ArticlePDF Available

Superstability of functional equations related to spherical functions

Authors:

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.
© 2017 Székelyhidi, published by De Gruyter Open.
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivs 3.0 License.
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
Unauthenticated
Download Date | 4/21/17 9:40 PM
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)
Unauthenticated
Download Date | 4/21/17 9:40 PM
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)
Unauthenticated
Download Date | 4/21/17 9:40 PM
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/ :
Adding to (7) we have
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/
Unauthenticated
Download Date | 4/21/17 9:40 PM
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/
Unauthenticated
Download Date | 4/21/17 9:40 PM
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.
References
[1] Dieudonné, J., Treatise on Analysis, Vol. VI., Academic Press, Inc., Harcourt Brace Jovanovich, Publishers, New York-London,
1978.
[2] Stetkær, H., d’Alembert’s equation and spherical functions, Aequationes Math., 1994, 48(2-3), 220–227.
[3] Stetkær, H., Functional equations and matrix-valued spherical functions, Aequationes Math., 2005, 69(3), 271–292.
[4] Stetkær, H., On operator-valued spherical functions, J. Funct. Anal., 2005, 224(2), 338–351.
[5] Székelyhidi, L., Stability of functional equations on hypergroups, Aequationes Math., 2015, 89(6), 1475–1483. (Erratum to:
Stability of functional equations on hypergroups, Aequationes Math., 2016, 90(2), 469–470.)
[6] Székelyhidi, L., The stability of the sine and cosine functional equations, Proc. Amer. Math. Soc., 1990, 110(1), 109–115.
[7] Székelyhidi, L., On a theorem of Baker, Lawrence and Zorzitto, Proc. Amer. Math. Soc., 1982, 84(1), 95–96.
Unauthenticated
Download Date | 4/21/17 9:40 PM
... In virtue of this theorem, as an application of Theorem 7.2, we obtain the following result on the superstability of functional equations related to spherical functions (see [18]). ...
... We mentioned above that the second equation of the system defining generalized moment function sequences is interesting on its own: we can consider it independently from the other equations. It has the form (18) f px˚yq " f pxqgpyq`gpxqf pyq . ...
... At this point we mention the open problem about the stability of the sine equation (18). Another related stability result in [25] is the following. ...
Chapter
Full-text available
This is a survey paper about functional equations on hypergroups. We show how some fundamental functional equations can be treated on some types of hypergroups. We also present stability and superstability results using invariant means and other tools.
... Fechner [11] and Gilányi [14] proved the Hyers-Ulam stability of the functional inequality (1.3). Park [18,19] defined additive ρ -functional inequalities and proved the Hyers-Ulam stability of the additive ρ -functional inequalities in Banach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors (see [6,7,9,10,18,24]). We recall a fundamental result in fixed point theory. ...
... Park [15,16,18] defined additive ρ-functional inequalities and proved the Hyers-Ulam stability of the additive ρ-functional inequalities in Banach spaces and non-Archimedean Banach spaces. The stability problems of various functional equations and functional inequalities have been extensively investigated by a number of authors (see [7,8,13,21,23]). ...
Preprint
In this paper, we introduce and solve the following additive-additive $(s,t)$-functional inequality \begin{eqnarray}\label{0.1} && \|g\left(x+y\right) -g(x) -g(y)\| +\| h(x+y) + h(x-y) -2 h(x) \| && \le \left\|s\left( 2 g\left(\frac{x+y}{2}\right)-g(x)-g(y)\right)\right\|+ \left\|t \left( 2h\left(\frac{x+y}{2}\right)+ 2h \left(\frac{x-y}{2}\right)- 2h (x)\right) \right\| , \nonumber \end{eqnarray} where $s$ and $t$ are fixed nonzero complex numbers with $|s| <1$ and $ |t| <1$. Using the direct method and the fixed point method, we prove the Hyers-Ulam stability of Lie bracket derivation-derivations in complex Banach algebras, associated to the additive-additive $(s,t)$-functional inequality (\ref{0.1}) and the following functional inequality \begin{eqnarray} \label{0.2}\| [g, h](xy)-[g,h](x) y- x [g,h](y) \| +\| h(xy) - h(x) y -x h(y) \| \le \varphi(x,y). \end{eqnarray}
... In particular, every solution of the Jensen additive functional equation is said to be a Jensen additive mapping. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [15,16,20,36]). ...
Chapter
This is a survey paper about the use of invariant means in the theory of Ulam type stability of functional equations. We give a summary about invariant means and we present some typical recent applications concerning stability.
Article
Full-text available
In this work the stability of the functional equations describing the addition theorems for sine and cosine is proved.
Article
Full-text available
In this paper we prove stability theorems for functional equations on hypergroups. Our proofs are based on superstability-type methods and on the method of invariant means.
Article
Full-text available
The result of J. Baker, J. Lawrence and F. Zorzitto on the stability of the equation $f(x + y) = f(x)f(y)$ is generalized by proving the following theorem: if $G$ is a semigroup and $V$ is a right invariant linear space of complex valued functions on $G$, and if $f, m$ are complex valued functions on $G$ for which the function $x \rightarrow f(xy) - f(x)m(y)$ belongs to $V$ for every $y$ in $G$, then either $f$ is in $V$ or $m$ is exponential.
Article
Full-text available
We show that any solution of a certain functional equation, generalizing such equations as Wilsons, Levi-Civits, Gajdas and the sine addition equation, can be expressed in terms of a matrix-valued spherical function. We compute this matrix-valued function for solutions of various functional equations, and use it to solve the equations.
Article
Full-text available
The observation that the solutions to d'Alembert's functional equation are Z2-spherical functions onR 2 gives us a natural way of extending d'Alembert's functional equation to groups. We deduce in this setting that the general solutions are joint eigenfunctions for a system of partial differential operators, and we find a formula for the bounded solutions.
Article
The result of J. Baker, J. Lawrence and F. Zorzitto on the stability of the equation f ( x + y ) = f ( x ) f ( y ) f(x + y) = f(x)f(y) is generalized by proving the following theorem: if G G is a semigroup and V V is a right invariant linear space of complex valued functions on G G , and if f f , m m are complex valued functions on G G for which the function x → f ( x y ) − f ( x ) m ( y ) x \to f(xy) - f(x)m(y) belongs to V V for every y y in G G , then either f f is in V V or m m is exponential.
Article
We consider the equation(1)in which a compact group K with normalized Haar measure dk acts on a locally compact abelian group (G,+). Let H be a Hilbert space, B(H) the bounded operators on H. Let Φ:G→B(H) any bounded solution of (0.1) with Φ(0)=I:(1) Assume G satisfies the second axiom of countability. If Φ is weakly continuous and takes its values in the normal operators, then , x∈G, where U is a strongly continuous unitary representation of G on H.(2) Assuming G discrete, K finite and the map x↦x-k·x of G into G surjective for each k∈K⧹{I}, there exists an equivalent inner product on H, such that Φ(x) for each x∈G is a normal operator with respect to it.Conditions (1) and (2) are partial generalizations of results by Chojnacki on the cosine equation.
d’Alembert’s equation and spherical functions, Aequationes Math
  • H Stetkær