WEIGHTED ADDITIVE INFORMATION MEASURES

We determine all measurable functions I,G,L:[0,1]→ℝ satisfying the functional equation ∑i=1n∑j=1mI(pi,qj)=∑i=1n∑j=1mG(pi)I(qj)


WEIGHTED ADDITIVE INFORMATION MEASURES WOLFGANG SANDER
Institute for Analysis University of Braunschweig Pockelsstr.14, D 3300 Braunschweig, Germany (Received January 26, 1989 and in revised form May 25, 1990)   ABSTRACT.
We determine all measurable functions I,G,L: [ I.This functional equation has interesting applications in information theory.
KEY WORDS AND PHRASES.Weighted additive information measures of sum form type, entropies of degree (e,8).

INTRODUCTION. k
Let F k {P (pl,...,pk i=1 We say that an information measure (n,m)-weighted additive (n,m e q) if there exist weight functions Gk,L k F k R k > 2 such that Inm(P.Q) Gn(P)Im(Q + In(P)Lm(Q) (1.8) The aim of this paper is to determine all measurable triples (I,G,L) satisfying. (1.3) for a fixed pair (n,m), n > 3, m > 3 where because of the known results and the convention (I .(1 .9) Thus we determine not only all measurable functions I of (I k) (see (I .2))but also all possible choices for G and L in (I .3).Therefore the results due to Kannappan [3][4][5], Losonczi [1], Sharma and Taneja  [2,6] are special cases of our main result.Moreover, if we assume that I is not constant and that G and L are continuous then we can interpret our result in the form that, without loss of generality, we may assume that G and L in (I .3)are continuous, non zero multi- plicative functions, that is they are non zero continuous solutions of the functional equation p,q e [O,I] (1.10)

MAIN RESULTS.
We make use of the following well known result (Kannappan, [5]).Now we are ready to prove our main result which is an extension of the results, mentioned above.THEOREM 2. Let I,G,L [O, lJ R be measurable and let I be non constant.Then I,G,L satisfy (1.9) and (1.3) for a fixed pair (n,m), n 3, m 3 if, and only if they are of one of the following forms pA[cos(clog p) bsin(clog p)]
In a further step we derive a functional equation for I', G and L in which no sums will occur.Setting F(p,q) I' (p-q) G(p)I' (q) L(q)I' (p) p,q [O,1] (2.16) we get from (2.14) we get from Lemma in Kannappan [3] (This Lemma is an application of the above Lemma that F can be represented in the form Thus (2.16) and (2.17) imply I' (p'q) G(p)I' (q) + L(q)I' (p) p,q e [O,1] (2.18) since (2.17), (1.9) and (2.7) yield F(p,O) O it is enough to solve (2.18) for all p,q e (O,13.Complex-valued functional equations of this type were intensively studied by Vincze [7-9 3. From these results we get the solutions of (2.18) for p,q e (O,1] (Ebanks,   (2.24) Since I has the form ) we can derive the solutions (I,G,L) of (1.9) and (1.3) from (2.19) to (2.24).Let us first consider the case that M(p) p in (2.19)   or if b MI(p) M2(p) p in (2.20).
In both cases we get the solution (2.3).Now we assume that n ----(G(Pi) pi O for some P e F or i=I n m (L(qj) qj) O for some Q e F 3= m" Then (2.12) and (2.13) imply that I(O) O so that in all cases where G(p) p or L(p) # p W SANDER we get that I is equal to I' and thus I is not dependent upon n and m (see (2.25)).Using G(1) L(1) and the hypothesis that I is not constant we obtain from (2.19) to (2.24) the remaining solutions (2.1), (2.2), (2.4) and (2.5) Thus the Theorem is proven.
It is clear that we can obtain from Theorem 2 some new characterization theorems for information measures.For instance, we remark that the functions G and L given by (2.4) or (2.5) cannot be continuous simultaneously.Thus we get the following extension of results in Kannappan [3,4], Sharma and Taneja [2,6].
If in addition to the hypotheses of Theorem 2, G and L are continuous then the only solutions (I,G,L) of (1.9) and (1.3) are given by (2.1), (2.2) and (2.3).
Corollary 3 implies immediately the following characterization theorem Let I k be an (n,m)-weighted additive information measure where I k, G k, L k have the sum property with continuous generating functions respectively.Thus we may consider Corollary 3 as a justification for the fact that in the literature only two special forms of G and L were considered, namely (1.4) and (1.5).
On the other hand, the condition b O in (2.1) and (2.2) im- plies that in Corollary 3 we may assume without loss of generality that G and L are continuous, non zero multiplicative functions.
This result is analogous to a result concerning recursive measures of multiplicative type (Aczl and Ng, [113).
j) ------G(Pi)I(qj) + -----L(qj)I(Pi) F n, Q e F m and for a fixed pair (n,m), n > 3, m > 3 where as usual P-Q (plql ,piqj,...,pnqm) e Fnm.If in addition I k, Gk, L k have the sum property with generating functions Iin the characterization of the entropies of degree a (Losonczi, [I) f a Hk(a,1) (p) (21 a 1)-I (Pi Pi a # we follow the conventions log log2 O.log O O and oa O a e I.

(2. 5 )
Here A,B,a,b,c,d are constants and we follow the conventions Oa.cos(log O) O Oa. sin(log O) O a e .

( 2 .
21)Here a,b,c are constants and M, MI, M2: (O,1] R are measurable multiplicative functions.Let us remark that the measurable solutions of (1.10) for p,q e (O, r',s,s' are constants.Because of (1.9) we arrive at G P) tt ,B)(p) or Ik(P) Ilk(p) P e F k.Here ,B,C are real constants with A B. Finally we give two interpretations of our result.If we put b O into (2.1),(2.2) and (2.3) then we get with unchanged I(p)