Bisymmetric functionals revisited or a converse of the Fubini theorem
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Aequationes Mathematicae
Bisymmetric functionals revisited or a converse of the Fubini theorem Maciej Sablik Dedicated to Professor J´ anos Acz´el on his 95th Birthday. Abstract. We observe that bisymmetry is in fact the assertion of the Fubini theorem and we describe the form of general bisymmetric operations on some function spaces. Mathematics Subject Classification. 39B52, 43A07, 47A67. Keywords. Functional equations, Bisymmetry, Fubini theorem, Representation of bisymmetric functionals.
1. Introduction Let us observe that the well known arithmetic mean may be treated as a functional on the space of all functions from Z2 to R. In fact, if x : Z2 −→ R is an arbitrary function then the arithmetic mean M (x) is given by x(0) + x(1) , x : Z2 −→ R. 2 It is rather easy to observe that M satisfies the following conditions: (a) M is linear, (b) M (x) ≤ sup x(G) (⇐⇒ M (x) ≥ inf x(G)), (c) M is translation invariant (i.e. M (xa ) = M (x) for every a ∈ G, where xa (t) = x(ta), t ∈ G), where G is an arbitrary group, in our case G = Z2 . Slightly more generally, if we fix arbitrary n ≥ 2, then a functional M satisfying (a), (b) and (c) and defined on B(Zn ; R) (i.e. the family of all bounded functions mapping Zn into R—in fact, all the mappings from Zn into R) has to be M (x) =
M. Sablik
AEM
M (c1) = c
(1.1)
(d) continuous; (e) reflexive, i. e. for every c ∈ R, here 1 : G −→ R is defined by 1(t) = 1, t ∈ G; (f) monotonic, i. e. x ≥ 0 =⇒ M (x) ≥ 0.
(1.2)
Now, from (a) we infer that M (x) =
n−1
wk x(k),
(1.3)
k=0
and (1.1) implies n−1
wk = 1.
k=0
Moreover, because of (f) ((1.2), which is a consequence of (b)) we obtain wk ≥ 0. k∈{0,...,n−1}
Further, because of (c) we get
wk+j = wk ,
k∈{0,...,n−1} j∈{0,...,n−1}
in other words k∈{0,...,n−1}
wk =
1 . n
Hence, if M satisfies (a), (f) and (c) then it has to be the arithmetic mean. Therefore, we have to develop some other instruments to investigate means on finite groups. It turns out that linearity is in a sense too much - it implies the form (1.3). We know several means that are not linear, but bisymmetric, i.e. satisfy the functional equation M [M (x, y) , M (u, v)] = M [M (x, u) , M (y, v)] .
(B)
1.1. History In the 1940’s Acz´el [1, p. 281] proved the following theorem on the form of bisymmetric means.
Bisymmetric functionals revisited
Theorem 1.1. The quasi-arithmetic mean ϕ(x) + ϕ(y) M (x, y) = ϕ−1 2 is the general continuous, reducible on both sides,1 real solution of M [M (x00 , x01 ) , M (x10 , x11 )] = M [M (x00 , x10 ) , M (x01 , x11 )] under the additional conditions x, y, M (x, y) ∈ [a, b], M (x, x) = x
for all x ∈ [a, b],
M (x, y) = M (y, x)
for all x, y ∈ [a, b].
(Without symmetry M is given by M (x, y) = ϕ−1 ((1 − q)ϕ(x) + qϕ(y)) where q ∈ (0, 1).) Remark 1.1. Without reflexivity and symmetry M is given by M (x, y) = ϕ−1 ((αϕ(x) + βϕ(y) + γ) for some α > 0, β > 0 and γ ∈ R such that u, v ∈ [a, b] =⇒ αu + βv + γ ∈ [a, b]. (Volkmann [12], Maksa [9]). Remark 1.2. It is noteworthy that the formula expressing bisymmetry is actu
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