On Arithmetic Means of Sequences Generated by a Periodic Function
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1 Caad Math Bll Vol 4 () 1999 pp O Arithmetic Meas of Seqeces Geerated by a Periodic Fctio Giovai Fiorito Abstract I this paper we prove the covergece of arithmetic meas of seqeces geerated by a periodic fctio ϕ(x) moreover if ϕ(x) satisfies a sitable symmetry coditio we prove that their limit is ϕ() Applicatios of previos reslts are give to stdy other meas of seqeces ad the behavior of a class of recrsive series Itrodctio Let Φ be the class of real fctios defied i R ad periodic of period For every ϕ(x) Φ we cosider the seqece { } give by = ϕ() N; We deote by {A ϕ } {G ϕ } ad {Mϕ} p respectively the seqece of arithmetic meas the seqece of geometric meas ad the seqece of power meas of order p (p R + )of { } Usig a kow fact we prove that the seqeces {A ϕ } {G ϕ } ad {Mϕ} p are coverget; moreover if ϕ(x) satisfies a sitable symmetry coditio the limit of these seqeces is ϕ() From these theorems we dedce the the covergece or the divergece of the class of recrsive series ϕ that we have cosidered i [3] ad i [4]; fially we give some examples to complete the theory We itrodce ow other otatios If x is a real mber we deote as sal by [x]the greatest iteger less tha or eqal to x; we deote by {τ } the seqece defied by settig τ = [ ] N If T Q + weset= v ad sppose that is prime to v 1 Arithmetic Meas ad Applicatios 184 Theorem 1 Let ϕ(x) Φ ad if T R + Q letϕ(x)be boded ad Riemaitegrable i [ ] The 1 lim A ϕ = 1 ϕ(x) dx if T R + Q ϕ(1) + ϕ() + +ϕ() if T Q + 1 If T R + Q ad ϕ(x) satisfies a other sitable hypothesis the reslt is kow (see [ p 48 Exercise 15]) Received by the editors Jly AMS sbject classificatio: 4A5 c Caadia Mathematical Society 1999
2 O Arithmetic Meas of Seqeces 185 Proof We sppose at first that T R + Q I this case the seqece { τ } is iformly distribted i [ 1] ad therefore for a kow fact (see for example [1 p 473] ad [7 p 3 Corollary 11]) we have (1) f ( τ 1 lim + f τ + + f τ ) 1 = f(x)dx where f (x) = ϕ(x) is Riema-itegrable i [ 1] O the other had we have also ϕ(1) + ϕ() + +ϕ() A ϕ = = ϕ(τ 1)+ϕ(τ )+ +ϕ(τ ) () = f ( τ 1 + f τ + + f τ ) ad 1 1 f (x) dx = ϕ(x)dx = 1 (3) ϕ(x) dx The from (1) () ad (3) the thesis follows easily If T Q + > wehave ϕ(1) + ϕ() + +ϕ()+ +ϕ() A ϕ = (4) [ ]( [ ϕ(1) + ϕ() + +ϕ() +ϕ ] ) ϕ() = The from (4) the desired coclsio follows ad the proof is completed Remark 1 Theorem 1 caot be exteded spposig that the fctio ϕ(x) is Lebesgeitegrable Ideed let s sppose that f (x) Φ (T R + Q) ad that f (x) is boded Lebesge-itegrable i [ ] ad sch that f (x) dx > The let s cosider the fctio ϕ(x) obtaied by extedig by periodicity i R the fctio { f (x) ifx [ ] E g(x) = ifx E where E is the rage of the seqece {τ } Therefore we have: ad ϕ(x) Φ ϕ() = g(τ ) = N A ϕ = N ϕ(x) dx = f (x) dx > Corollary 1 Let ϕ(x) satisfy the hypotheses of Theorem 1; moreover let ϕ(x) satisfy the symmetry coditio ϕ(x) +ϕ( x) = ϕ(t) x [ T] The T R + the seqece {A ϕ } is coverget to ϕ() This coditio is eqivalet to sayig that the fctio ϕ(x) ϕ() is a odd fctio
3 186 Giovai Fiorito Proof We observe at first that from the symmetry coditio it follows that k N ad ϕ(kt) = ϕ() ϕ(x)+ϕ(kt x) = ϕ(t) x [ kt] Therefore if T Q + ad is eve takig ito accot that = vwehave ϕ(1) + ϕ() + +ϕ() ϕ(1) + +ϕ(vt 1) + ϕ(vt)+ϕ(vt +1)+ +ϕ(v 1) + ϕ(v) = = ϕ(t) = ϕ() If T Q + ad is odd we have ϕ(1) + ϕ() + +ϕ() = ϕ(1) + +ϕ( vt 1 = ϕ(t) = ϕ() Fially if T R + Q wehaveeasily + ϕ vt+1 ) + +ϕ(v 1) + ϕ(v) ϕ(x)dx = ϕ(t) = ϕ() ad this completes the proof Corollary Let ϕ(x) x R ad let [ϕ(x)] p (p R + ) verify the hypotheses of Corollary 1 The the seqece {Mϕ} p coverges to ϕ() Proof Sice (( ) p ( ) p ( ) p ϕ(1) + ϕ() + + ϕ() Mϕ p = )1 p it is sfficiet to otice that the seqece {A ϕ p} coverges to ( ϕ() ) p Corollary 3 Let ϕ(x) Φ if ϕ(x) > ad if T R + Q letϕ(x)be boded ad Riema-itegrable i [ ] The lim G ϕ = { 1 e ϕ(1) ϕ() ϕ() if T Q + log ϕ(x) dx if T R + Q
4 O Arithmetic Meas of Seqeces 187 Proof It is sfficiet to observe that the fctio log ϕ(x) satisfies the hypotheses of Theorem 1 ad the the seqece {A log ϕ } coverges From this ideed ad from the relatio 1 ϕ(1)ϕ() ϕ()=e log ϕ(i) i=1 N the desired coclsio follows immediately Corollary 4 Let ϕ(x) verify the hypotheses of Corollary 3; moreover let ϕ(x) verify the symmetry coditio ϕ(x)ϕ( x) = ( ϕ(t) ) x [ T] The the seqece {G ϕ } coverges to ϕ() Proof It is sfficiet to observe that the fctio log ϕ(x) satisfies the hypotheses of Corollary 1 ad therefore the seqece {A log ϕ } coverges to log ϕ() Corollary 5 Let ϕ(x) verify the hypotheses of Corollary 4 The the series 3 ϕ coverges or diverges to + accordig as ϕ() < 1 or ϕ() > 1 Remark Theorem 1 ad the followig corollaries are sefl if the seqece { } is ot coverget becase i this case the well kow theorems of Cesàro caot be applicable This case really happes for example if ϕ(x) Φ C (R) T R + Q ad ϕ(x) is ot costat Ideed let x 1 x ] [ ad sch that ϕ(x 1 ) <ϕ(x ) The by the cotiity of ϕ(x) thereexistsδ>schthat ad moreover we have [x 1 δ x 1 + δ] [ ] [x δ x + δ] [ ] [x 1 δ x 1 + δ] [x δ x + δ] = x [x 1 δ x 1 + δ] ad x [x δ x + δ] ϕ(x ) ϕ(x ) > ϕ(x ) ϕ(x 1 ) (5) O the other had for Kroecker s theorem (see for example [5 p 373]) η >thereexist m N [η + [ sch that (6) τ [x 1 δ x 1 + δ] ad τ m [x δ x + δ] 3 ϕ is the series whose terms are defied recrsively by settig { a 1 = R + a +1 = ϕ()a N
5 188 Giovai Fiorito From (5) ad (6) it follows ϕ(m) ϕ() = ϕ(τ m ) ϕ(τ ) > ϕ(x ) ϕ(x 1 ) ad this proves obviosly that the seqece { } is ot reglar Corollary 6 Let f (x) be boded ad Riema-itegrable i the iterval [a b] ad let T R + Q The 4 (7) b a f (x) dx = (b a) lim A ϕ where ϕ(x) is obtaied by extedig by periodicity i R the fctio f ( b a x + a) x [ ] Proof We have easily b a f (x) dx = b a f ( ) b a t + a dt = b a ϕ(x) dx Ad from this relatio the desired coclsio follows immediately by virte of Theorem 1 Example 1 Let ϕ 1 (x) = si x We see easily that ϕ 1 (x) satisfies the hypotheses of Corollary 1 (with T = π) the the seqece {A ϕ1 } coverges to We have so obtaied by other meas a well kow reslt (see [6 p 316 Example 5]) Example Let s cosider the fctio { x + πx + f (x) = x 3πx+π + for x [π] for x ]π π] where R Let ϕ (x) be the fctio obtaied by extedig f (x) by periodicity i R We see easily that ϕ (x) satisfies the hypotheses of Corollary 1 (with T = π) therefore the seqece {A ϕ } coverges to Example 3 Let s cosider the fctio ϕ 3 (x) = ke si x where k R + We see easily that ϕ 3 (x) satisfies the hypotheses of Corollaries 4 ad 5 (with T = π) Therefore the seqece {G ϕ3 } coverges to k ad the series ϕ 3 coverges or diverges to + accordig as 5 k < 1ork>1 4 The formla (1) may be tilized to compte a approximate vale of the itegral b a f (x) dx 5 For k = 1 the series ϕ 3 diverges to + becase the geeral term does ot ted to
6 O Arithmetic Meas of Seqeces 189 Example 4 Let x + πx + k g(x) = k x +3πx π +k for x [π] for x ]π π] where k R + ad let s cosider the fctio ϕ 4 (x) obtaied by extedig g(x) byperiodicity i R We see easily that the fctio ϕ 4 (x) satisfies the hypotheses of Corollaries 4 ad 5 (with T = π) Therefore the seqece {G ϕ4 } coverges to k ad the series ϕ 4 coverges or diverges to + accordig as 6 k < 1ork>1 Example 5 Let f (x) = arcta x x [ 3 [ ad let s cosider the fctio ϕ 5 (x) obtaied by extedig f (x) by periodicity i R We see easily that the fctio ϕ 5 (x) satisfies the hypotheses of Theorem 1 (with = 3 ) Therefore the seqece {A ϕ5 } coverges to ϕ 5 (1) + ϕ 5 () + ϕ 5 (3) 3 = π 4 + arcta 1 3 Refereces [1] NKBaryA Treatise o Trigoometric Series Vol II Pergamo Press 1964 [] R E Edwards Forier Series A Moder Itrodctio Vol I Holt Riehart ad Wisto Ic 1967 [3] G Fiorito R Msmeci ad M Strao Diophatie approximatios ad covergece of series i Baach spaces Matematiche (Cataia) XLVIII(1993) Fasc II [4] Uiforme distribzioe ed applicazioi ad a classe di serie ricorreti Matematiche (Cataia) XLVIII(1993) Fasc I [5] G Hardy ad E Wright A itrodctio to the theory of mbers Claredo Press Oxford 1954 [6] K Kopp Theory ad applicatio of ifiite series Hafer Pblishig Compay New York 1971 [7] L Kipers ad H Niederreiter Uiform distribtio of seqeces J Wiley & Sos New York 1974 Dipartimeto di Matematica Uiversità di Cataia viale Adrea Doria 6 Cataia I-9515 Italy Fiorito@dipmatictit 6 For k = 1 the problem of determiig the behavior of the series ϕ 4 is ope
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