z))dmK(k)dmK(l)=KKf(xk(yl(z)))dmK(k)dmK(l).     □

The next theorem is about the superstability of the functional equation of type (1) where f and g are equal and p = 0.

Theorem 2

Suppose that f, g:G → ℂ are continuous functions such that the function xKf(xk(y))dmK(k)f(x)g(y) 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 F(x,y)=Kf(xk(y))dmK(k)f(x)g(y) for each x, y in G. Then F : G × G → ℂ is continuous and it satisfies |F(x, y)| ≤ A(y) with some function A : G → ℂ for each x, y in G.

Substituting yl(z) for y and using the fact that lF(x, yl(z)) is continuous, hence integrable on K, we have, by Lemma 1 KKf(xk(y)l(z))dmK(k)dmK(l)f(x)Kg(yl(z))dmK(l)=KKf(xk(yl(z))dmK(k)dmK(l)f(x)Kg(yl(z))dmK(l)=KF(x,yl(z))dmK(l).(4)

On the other hand, substituting xk(y) for x and z for y we obtain KKf(xk(y)l(z))dmK(k)dmK(l)g(z)Kf(xk(y))dmK(k)=KF(xk(y),z)dmK(k).(5)

Moreover, we have g(z)Kf(xk(y))dmK(k)f(x)g(y)g(z)=g(z)F(x,y) which implies, together with (5) KKf(xk(y)l(z))dmK(k)dmK(l)f(x)g(y)g(z)=KF(xk(y),z)dmK(k)+g(z)F(x,y).(6)

Now, from (4) and (6) we derive f(x)(Kg(yl(z))dmK(l)g(y)g(z))=KF(x,yl(z))dmK(l)+KF(xk(y),z)dmK(k)+g(z)F(x,y) 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 Kg(yl(z))dmK(l)=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 : G → ℂ is a continuous function such that the function xKf(xk(y))dmK(k)f(x)f(y) is bounded on G for each y in G. Then either f is bounded or it is a generalized K-spherical function.

Now we are in the position to prove the general superstability-type result for equation (1) in the case p = 0.

Theorem 4

Suppose that f, g, h : G → ℂ are continuous functions such that h is nonzero, and the function xKf(xk(y))dmK(k)g(x)h(y) is bounded on G for each y in G. Then either g is bounded or h(e) ≠ 0 and h/h(e) is a generalized K-spherical function.

Proof

Suppose that h(e) = 0, where e is the identity of G. Then we have Kf(xk(e))dmK(k)g(x)h(e)=f(x), it follows that f is bounded, which implies immediately that the function xg(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) = 1.

We use similar ideas like above. We introduce the function F(x,y)=Kf(xk(y))dmK(k)g(x)h(y) for each x, y in G, then F : G × G → ℂ is continuous, and it satisfies |F(x,y)|A(y) for each x, y in G with some function A : G → ℂ. We have then KKf(xk(y)l(z))dmK(k)dmK(l)h(z)Kg(xk(y))dmK(k)=KF(xk(y),z)dmK(k)(7) and KKf(xk(y)l(z))dmK(k)dmK(l)g(x)Kh(yl(z))dmK(l)=KF(x,yl(z))dmK(l)(8) for each x, y, z in G and k, l in K. It follows h(z)Kg(xk(y))dmK(k)g(x)Kh(yl(z))dmK(l)=K[F(x,yk(z))F(xk(y),z)]dmK(k) for each x, y, z in G. Substituting z = e and using h(e) = 1 we obtain Kg(xk(y))dmK(k)g(x)h(y)=F(x,y)KF(xk(y),e)dmK(k), and h(z)Kg(xk(y)dmK(k)g(x)h(y)h(z)=h(z)[F(x,y)KF(xk(y),e)dmK(k)].

Adding to (7) we have KKf(xk(y)l(z))dmK(k)dmK(l)g(x)h(y)h(z)=KF(xk(y),z)dmK(k)+h(z)[F(x,y)KF(xk(y),e)dmK(k)].(9)

Finally, we subtract (8) from (9) to get g(x)(Kh(yl(z))dmK(l)h(y)h(z))=KF(xk(y),z)dmK(k)+h(z)[F(x,y)KF(xk(y),e)dmK(k)]KF(x,yk(z))dmK(k), and the right hand side is a bounded function of x. Hence if g is unbounded, then we must have Kh(yl(z))dmK(l)=h(y)h(z) for each y, z in G, which is our statement.     □

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 = 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.

Theorem 5

Suppose that G is an amenable group, K is finite and let f, g, h, p be continuous functions with f and h unbounded. Then the function (x,y)Kf(xk(y))dmK(k)g(x)h(y)p(y) is bounded if and only if we have f(x)=h(e)[φ(x)+ψ(x)]+b1(x)g(x)=φ(x)+ψ(x)h(x)=h(e)ω(x)p(x)=h(e)φ(x)+b2(x) where ω : G → ℂ is a generalized K-spherical function, b1, b2 : G → ℂ are bounded functions, h(e) is a nonzero complex number, φ : G → ℂ is a function satisfying Kφ(xk(y))dmK(k)=φ(x)ω(y)+φ(y)(10) and ψ : G → ℂ is a function satisfying Kψ(xk(y))dmk(k)=ψ(x)ω(y)(11) for each x, y in G.

Proof

As f is unbounded, hence g is unbounded, too, and the function xKf(xk(y))dmK(k)g(x)h(y) is bounded for every fixed y in G. By Theorem 4, it follows that h = cω, where c = h(e) ≠ 0, and ω is a generalized K-spherical function on G. Replacing h by h/h(e) we may suppose that h(e) = 1. Putting y = e in the condition we have that fg is bounded. Let M be a right invariant mean on G and we define φ(y)=MX[Kg(xk(y))dmK(k)g(x)ω(y)] for each y in G. Here Mx means that the mean M is applied to the expression in the bracket as a function of x while y is kept fixed. Then, since ω is a generalized K-spherical function, we have Kφ(yl(z))dmK(l)φ(y)ω(z)φ(z)=KMx[Kg(xk(yl(z))dmK(k)g(x)ω(yl(z))]dmK(l)ω(z)Mx[Kg(xk(y))dmK(k)g(x)ω(y)]Mx[Kg(xk(z))dmK(k)g(x)ω(z)]=KMx[Kg(xk(y)l(z))dmK(l)g(xk(y))ω(z)]dmK(k)KMx[Kg(xl(z))dmK(l)g(x)ω(z)]dmK(k)=0, by Lemma 1 and by the right invariance of the mean M. Now we obtain φ(y)p(y)=Mx[Kg(xk(y))dmK(k)g(x)ω(y)l(y)]=Mx[Kf(xk(y))dmK(k)g(x)ω(y)l(y)]+Mx[K(g(xk(y))f(xk(y))dmK(k)] and here both terms are bounded. It follows that lφ is bounded.

As fg is bounded we have (x,y)Kf(xk(y))dmK(k)Kg(xk(y))dmK(k) is bounded, hence we have that the function (x,y)Kg(xk(y))dmK(k)g(x)ω(y)φ(y) is bounded, too. We let |Kg(xk(y))dmK(k)g(x)ω(y)φ(y)|L for each x, y in G with some constant L. It follows |KKg(xl(y)k(z))dmK(k)dmK(l)ω(z)Kg(xl(y))dmK(l)φ(z)|L and |KKg(xl(yk(z))dmK(l)dmK(k)g(x)Kω(yk(z))dmK(k)Kφ(yk(z))dmK(k)|L.

From these two inequalities, by (10) and the property of ω, we infer |ω(z)(Kg(xl(y))dmK(l)g(x)ω(y)φ(y))|2L for each x, y, z in G. As ω = h is unbounded it follows that we have Kg(xl(y))dmK(l)=g(x)ω(y)+φ(y) for each x, y in G. Hence and from (10), we have K(g(xl(y))φ(xl(y)))dmK(l)=(g(x)φ(x))ω(y), that is, g = φ + ψ, where ψ: 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.

Acknowledgement

The research was supported by the Hungarian National Foundation for Scientific Research (OTKA), Grant No. K111651.

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About the article

Received: 2016-01-12

Accepted: 2016-03-02

Published Online: 2017-04-17


Citation Information: Open Mathematics, Volume 15, Issue 1, Pages 427–432, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2017-0038.

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© 2017 Székelyhidi. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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