Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 9 REVIEW ARTICLE Itepolatio of geealized Biaxisyetic potetials D Kua* GL `Reddy** ABSTRACT I this pape we study the chebyshev ad itepolatio eo fo a eal valued geealized biaxisyetic Potetial (GBASP which is egula i the ope hype sphee about the oigi The lowe (p q ode ad lowe geealized (p q type have bee chaacteized i tes of these appoxiatio eos Key wods: Tasfiite diaete poxiate ode Lagage itepolatio polyoials ad exteal polyoials *00 Matheatics Subject classificatio: Piay 30 E0 Secoday 4A0 o EOL fo Uivesity of Hydeabad Idia e-ail: ggleddy@yahooco **Depatet of Matheatics Addis Ababa Uivesity Addis Ababa Ethiopia
Ethiop J Educ & Sc Vol No Septebe 006 30 INTRODUCTION Let F αβ be a eal-valued egula solutio to the geealized biaxially syetic potetial equatio x + y + ( + (β + α β α x F x + x F y y F y = 0 α > β > - subject to the Cauchy data (0 y = ( x y = 0 which is satisfied a the sigula lies i the ope hype sphee ; x + y < Such fuctios with eve haoic extesios ae efeed to as geealized biaxisyetic potetials (GBASP havig local expasios of the fo F ( x y = =0 ites of the coplete set a R ( x y R ( x y P x y x + y = (x + y [ ] ( /( / ( = 0 3 P of biaxisyetic haoic potetials whee Let the opeato f(z = =0 K uiquely associated eve aalytic fuctio z P ae Jacobi Polyoials ([] [5] = 0 a z = x + iy C/ oto GBASP F α β ( x y = a R ( x y Followig McCoy [] fo Kaoowide s itegal fo Jacobi polyoials F ( x y π = K α β (f = 0 0 f ( ς µ α β (t s dsdt
Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 3 Whee µ α β (t s = γ ζ = x y t ixyt + cos s ( t α - β - t β+ (sis α γ = (α + (α - β + β The ivese opeato K applies othogoality of Jacobi polyoials ([] p8 ad Poisso Keel ([] p to uiquely defie the tasfo f(z = + K (F αβ β = α ( ξ ( ξ whee υ αβ ( I ξ = S αβ (Iξ ( ξ α ( + ξ β I α ( + I S αβ (I ξ = η α β F + + η αβ = (α + β + / α+β+ (α+ (β + z F υ α β ( ξ dξ β α + β + α + β + 3 I( + ξ ; ; β + ; ( + I Hee the oalizatio K αβ ( = K ( = is tae place The Keel S αβ (I ξ is aalytic o I < fo - ξ Let E be a copact set is the coplex plae ad ξ ( = {ξ 0 ξ ξ } be a syste of ( + poits of the set E such that V(ξ ( = o j ξ j ξ ad (j (ξ ( = = 0 j ξ j ξ j = 0 Agai let η ( = {η 0 η η η } be the syste of ( + poits i E such that V V(η ( = sup ξ ( E V(ξ ( ad (0 (η ( (j (η ( fo j = Such a syste always exists ad is called the th exteal syste of E The polyoials L j (z η ( = = 0 j z η ηj η j = 0
Ethiop J Educ & Sc Vol No Septebe 006 3 ae called Lagage exteal polyoials ad the liit d d(e = called the tasfiite diaete of E li ( + V is Let C(E (f C(E is holoophic i the iteio of E ad cotiuous o E deote the algeba of aalytic fuctios o the set E Let the Chebyshev o be defied fo f C(E ad F α β C as follows: e (f: E e (f = whee f g = ad E α β ; if f g g h sup f(x g(x x E F E (F α β = if{ F αβ - G αβ G αβ F αβ - G αβ = sup x + y = F αβ (x y G αβ (x y H } = 0 The set h cotais all eal polyoials of degee at ost ad the set H cotais all eal biaxisyetic haoic polyoials of degee at ost The opeatos K αβ ad K establish oe-oe equivalece of sets h ad H McCoy [] coected classical ode ad type of eal-valued etie fuctios GBASP F αβ ad the associate f espectively with eve polyoial appoxiatio eo defied i [- ] He obtaied the esults fo GBASP of Sato idex [3] usig the esults obtaied by Reddy [] It has bee oticed that these esults fail to copae the gowth of those etie GBASPs which have sae positive fiite ode but thei types ae ifiity Fo the view poit of icludig this class of etie GBASPS we shall utilize the cocept of poxiate ode Recetly Kua ad Kasaa [9] studied the (p q-ode ad geealized (p q-type of GBASP fo eve polyoial appoxiatio eo defied o E to iclude the eal valued etie GBASPs of slow gowth ad fast gowth These esults obviously leave a big class of eal-valued etie GBASP such as study of lowe (p q-ode ad geealized
Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 33 lowe (p q type which have ot bee cosideed ealie The ai of this pape is to exted the esults of [9] to lowe (p q-ode ad geealized lowe (p q-type The axiu odulii of GBASP ad associate ae defied as i coplex fuctio theoy M( f = ax f(z ad M( F α β = z = ax y x + = F αβ (x y A eal etie GBASP is said to be of (p q-ode ρ(p q ad lowe (p q ode λ(p q if it is of idex-pai (p q such that li sup if [ p] M ( F [ q ] = ρ( p q λ( p q ad the fuctio f(z havig (p q-ode ρ(p q (b < ρ(p q < is said to be of (p q type T(p q ad lowe (p q type t(p q if li sup if [ p ] M ( [ q ] ( whee b = if p = q b = 0 if p > q F ρ ( p q = T ( p q t( p q 0 t (p q T(p q A positive fuctio ρ p q ( defied o [ 0 0 > exp [q-] is said to be poxiate ode of a etie fuctio GBASP with idex-pai (p q if (i ρ p q ( ρ(p q as b < ρ < : (ii [q] ( ρ p q ( 0 as whee ρ p q ( deotes the deivative of ρ pq ( ad fo coveiece [q] ( = =0 i q [i] The existece of such copaiso fuctios o (p q-scale has bee established i [8] We ow defie geealized (p q-type T*(p q ad geealized lowe (p q-type t*(p q of F αβ with espect to a give poxiate ode ρ p q ( as li sup if [ p ] M ( [ q ] ( F ρ p a ( = T * ( p q t * ( p q 0 t* (p q T*(p q If the quatity t*(p q is diffeet fo zeo ad ifiite the ρ pq ( is said to be the lowe poxiate ode of a give etie fuctio GBASP ad t*(p q as its geealized lowe
Ethiop J Educ & Sc Vol No Septebe 006 34 (p q-type Clealy lowe poxiate ode ad coespodig geealized lowe (p q- type of F αβ ae ot uiquely deteied Fo exaple if we add c/ [q] 0< c < to the poxiate ode ρ p q ( the ρ p q ( + c/ [q] is also a poxiate ode satisfyig (i ad (ii ad cosequetly the geealized lowe (p q-type tus out to be e c t*(p q [ q ] ρ ( A Sice ( is a ootoically iceasig fuctio [] of fo > 0 we ca defie φ(x to be the uique solutio of the equatio [ q ] ρ ( A x = ( if ad oly if φ(x = [q-] whee A = if (p q = ( ad A = 0 othewise Cosequetly it ca be show that [] φ( ηx ( ρ p q A li = η uifoly fo evey η 0 < η < x φ( x Let E be the lagest equipotetial cuve of E defied by E = {z C/ / ϕ(z d = } (if = d the E = E whee w = ϕ(z is holoophic ad aps the ubouded copoet of the copleet of E o w > such that ϕ( = ad ϕ ( > 0 Also we set M ( F = sup F αβ (z 0 fo > ad M ( f z E sup f(z z E Soe Basic Results Lea Let F αβ be eal valued etie fuctios GBASP with ap K αβ associate f The the (p q odes ad lowe (p q-odes of F αβ ad f ae idetical Futhe the espective geealized (p q-type ad geealized lowe (p q-type of F αβ ad f ae also equal Poof: Let us coside the elatio N αβ (ϒ = ax{ η s ( γ β / ξ } Fo ay ε > 0 we have [ p58] ( M( F αβ M( f M(ε - F αβ N αβ (ε Usig ( ad defiitio of (p q-ode lowe (p q-ode geealized (p q-type ad geealized lowe (p q-type of F αβ ad f we coclude the esult
Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 35 Lea : Let F αβ be a eal valued etie fuctio GBASP of (p q-ode ρ(p q ad lowe (p q-ode λ(p q The li sup if [ p ] M ( F [ q] = ρ( p q λ( p q ad fo ρ(p q (b < ρ(p q < T*(p q ad t*ip q ae give by li sup if [ p ] M ( [ q ] ( F ρ p q ( = T * ( p q t * ( p q 0 t* (p q T*(p q Poof: Taig the defiitio of (p q-ode lowe (p q-ode geealized (p q-type ad geealized lowe (p q type of etie GBASP ito accout the poof follows o the lies of Lea i [6] Theoe A If f C(E ca be exteded to a etie fuctio with idex-pai (p q lowe (p q-ode λ(p q (b < λ(p q < ad geealized lowe (p q-type t*(p q the fo e (f thee exists a etie fuctio g(z = = 0 + e ( f z such that λ(p q f = λ(p q g ad t* (p q f = β*t*(p q f whee β* = d -ρ(p q fo q = β* = fo q > Poof: It has bee show i Lea 3 ad 4 of [6] that the fuctio g(z = = 0 + e ( f z is a etie fuctio Wiiasi [6 p 66] has poved that fo ay ε > 0 ( e (f K M ( f ε de whee K is a costat ad d > 0 is the tasfiite diaete of E Usig ( i the powe seies expasio of g(z it is ifeed that g de ε = =0 e ( f de ε + KM ( f ε de =0 e ε KM ( f de ε e ε (
Ethiop J Educ & Sc Vol No Septebe 006 36 o g z de 0( + M ( f + Thus i view of above iequality ad Lea fo p ad q = λ(p g λ(p f ad t*(p g e ε ρ(p d ρ(p t*(p f ad fo p ad q > λ(p q g λ (p q f ad t*(p q g t*(p q f Sice ε is abitay both iequalities iply that fo all (p q (3 λ(p q g λ(p q f ad β*t*(p q g t*(p q f Futhe usig the iequality M ( f a 0 g(/d we obseve that fo q = λ(p f λ(p g ad t*(p f d -ρ(p t*(p g ad fo q > λ(p q f λ(p q g ad t*(p f t*(p q g Hece fo all idex-pais (p q (4 λ(p q f λ(p q g ad t*(p q f β*t*(p q g Cobiig (3 ad (4 we have λ(p q f = λ(p q g ad t*(p q f = β*t*(p q g Theoe B: Let f(z C(E The f(z ca be exteded to a etie fuctio of lowe (p q-ode λ(p q (b < λ(p q < if ad oly if fo (p q ( (5 λ(p q = (6 λ(p q = whee l(p q = ax [P χ (l(p q] ad { } ax [P χ (l*(p q] { } [ p ] li [ q ] e ( f ad l*(p q = li [ q ] [ p ] e e ( f ( f
Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 37 such that P χ (L(p q = ad χ χ { } = L( p q χ + L( p q ax( L( p q li if q < p < if p = q = if 3 p = q if p = q = Futhe (5 ad (6 hold fo (p q = ( also povided { } be the sequece of picipal idices such that - as Poof: By Lea 3 & 4 of [6] we coclude that f C(E ca be exteded to a etie fuctio if ad oly if g(z is a etie fuctio Moeove by Theoe A f(z ad g(z have the sae lowe (p q-ode Applyig Theoe by Jueja et al [4 p 6] to the fuctio g(z = = 0 + e ( f z the esult follows Reas (a Fo E = [- ] ad (p q = ( the esult (5 icludes a Theoe by Sigh [4] ad a esult ( by Massa [0] ad i additio fo (p q = ( (6 gives Theoe 5 by Reddy [] (b Also fo E = [- ] the esults (5 ad (6 give Theoe ad by Jueja [3] fo etie fuctios of Sato gowth [3] ie (p q = (p Theoe C Let f(z C(E The f ca be exteded to a etie fuctio of (p q-ode ρ(p q (b < ρ(p q < ad geealized lowe (p q-type t*(p q (0 < t*(p q < if ad oly if (7 t*(p q = β* ax { } li φ [ p ] ( [ q ] [ q ] e ( f ρ ( p q p
Ethiop J Educ & Sc Vol No Septebe 006 38 ad futhe if the sequece of picipal idices { } satisfies - as the fo p = t * ( q ( q = β* ax { } li if φ( [ q ] [ A] e ( f ρ ( q A whee axiu is tae ove all iceasig sequece of positive iteges ad Poof: (p q = ( ρ( eρ( ρ ( /( ρ( ρ ( if if ( p q = ( ( p q = ( othewise Applyig Theoe of Kasaa et al [7] to the fuctio g(z = ad the esultig chaacteizatio of t*(p q g i tes of e (f ad the elatio t*(pqf = β*t*(p q g taig togethe pove the theoe = 0 e ( f z + Taig ρ pq ( = ρ(p q fo all > 0 ad φ(x = x ( ρ ( p q A we have the followig coollay which gives a foula fo lowe (p q-type t(p q i tes of appoxiatio eos of a etie fuctio f(z Coollay Let f(z C(E The f(z is the estictio of a etie fuctio havig (p q-ode ρ(p q (b < ρ(p q < ad lowe (p q-type t(p q (0 < t(p q < if ad oly if t( p q ( p q β* ax { } li if [ q ] e [ p ] ρ ( q A ( f O the doai E = [- ] ad fo appoxiatio eo e (f this coollay also icludes soe esults of Reddy[] espectively fo (p q = ( ad (p q = (
Itepolatio of Geealized Biaxisyetic potetials D Kua ad GL Reddy 39 3 Mai Results: Polyoial Appoxiatio of GBASP I this sectio we exaie the global existece of GBASP F αβ ad gowth of its c- o E (F αβ McCoy[] poved the followig theoe Theoe E Fo each GBASP F α β egula i the hype sphee thee is a uique ap K αβ associated with a eve fuctio f aalytic i the disc D R ad covesely Now we pove the followig Theoe Let F α β be eal valued GBASP egula i ad cotiuous o The F α β ca be exteded to a etie fuctio G ASP of lowe (p q-ode λ(p q (b < λ(p q < if ad oly if fo (p q ( 3 λ(p q = 3 λ(p q = whee l (p q = ax [p χ* (l (p q] ad { } ax [p χ * (l**(p q] { } if li [ q ] [ p ] E ( F ad l**(p q = li if [ p ] [ ] ( E F q E ( F such tht χ * χ *{ } = li if
Ethiop J Educ & Sc Vol No Septebe 006 40 Futhe (3 ad (3 hold fo (p q = ( also povide { } be the sequece of picipal idices such that - as Poof: By Theoe E F αβ is etie if ad oly if the associate f is etie Moeove lowe (p q-ode of the associate agees by Theoe B Usig Coollay ([4] p 6 ad Lea Theoe follows Theoe Let F αβ be eal valued GBASP egula i ad cotiuous o The F αβ ca be exteded to a etie fuctio GBASP of (p q-ode ρ(p q (b< ρ(p q < ad geealized lowe (p q-type t*(p q (0 < t*(p q < if ad oly if t*(p q = β* ax { } φ( li if [ q ] E [ p ] ad futhe if the sequece of picipal idices { } satisfies - as the fo p = t( q ( q = β* ax { } ( F α β ρ ( q p 3 ρ ( q A φ( li if [ A] E ( F whee axiu is tae ove all iceasig sequece of positive iteges Poof: Iequalities (4 ad (8 of [9] show that the associate f eets the sae liitig equieets as GBASP Usig Theoe C fo eve f ad Lea the poof of the theoe follows REFERENCES Asey R Othogoal Polyoials ad Special Fuctios Regioal cofeece seies i Applied Matheatics SIAM Philadelphia 975
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