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Superstability of functional equations related to spherical functions

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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.
Open Math. 2017; 15: 427–432
Open Mathematics Open Access
Research Article
László Székelyhidi*
Superstability of functional equations
related to spherical functions
DOI 10.1515/math-2017-0038
Received January 12, 2016; accepted March 2, 2016.
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: 39B82, 39B52, 43A90
1 Introduction
In this paper Cdenotes the set of complex numbers. We suppose that Gis a topological group and Kis a compact
topological group of continuous automorphisms of G. Hence, as a group Kis a subgroup of the group of Aut .G/
of all continuous automorphisms of G. We also assume that the mapping k7! k.x/ from Kinto Gis continuous
for each xin G. The normed Haar measure on Kis denoted by mK. Hence mKis right and left invariant and
mK.K/ D1. We shall consider the functional equation
Z
K
f .x k.y//d mK.k/ Dg.x/h.y/ Cp.y/ ; (1)
where f; g; h; p WG!Care continuous functions, and fis non-identically zero. Important special cases are
Z
K
f .x k.y//d mK.k/ Df .x / Cf .y / (2)
corresponding to the case hD1,fDgDp, and
Z
K
f .x k.y//d mK.k/ Df .x /f .y/ (3)
corresponding to the case pD0,fDgDh. Nonzero solutions fof the latter equation are called generalized
K-spherical functions. We note that if fis 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 [2–4]. In the
case Gis a discrete group and KD fidGgthen (1) reduces to
f .xy/ Dg.x/h.y/ Cp.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
[5–7].
*Corresponding Author: László Székelyhidi: Institute of Mathematics, University of Debrecen, Egyetem tér 1., P.O. Box 12, H 4010
Debrecen, Hungary and Department of Mathematics, University of Botswana, E-mail: lszekelyhidi@gmail.com
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428 L. Székelyhidi
2 Superstability of the functional equation (1) when pD0
The following lemma is crucial.
Lemma 1. Let fWG!Cbe continuous, then we have
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ DZ
K
Z
K
f .x k.yl .z///d mK.k/d mK.l/
for each x; y; z in G.
Proof. We apply Fubini’s Theorem and the invariance of mKto get
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ DZ
K
Z
K
f .x k.y/l .z//d mK.l /d mK.k/ D
Z
K
Z
K
f .x k.y/.k l /.z//d mK.k/d mK.l/ DZ
K
Z
K
f .x k.yl .z///d mK.k/d mK.l/:
The next theorem is about the superstability of the functional equation of type (1) where fand gare equal and
pD0.
Theorem 2. Suppose that f; g WG!Care continuous functions such that the function
x7! Z
K
f .x k.y//d mK.k/ f .x /g.y/
is bounded on Gfor each yin G. Then either fis bounded or gis a generalized K-spherical function.
Proof. We let
F .x; y / DZ
K
f .x k.y//d mK.k/ f .x /g.y/
for each x; y in G. Then FWGG!Cis continuous and it satisﬁes jF.x ; y/j  A.y / with some function
AWG!Cfor each x; y in G.
Substituting yl.z/ for yand using the fact that l7! F .x ; y l.z// is continuous, hence integrable on K, we have,
by Lemma 1
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ f .x/ Z
K
g.y l.z//d mK.l / D(4)
Z
K
Z
K
f .x k.yl .z//d mK.k/d mK.l/ f .x/ Z
K
g.y l.z//d mK.l / DZ
K
F .x; y l.z //d mK.l/:
On the other hand, substituting xk.y/ for xand zfor ywe obtain
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ g.z / Z
K
f .x k.y//d mK.k/ DZ
K
F .xk .y/; z /d mK.k/: (5)
Moreover, we have
g.z/ Z
K
f .x k.y//d mK.k/ f .x /g.y/g .z/ Dg.z/F .x; y /
which implies, together with (5)
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ f .x/g .y/g.z / DZ
K
F .xk .y/; z /d mK.k/ Cg.z /F .x; y /: (6)
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Superstability of functional equations related to spherical functions 429
Now, from (4) and (6) we derive
f .x/Z
K
g.y l.z//d mK.l / g.y /g.z/D Z
K
F .x; y l.z //d mK.l/ CZ
K
F .xk .y/; z /d mK.k/ Cg.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 fis unbounded,
then we must have Z
K
g.y l.z//d mK.l / Dg.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 fWG!Cis a continuous function such that the function
x7! Z
K
f .x k.y//d mK.k/ f .x /f .y/
is bounded on Gfor each yin G. Then either fis 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 pD0.
Theorem 4. Suppose that f; g; h WG!Care continuous functions such that his nonzero, and the function
x7! Z
K
f .x k.y//d mK.k/ g.x/h.y /
is bounded on Gfor each yin G. Then either gis bounded or h.e/ ¤0and h= h.e/ is a generalized K-spherical
function.
Proof. Suppose that h.e/ D0, where eis the identity of G. Then we have
Z
K
f .x k.e//d mK.k/ g.x/h.e/ Df .x/;
it follows that fis bounded, which implies immediately that the function x7! g.x/h.y / is bounded for each y,
too. As h¤0we infer that gis bounded. This means that we may assume that h.e/ ¤0. In this case, obviously, we
may replace hby h=h.e/, that is we assume h.e/ D1.
We use similar ideas like above. We introduce the function
F .x; y / DZ
K
f .x k.y//d mK.k/ g.x/h.y /
for each x; y in G, then FWGG!Cis continuous, and it satisﬁes
jF .x; y /j  A.y/
for each x; y in Gwith some function AWG!C. We have then
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ h.z/ Z
K
g.x k.y//d mK.k/ DZ
K
F .xk .y/; z /d mK.k/ (7)
and Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ g.x / Z
K
h.yl .z//d mK.l/ DZ
K
F .x; y l.z //d mK.l/ (8)
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430 L. Székelyhidi
for each x; y ; z in Gand k ; l in K. It follows
h.z/ Z
K
g.x k.y//d mK.k/ g .x/ Z
K
h.yl .z//d mK.l/ DZ
K
ŒF .x; y k.z// F .xk.y /; z /d mK.k/
for each x; y ; z in G. Substituting zDeand using h.e/ D1we obtain
Z
K
g.x k.y//d mK.k/ g.x/h.y / DF .x ; y/ Z
K
F .xk .y/; e /d mK.k/ ;
and
h.z/ Z
K
g.x k.y/d mK.k/ g.x/h.y /h.z/ Dh.z/ŒF .x ; y/ Z
K
F .xk .y/; e /d mK.k/ :
Z
K
Z
K
f .x k.y/l .z//d mK.k/d mK.l/ g.x/h.y/h.z / D(9)
Z
K
F .xk .y/; z /d mK.k/ Ch.z/ŒF .x ; y / Z
K
F .xk .y/; e /d mK.k/:
Finally, we subtract (8) from (9) to get
g.x/Z
K
h.yl .z//d mK.l/ h.y/h.z/DZ
K
F .xk .y/; z /d mK.k/C
h.z/ŒF .x ; y/ Z
K
F .xk .y/; e /d mK.k/ Z
K
F .x; y k.z//d mK.k/ ;
and the right hand side is a bounded function of x. Hence if gis unbounded, then we must have
Z
K
h.yl .z//d mK.l/ Dh.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 hD1. 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 Gis an amenable group, Kis ﬁnite and let f; g; h; p be continuous functions with fand
hunbounded. Then the function
.x; y / 7! Z
K
f .x k.y//d mK.k/ g.x/h.y / p.y/
is bounded if and only if we have
f .x/ Dh.e/Œ'.x/ C .x/ Cb1.x/
g.x/ D'.x/ C .x/
h.x/ Dh.e/!.x/
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Superstability of functional equations related to spherical functions 431
p.x/ Dh.e/'.x/ Cb2.x/
where !WG!Cis a generalized K-spherical function, b1; b2WG!Care bounded functions, h.e/ is a nonzero
complex number, 'WG!Cis a function satisfying
Z
K
'.x k.y//d mK.k/ D'.x/!.y/ C'.y/ (10)
and WG!Cis a function satisfying
Z
K
.xk.y//d mK.k/ D .x/!.y/ (11)
for each x; y in G.
Proof. As fis unbounded, hence gis unbounded, too, and the function
x7! Z
K
f .x k.y//d mK.k/ g.x/h.y /
is bounded for every ﬁxed yin G. By Theorem 4, it follows that hDc !, where cDh.e/ ¤0, and !is a
generalized K-spherical function on G. Replacing hby h=h.e/ we may suppose that h.e/ D1. Putting yDein the
condition we have that fgis bounded. Let Mbe a right invariant mean on Gand we deﬁne
'.y/ DMxŒZ
K
g.x k.y//d mK.k/ g.x/!.y/
for each yin G. Here Mxmeans that the mean Mis applied to the expression in the bracket as a function of xwhile
yis kept ﬁxed. Then, since !is a generalized K-spherical function, we have
Z
K
'.y l.z //d mK.l/ '.y /!.z / '.z / D
Z
K
MxŒZ
K
g.x k.yl .z//d mK.k/ g .x/!.y l.z //d mK.l/
!.z/MxŒZ
K
g.x k.y//d mK.k/ g.x/!.y/ MxŒZ
K
g.x k.z//d mK.k/ g.x/!.z/ D
Z
K
MxŒZ
K
g.x k.y/l .z//d mK.l/ g.x k.y//! .z/d mK.k/
Z
K
MxŒZ
K
g.x l.z//d mK.l/ g.x /!.z /d mK.k/ D0;
by Lemma 1 and by the right invariance of the mean M. Now we obtain
'.y/ p.y/ DMxŒZ
K
g.x k.y//d mK.k/ g.x/!.y/ l.y/ D
MxŒZ
K
f .x k.y//d mK.k/ g.x/!.y/ l.y/ CMxŒZ
K
.g.xk.y// f .x k.y//d mK.k/
and here both terms are bounded. It follows that l'is bounded.
As fgis bounded we have
.x; y / 7! Z
K
f .x k.y//d mK.k/ Z
K
g.x k.y//d mK.k/
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432 L. Székelyhidi
is bounded, hence we have that the function
.x; y / 7! Z
K
g.x k.y//d mK.k/ g.x/!.y/ '.y/
is bounded, too. We let
jZ
K
g.x k.y//d mK.k/ g.x/!.y/ '.y/j  L
for each x; y in Gwith some constant L. It follows
jZ
K
Z
K
g.x l.y/k .z//d mK.k/d mK.l/ !.z / Z
K
g.x l.y//d mK.l / '.z/j  L
and
jZ
K
Z
K
g.x l.yk .z//d mK.l /d mK.k/ g.x/ Z
K
!.y k.z//d mK.k/ Z
K
'.y k.z//d mK.k/j  L:
From these two inequalities, by (10) and the property of !, we infer
j!.z/Z
K
g.x l.y//d mK.l / g.x /!.y/ '.y/j  2L
for each x; y ; z in G. As !Dhis unbounded it follows that we have
Z
K
g.x l.y//d mK.l / Dg.x /!.y/ C'.y/
for each x; y in G. Hence and from (10), we have
Z
K
.g.xl.y// '.x l.y ///d mK.l/ D.g.x / '.x //!.y/;
that is, gD'C , where WG!Csatisﬁes (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 Gand Kare just some types of topological
semigroups satisfying reasonable conditions so that the existence of an invariant integral on Kand – in the case of
the general equation (2) – an invariant mean on Gis guaranteed.
Acknowledgement: The research was supported by the Hungarian National Foundation for Scientiﬁc Research
(OTKA), Grant No. K111651.
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[4] Stetkær, H., On operator-valued spherical functions, J. Funct. Anal., 2005, 224(2), 338–351.
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Unauthenticated
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• H Stetkær