## 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.

Show Summary Details# Superstability of functional equations related to spherical functions

#### Open Access

## Abstract

## 1 Introduction

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

#### Lemma 1

#### Proof

#### Theorem 2

#### Proof

#### Corollary 3

#### Theorem 4

#### Proof

## 3 Stability of the functional equation (1)

#### Theorem 5

#### Proof

## Acknowledgement

## References

## About the article

More options …# Open Mathematics

### formerly Central European Journal of Mathematics

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

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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

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* ↦ *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

In this paper we study stability properties of functional equations of type (1). The ideas are similar to those in [5–7].

The following lemma is crucial.

*Let* *f* : *G* → ℂ *be continuous*, *then we have*
$$\begin{array}{}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)=\underset{K}{\int}\underset{K}{\int}f(xk(yl(z)))d{m}_{K}(k)d{m}_{K}(l)}\end{array}$$
*for each* *x*, *y*, *z* *in* *G*.

We apply Fubini’s Theorem and the invariance of *m _{K}* to get
$$\begin{array}{}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)=\underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(l)d{m}_{K}(k)=}\\ {\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)(kl)(z))d{m}_{K}(k)d{m}_{K}(l)=\underset{K}{\int}\underset{K}{\int}f(xk(yl(z)))d{m}_{K}(k)d{m}_{K}(l).}\end{array}$$ □

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

*Suppose that* *f*, *g*:*G* → ℂ *are continuous functions such that the function*
$$\begin{array}{}{\displaystyle x\mapsto \underset{K}{\int}f(xk(y))d{m}_{K}(k)-f(x)g(y)}\end{array}$$
*is bounded on* *G* *for each* *y* *in G*. *Then either* *f* *is bounded or* *g* *is a generalized* *K*-*spherical function*.

We let
$$\begin{array}{}F(x,y)={\displaystyle \underset{K}{\int}f(xk(y))d{m}_{K}(k)-f(x)g(y)}\end{array}$$
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 *l* ↦ *F*(*x*, *yl*(*z*)) is continuous, hence integrable on *K*, we have, by Lemma 1
$$\begin{array}{}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)-f(x)\underset{K}{\int}g(yl(z))d{m}_{K}(l)=}\\ {\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(yl(z))d{m}_{K}(k)d{m}_{K}(l)-f(x)\underset{K}{\int}g(yl(z))d{m}_{K}(l)=\underset{K}{\int}F(x,yl(z))d{m}_{K}(l).}\end{array}$$(4)

On the other hand, substituting *xk*(*y*) for *x* and *z* for *y* we obtain
$$\begin{array}{}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)-g(z)\underset{K}{\int}f(xk(y))d{m}_{K}(k)=\underset{K}{\int}F(xk(y),z)d{m}_{K}(k).}\end{array}$$(5)

Moreover, we have $$\begin{array}{}{\displaystyle g(z)\underset{K}{\int}f(xk(y))d{m}_{K}(k)-f(x)g(y)g(z)=g(z)F(x,y)}\end{array}$$ which implies, together with (5) $$\begin{array}{}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)-f(x)g(y)g(z)=\underset{K}{\int}F(xk(y),z)d{m}_{K}(k)+g(z)F(x,y).}\end{array}$$(6)

Now, from (4) and (6) we derive
$$\begin{array}{}{\displaystyle f(x)(\underset{K}{\int}g(yl(z))d{m}_{K}(l)-g(y)g(z))=-\underset{K}{\int}F(x,yl(z))d{m}_{K}(l)+\underset{K}{\int}F(xk(y),z)d{m}_{K}(k)+g(z)F(x,y)}\end{array}$$
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
$$\begin{array}{}{\displaystyle \underset{K}{\int}g(yl(z))d{m}_{K}(l)=g(y)g(z)}\end{array}$$
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.

*Suppose that* *f* : *G* → ℂ *is a continuous function such that the function*
$$\begin{array}{}x\mapsto {\displaystyle \underset{K}{\int}f(xk(y))d{m}_{K}(k)-f(x)f(y)}\end{array}$$
*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.

*Suppose that* *f*, *g*, *h* : *G* → ℂ *are continuous functions such that* *h* *is nonzero*, *and the function*
$$\begin{array}{}{\displaystyle x\mapsto \underset{K}{\int}f(xk(y))d{m}_{K}(k)-g(x)h(y)}\end{array}$$
*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*.

Suppose that *h*(*e*) = 0, where *e* is the identity of *G*. Then we have
$$\begin{array}{}{\displaystyle \underset{K}{\int}f(xk(e))d{m}_{K}(k)-g(x)h(e)=f(x),}\end{array}$$
it follows that *f* is bounded, which implies immediately that the function *x* ↦ *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*) = 1.

We use similar ideas like above. We introduce the function
$$\begin{array}{}F(x,y)={\displaystyle \underset{K}{\int}f(xk(y))d{m}_{K}(k)-g(x)h(y)}\end{array}$$
for each *x*, *y* in *G*, then *F* : *G* × *G* → ℂ is continuous, and it satisfies
$$\begin{array}{}|F(x,y)|\le A(y)\end{array}$$
for each *x*, *y* in *G* with some function *A* : *G* → ℂ. We have then
$$\begin{array}{}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)-h(z)\underset{K}{\int}g(xk(y))d{m}_{K}(k)=\underset{K}{\int}F(xk(y),z)d{m}_{K}(k)}\end{array}$$(7)
and
$$\begin{array}{}{\displaystyle \underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)-g(x)\underset{K}{\int}h(yl(z))d{m}_{K}(l)=\underset{K}{\int}F(x,yl(z))d{m}_{K}(l)}\end{array}$$(8)
for each *x*, *y*, *z* in *G* and *k*, *l* in *K*. It follows
$$\begin{array}{}{\displaystyle h(z)\underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)\underset{K}{\int}h(yl(z))d{m}_{K}(l)=\underset{K}{\int}[F(x,yk(z))-F(xk(y),z)]d{m}_{K}(k)}\end{array}$$
for each *x*, *y*, *z* in *G*. Substituting *z* = *e* and using *h*(*e*) = 1 we obtain
$$\begin{array}{}{\displaystyle \underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)h(y)=F(x,y)-\underset{K}{\int}F(xk(y),e)d{m}_{K}(k),}\end{array}$$
and
$$\begin{array}{}{\displaystyle h(z)\underset{K}{\int}g(xk(y)d{m}_{K}(k)-g(x)h(y)h(z)=h(z)[F(x,y)-\underset{K}{\int}F(xk(y),e)d{m}_{K}(k)].}\end{array}$$

Adding to (7) we have $$\begin{array}{}{\displaystyle \phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}\underset{K}{\int}\underset{K}{\int}f(xk(y)l(z))d{m}_{K}(k)d{m}_{K}(l)-g(x)h(y)h(z)=}\\ {\displaystyle \underset{K}{\int}F(xk(y),z)d{m}_{K}(k)+h(z)[F(x,y)-\underset{K}{\int}F(xk(y),e)d{m}_{K}(k)].}\end{array}$$(9)

Finally, we subtract (8) from (9) to get
$$\begin{array}{}{\displaystyle g(x)(\underset{K}{\int}h(yl(z))d{m}_{K}(l)-h(y)h(z))=\underset{K}{\int}F(xk(y),z)d{m}_{K}(k)+}\\ h(z)[F(x,y)-{\displaystyle \underset{K}{\int}F(xk(y),e)d{m}_{K}(k)]-\underset{K}{\int}F(x,yk(z))d{m}_{K}(k),}\end{array}$$
and the right hand side is a bounded function of *x*. Hence if *g* is unbounded, then we must have
$$\begin{array}{}{\displaystyle \underset{K}{\int}h(yl(z))d{m}_{K}(l)=h(y)h(z)}\end{array}$$
for each *y*, *z* in *G*, which is our statement. □

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.

*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*
$$\begin{array}{}{\displaystyle (x,y)\mapsto \underset{K}{\int}f(xk(y))d{m}_{K}(k)-g(x)h(y)-p(y)}\end{array}$$
*is bounded if and only if we have*
$$\begin{array}{}{\displaystyle f(x)=h(e)[\phi (x)+\psi (x)]+{b}_{1}(x)}\\ g(x)=\phi (x)+\psi (x)\\ h(x)=h(e)\omega (x)\\ p(x)=h(e)\phi (x)+{b}_{2}(x)\end{array}$$
*where* *ω* : *G* → ℂ *is a generalized* *K*-*spherical function*, *b*_{1}, *b*_{2} : *G* → ℂ *are bounded functions*, *h*(*e*) *is a nonzero complex number*, *φ* : *G* → ℂ *is a function satisfying*
$$\begin{array}{}{\displaystyle \underset{K}{\int}\phi (xk(y))d{m}_{K}(k)=\phi (x)\omega (y)+\phi (y)}\end{array}$$(10)
*and* *ψ* : *G* → ℂ *is a function satisfying*
$$\begin{array}{}{\displaystyle \underset{K}{\int}\psi (xk(y))d{m}_{k}(k)=\psi (x)\omega (y)}\end{array}$$(11)
*for each* *x*, *y* *in* *G*.

As *f* is unbounded, hence *g* is unbounded, too, and the function
$$\begin{array}{}{\displaystyle x\mapsto \underset{K}{\int}f(xk(y))d{m}_{K}(k)-g(x)h(y)}\end{array}$$
is bounded for every fixed *y* in *G*. By Theorem 4, it follows that *h* = *c**ω*, 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 *f* – *g* is bounded. Let *M* be a right invariant mean on *G* and we define
$$\begin{array}{}{\displaystyle \phi (y)={M}_{X}[\underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)\omega (y)]}\end{array}$$
for each *y* in *G*. Here *M*_{x} 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
$$\begin{array}{}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{thinmathspace}{0ex}}{\displaystyle \underset{K}{\int}\phi (yl(z))d{m}_{K}(l)-\phi (y)\omega (z)-\phi (z)=}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{\displaystyle \underset{K}{\int}{M}_{x}[\underset{K}{\int}g(xk(yl(z))d{m}_{K}(k)-g(x)\omega (yl(z))]d{m}_{K}(l)-}\\ {\displaystyle \omega (z){M}_{x}[\underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)\omega (y)]-{M}_{x}[\underset{K}{\int}g(xk(z))d{m}_{K}(k)-g(x)\omega (z)]=}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{\displaystyle \underset{K}{\int}{M}_{x}[\underset{K}{\int}g(xk(y)l(z))d{m}_{K}(l)-g(xk(y))\omega (z)]d{m}_{K}(k)-}\\ \phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}{\displaystyle \underset{K}{\int}{M}_{x}[\underset{K}{\int}g(xl(z))d{m}_{K}(l)-g(x)\omega (z)]d{m}_{K}(k)=0,}\end{array}$$
by Lemma 1 and by the right invariance of the mean *M*. Now we obtain
$$\begin{array}{}\phantom{\rule{2em}{0ex}}\phantom{\rule{2em}{0ex}}\phantom{\rule{1em}{0ex}}{\displaystyle \phi (y)-p(y)={M}_{x}[\underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)\omega (y)-l(y)]=}\\ {\displaystyle {M}_{x}[\underset{K}{\int}f(xk(y))d{m}_{K}(k)-g(x)\omega (y)-l(y)]+{M}_{x}[\underset{K}{\int}(g(xk(y))-f(xk(y))d{m}_{K}(k)]}\end{array}$$
and here both terms are bounded. It follows that *l* − *φ* is bounded.

As *f* − *g* is bounded we have
$$\begin{array}{}{\displaystyle (x,y)\mapsto \underset{K}{\int}f(xk(y))d{m}_{K}(k)-\underset{K}{\int}g(xk(y))d{m}_{K}(k)}\end{array}$$
is bounded, hence we have that the function
$$\begin{array}{}{\displaystyle (x,y)\mapsto \underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)\omega (y)-\phi (y)}\end{array}$$
is bounded, too. We let
$$\begin{array}{}{\displaystyle |\underset{K}{\int}g(xk(y))d{m}_{K}(k)-g(x)\omega (y)-\phi (y)|\le L}\end{array}$$
for each *x*, *y* in *G* with some constant *L*. It follows
$$\begin{array}{}{\displaystyle |\underset{K}{\int}\underset{K}{\int}g(xl(y)k(z))d{m}_{K}(k)d{m}_{K}(l)-\omega (z)\underset{K}{\int}g(xl(y))d{m}_{K}(l)-\phi (z)|\le L}\end{array}$$
and
$$\begin{array}{}{\displaystyle |\underset{K}{\int}\underset{K}{\int}g(xl(yk(z))d{m}_{K}(l)d{m}_{K}(k)-g(x)\underset{K}{\int}\omega (yk(z))d{m}_{K}(k)-\underset{K}{\int}\phi (yk(z))d{m}_{K}(k)|\le L.}\end{array}$$

From these two inequalities, by (10) and the property of *ω*, we infer
$$\begin{array}{}{\displaystyle |\omega (z)(\underset{K}{\int}g(xl(y))d{m}_{K}(l)-g(x)\omega (y)-\phi (y))|\le 2L}\end{array}$$
for each *x*, *y*, *z* in *G*. As *ω* = *h* is unbounded it follows that we have
$$\begin{array}{}{\displaystyle \underset{K}{\int}g(xl(y))d{m}_{K}(l)=g(x)\omega (y)+\phi (y)}\end{array}$$
for each *x*, *y* in *G*. Hence and from (10), we have
$$\begin{array}{}{\displaystyle \underset{K}{\int}(g(xl(y))-\phi (xl(y)))d{m}_{K}(l)=(g(x)-\phi (x))\omega (y),}\end{array}$$
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.

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

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**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.

© 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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