SARAJEVO JOURNAL OF MATHEMATICS Vol.5 (17 (2009, 3 12 RIEMANN-LIOUVILLE AND CAPUTO FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE GEORGE A. ANASTASSIOU Abstrct. Here re estblished vrious tight probbilistic inequlities tht give nerly best estimtes for the Csiszr s f-divergence. These involve Riemnn-Liouville nd Cputo frctionl derivtives of the directing function f. Also lower bound is given for the Csiszr s distnce. The Csiszr s discrimintion is the most essentil nd generl mesure for the comprison between two probbility mesures. This is continution of [4]. 1. Preliminries Throughout this pper we use the following. I Let f be convex function from (0, + into R which is strictly convex t 1 with f (1 = 0. Let (, A, λ be mesure spce, where λ is finite or σ-finite mesure on (, A. And let µ 1, µ 2 be two probbility mesures on (, A such tht µ 1 λ, µ 2 λ (bsolutely continuous, e.g. λ = µ 1 + µ 2. Denote by p = dµ 1 dλ, q = dµ 2 dλ the (densities Rdon-Nikodym derivtives of µ 1, µ 2 with respect to λ. Here we ssume tht The quntity 0 < p q Γ f (µ 1, µ 2 = b,.e. on nd 1 b. q (x f ( p (x dλ (x, (1 q (x ws introduced by I. Csiszr in 1967, see [7], nd is clled f-divergence of the probbility mesures µ 1 nd µ 2. By Lemm 1.1 of [7], the integrl (1 is well-defined nd Γ f (µ 1, µ 2 0 with equlity only when µ 1 = µ 2. In [7] the uthor without proof mentions tht Γ f (µ 1, µ 2 does not depend on the choice of λ. 2000 Mthemtics Subject Clssifiction. 26A33, 26D15, 28A25, 60E15. Key words nd phrses. Csiszr s discrimintion, Csiszr s distnce, frctionl clculus, Riemnn-Liouville nd Cputo frctionl derivtives.
4 GEORGE A. ANASTASSIOU For proof of the lst see [4], Lemm 1.1. The concept of f-divergence ws introduced first in [6] s generliztion of Kullbck s informtion for discrimintion or I-divergence (generlized entropy [11], [12] nd of Rényi s informtion gin (I-divergence of order α [13]. In fct the I-divergence of order 1 equls Γ u log2 u (µ 1, µ 2. The choice f (u = (u 1 2 produces gin known mesure of difference of distributions tht is clled κ 2 divergence, of course the totl vrition distnce µ 1 µ 2 = p (x q (x dλ (x equls Γ u 1 (µ 1, µ 2. Here by ssuming f (1 = 0 we cn consider Γ f (µ 1, µ 2 s mesure of the difference between the probbility mesures µ 1, µ 2. The f-divergence is in generl symmetric in µ 1 nd µ 2. But since f is convex nd strictly convex t 1 (see Lemm 2, [4] so is ( 1 f (u = uf (2 u nd s in [7] we get Γ f (µ 2, µ 1 = Γ f (µ 1, µ 2. (3 In Informtion Theory nd Sttistics mny other concrete divergences re used which re specil cses of the bove generl Csiszr f-divergence, e.g. Hellinger distnce D H, α-divergence D α, Bhttchryy distnce D B, Hrmonic distnce D Hα, Jeffrey s distnce D J, tringulr discrimintion D, for ll these see, e.g. [5], [9]. The problem of finding nd estimting the proper distnce (or difference or discrimintion of two probbility distributions is one of the mjor ones in Probbility Theory. The bove f-divergence mesures in their vrious forms hve been lso pplied to Anthropology, Genetics, Finnce, Economics, Politicl Science, Biology, Approximtion of Probbility distributions, Signl Processing nd Pttern Recognition. A gret inspirtion for this rticle hs been the very importnt monogrph on the topic by S. Drgomir [9]. II Here we follow [8]. We strt with Definition 1. Let ν 0, the opertor J ν, defined on L 1 (, b by J ν f (x := 1 Γ (ν x (x t ν 1 f (t dt (4 for x b, is clled the Riemnn-Liouville frctionl integrl opertor of order ν. For ν = 0, we set J 0 := I, the identity opertor. Here Γ stnds for the gmm function.
FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE 5 Let α > 0, f L 1 (, b,, b R, see [8]. Here [ ] stnds for the integrl prt of the number. We define the generlized Riemnn-Liouville frctionl derivtive of f of order α by ( D α 1 d m s f (s := (s t m α 1 f (t dt, Γ (m α ds where m := [α] + 1, s [, b], see lso [1], Remrk 46 there. In ddition, we set We need D 0 f := f, J α f := D α f, if α > 0, D α f := J α f, if 0 < α 1, D n f = f (n, for n N. (5 Definition 2. ([3] We sy tht f L 1 (, b hs n L frctionl derivtive D α f (α > 0 in [, b],, b R, iff D α k f C ([, b], k = 1,..., m := [α]+ 1, nd D α 1 f AC ([, b] (bsolutely continuous functions nd D α f L (, b. Lemm 3. ([3] Let β > α 0, f L 1 (, b,, b R, hve L frctionl derivtive D β f in [, b], let D β k f ( = 0 for k = 1,..., [β] + 1. Then D α f (s = 1 Γ (β α s (s t β α 1 D β f (t dt, s [, b]. (6 Here D α f AC ([, b] for β α 1, nd D α f C ([, b] for β α (0, 1. Here AC n ([, b] is the spce of functions with bsolutely continuous (n 1-st derivtive. We need to mention Definition 4. ([8] Let ν 0, n := ν, is ceiling of the number, f AC n ([, b]. We cll Cputo frctionl derivtive D ν f (x := 1 Γ (n ν x (x t n ν 1 f (n (t dt, x [, b]. (7 The bove function D ν f (x exists lmost everywhere for x [, b]. We need Proposition 5. ([8] Let ν 0, n := ν, f AC n ([, b]. Then D ν f exists iff the generlized Riemnn-Liouville frctionl derivtive D ν f exists.
6 GEORGE A. ANASTASSIOU Proposition 6. ([8] Let ν 0, n := ν. Assume tht f is such tht both D ν f nd D ν f exist. Suppose tht f (k ( = 0 for k = 0, 1,..., n 1. Then In conclusion D ν f = D ν f. (8 Corollry 7. ([2] Let ν 0, n := ν, f AC n ([, b], D ν f exists or D ν f exists, nd f (k ( = 0, k = 0, 1,..., n 1. Then We need D ν f = D ν f. (9 Theorem 8. ([2] Let ν 0, n := ν, f AC n ([, b] nd f (k ( = 0, k = 0, 1,..., n 1. Then We lso need f (x = 1 Γ (ν x (x t ν 1 D ν f (t dt. (10 Theorem 9. ([2] Let ν γ + 1, γ 0. Cll n := ν. Assume f AC n ([, b] such tht f (k ( = 0, k = 0, 1,..., n 1, nd D f ν L (, b. Then D f γ AC ([, b], nd D γ f (x = 1 Γ (ν γ x (x t ν γ 1 D ν f (t dt, x [, b]. (11 Theorem 10. ([2]Let ν γ +1, γ 0, n := ν. Let f AC n ([, b] such tht f (k ( = 0, k = 0, 1,..., n 1. Assume Df ν (x R, x [, b], nd Df ν L (, b. Then Df γ AC ([, b], nd D γ f (x = 1 Γ (ν γ x (x t ν γ 1 D ν f (t dt, x [, b]. (12 2. Results Here f nd the whole setting is s in 1. Preliminries (I. We present first results regrding the Riemnn-Liouville frctionl derivtive. Theorem 11. Let β > 0, f L 1 (, b, hve L frctionl derivtive D β f in [, b], let D β k f ( = 0 for k = 1,..., [β] + 1. Also ssume 0 < p(x q(x b,.e. on, < b. Then D β f,[,b] Γ f (µ 1, µ 2 q (x 1 β (p (x q (x β dλ (x. (13 Γ (β + 1
FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE 7 Proof. By (6, α = 0, we get Then f (s = 1 s (s t β 1 D β f (t dt, ll s b. (14 s f (s 1 (s t β 1 D β f (t dt D β f s,[,b] (s t β 1 dt D β f,[,b] (s β = β D β f,[,b] = (s β, ll s b. (15 Γ (β + 1 I.e. we hve tht D β f,[,b] f (s (s β, ll s b. (16 Γ (β + 1 Consequently we obtin Γ f (µ 1, µ 2 = q (x f D β f,[,b] Γ (β + 1 D β f,[,b] = Γ (β + 1 proving the clim. Next we give n L δ result. ( p (x dλ (x q (x ( p (x β q (x q (x dλ (x q (x 1 β (p (x q (x β dλ (x, (17 Theorem 12. Sme ssumptions s in Theorem 11. Let γ, δ > 1 : 1 γ + 1 δ = 1 nd γ (β 1 + 1 > 0. Then D β f δ,[,b] Γ f (µ 1, µ 2 (γ (β 1 + 1 1/γ q (x 2 β 1 γ (p (x q (x β 1+ 1 γ dλ (x. (18
8 GEORGE A. ANASTASSIOU Proof. By (6, α = 0, we get gin Hence Tht is f (s = 1 f (s 1 s s (s t β 1 D β f (t dt, ll s b. (19 (s t β 1 D β f (t dt 1 ( s 1/γ ( s (s t γ(β 1 dt D β f (t δ 1/δ dt D β f δ,[,b] (s β 1+ 1 γ, ll s b. (20 1/γ (γ (β 1 + 1 D β f f (s δ,[,b] Consequently we obtin ( Γ f (µ 1, µ 2 q p f dλ q D β f δ,[,b] (γ (β 1 + 1 1/γ D β f δ,[,b] = (γ (β 1 + 1 1/γ proving the clim. An L 1 estimte follows. (s β 1+ 1 γ, ll s b. (21 1/γ (γ (β 1 + 1 ( p β 1+ 1 q q γ dλ q 2 β 1 γ (p q β 1+ 1 γ dλ, (22 Theorem 13. Sme ssumptions s in Theorem 11. Let β 1. Then D β f ( 1,[,b] Γ f (µ 1, µ 2 (q (x 2 β (p (x q (x β 1 dλ (x (23 Proof. By (19 we hve f (s 1 s (s t β 1 D β f (t dt (s b β 1 D β f (t dt = (s β 1 D β f. (24 1,[,b]
I.e. FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE 9 f (s (s β 1 D β f for ll s in [, b]. Therefore ( Γ f (µ 1, µ 2 q p D β f f dλ q D β f ( 1,[,b] = proving the clim. 1,[,b], (25 1,[,b] ( p β 1 q q dλ q 2 β (p q β 1 dλ, (26 We continue with results regrding the Cputo frctionl derivtive. Theorem 14. Let ν > 0, n := ν, f AC n ([, b] nd f (k ( = 0, k = 0, 1,..., n 1. Assume D f ν L (, b, 0 < p(x q(x b,.e. on, < b. Then Γ f (µ 1, µ 2 Dν f,[,b] q (x 1 ν (p (x q (x ν dλ (x. (27 Γ (ν + 1 Proof. Similr to Theorem 11, using Theorem 8. Next we give n L δ result. Theorem 15. Assume ll s in Theorem 14. Let γ, δ > 1 : 1 γ + 1 δ = 1 nd γ (ν 1 + 1 > 0. Then D ν f δ,[,b] Γ f (µ 1, µ 2 Γ (ν (γ (ν 1 + 1 1/γ q (x 2 ν 1 γ (p (x Proof. Similr to Theorem 12, using Theorem 8. It follows n L 1 estimte. q (x ν 1+ 1 γ dλ (x. (28 Theorem 16. Assume ll s in Theorem 14. Let ν 1. Then ( Γ f (µ 1, µ 2 Dν f 1,[,b] (q (x 2 ν (p (x q (x ν 1 dλ (x. Γ (ν (29 Proof. Similr to Theorem 13, using Theorem 8. Regrding gin the Riemnn-Liouville frctionl derivtive we need:
10 GEORGE A. ANASTASSIOU Corollry 17. Let ν 0, n := ν, f AC n ([, b], Df ν (x R, x [, b], f (k ( = 0, k = 0, 1,..., n 1. Then f (x = 1 Γ (ν x Proof. By Corollry 7 nd Theorem 8. (x t ν 1 D ν f (t dt. (30 We continue with results gin regrding the Riemnn-Liouville frctionl derivtive. Theorem 18. Let ν > 0, n := ν, f AC n ([, b], Df ν (x R, x [, b], f (k ( = 0, k = 0, 1,..., n 1. Assume Df ν L (, b, 0 < p(x q(x b,.e. on, < b. Then Γ f (µ 1, µ 2 Dν f,[,b] q (x 1 ν (p (x q (x ν dλ (x. (31 Γ (ν + 1 Proof. Similr to Theorem 11, using Corollry 17. Next we give the corresponding L δ result. Theorem 19. Assume ll s in Theorem 18. Let γ, δ > 1 : 1 γ + 1 δ = 1 nd γ (ν 1 + 1 > 0. Then D ν f δ,[,b] Γ f (µ 1, µ 2 Γ (ν (γ (ν 1 + 1 1/γ q (x 2 ν 1 γ (p (x Proof. Similr to Theorem 12, using Corollry 17. It follows the L 1 estimte. q (x ν 1+ 1 γ dλ (x. (32 Theorem 20. Assume ll s in Theorem 18. Let ν 1. Then ( Γ f (µ 1, µ 2 Dν f 1,[,b] (q (x 2 ν (p (x q (x ν 1 dλ (x. (33 Γ (ν Proof. Similr to Theorem 13, using Corollry 17. We need Theorem 21. (Tylor expnsion for Cputo derivtives, [8], p. 40 Assume ν 0, n = ν, nd f AC n ([, b]. Then f (x = n 1 k=0 f (k ( k! (x k + 1 Γ (ν x (x t ν 1 D ν f (t dt, x [, b]. (34
We mke FRACTIONAL APPROIMATION OF CSISZAR S f DIVERGENCE 11 Remrk 22. Let ν > 0, n = ν, nd f AC n ([, b]. If D f ν 0 over [, b], then ( 0 x By (34 then we obtin x [, b]. Hence ( p qf q Consequently we get (x t ν 1 D ν f (t dt 0 ( 0 f (x Γ f (µ 1, µ 2 We hve estblished n 1 ( k=0 n 1 ( n 1 ( k=0 f (k ( k! on [, b]. (x k, (35 f (k ( ( p k q k! q,.e. on. (36 k=0 f (k ( k! q 1 k (p q k dλ. (37 Theorem 23. Let ν > 0, n = ν, nd f AC n ([, b]. If D f ν 0 on ( 0 [, b], then Γ f (µ 1, µ 2 We finish with n 1 ( k=0 f (k ( k! ( (q (x 1 k (p (x q (x k dλ (x. (38 Remrk 24. Using Lemm 3, Theorem 9 nd Theorem 10 nd in their settings, for g ny of D α f, D γ f, D γ f, which fulfill the conditions nd ssumptions of 1. Preliminries (I, we cn find s bove similr estimtes for Γ g (µ 1, µ 2. References [1] G. Anstssiou, Frctionl Poincré type inequlities, submitted, 2007. [2] G. Anstssiou, Cputo frctionl multivrite Opil type inequlities on sphericl shells, submitted, 2007.
12 GEORGE A. ANASTASSIOU [3] G. Anstssiou, Riemnn-Liouville frctionl multivrite Opil type inequlities on sphericl shells, Bull. Allhbd Mth. Soc., 23 (2008, 65 140. [4] G. Anstssiou, Frctionl nd other pproximtion of Csiszr s f-divergence, Rend. Circ. Mt. Plermo, Serie II, Suppl. 99 (2005, 5 20. [5] N. S. Brnett, P. Cerone, S. S. Drgomir, nd A. Sofo, Approximting Csiszr s f- divergence by the use of Tylor s formul with integrl reminder, (pper #10, pp. 16, in Inequlities for Csiszr f-divergence in Informtion Theory, S. S. Drgomir (ed., Victori University, Melbourne, Austrli, 2000. On line: http://rgmi.vu.edu.u [6] I. Csiszr, Eine Informtionstheoretische Ungleichung und ihre Anwendung uf den Beweis der Ergodizität von Mrkoffschen Ketten, Mgyr Trud. Akd. Mt. Kutto Int. Közl. 8 (1963, 85 108. [7] I. Csiszr, Informtion-type mesures of difference of probbility distributions nd indirect observtions, Stud. Sci. Mth. Hung., 2 (1967, 299 318. [8] Ki Diethelm, Frctionl Differentil Equtions, on line: http://www.tu-bs.de/ diethelm/lehre/f-dgl02/fde-skript.ps.gz [9] S. S. Drgomir (Ed., Inequlities for Csiszr f-divergence in Informtion Theory, Victori University, Melbourne, Austrli, 2000. On line: http://rgmi.vu.edu.u [10] S. S. Drgomir nd C. E. M. Perce, Selected Topics on Hermite-Hdmrd Inequlities nd Applictions, Victori University, Melbourne, Adelide, Austrli, on line:http://rgmi.vu.edu.u [11] S. Kullbck nd R. Leibler, On informtion nd sufficiency, Ann. Mth. Stt., 22 (1951, 79 86. [12] S. Kullbck, Informtion Theory nd Sttistics, Wiley, New York, 1959. [13] A. Rényi, On mesures of entropy nd informtion, in Proceedings of the 4th Berkeley Symposium on Mthemticl Sttistics nd Probbility, I, Berkeley, CA, 1960, 547 561. (Received: Jnury 14, 2008 Deprtment of Mthemticl Sciences University of Memphis Memphis, TN 38152, USA E mil: gnstss@memphis.edu