ON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC POTENTIALS IN L p -NORM, p 1

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1 TJMM 3 (211), No. 2, ON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC POTENTIALS IN L p -NORM, p 1 HUZOOR H. KHAN AND RIFAQAT ALI Abstract. The real valued regular solutio of geeralized biaxially symmetric potetial equatio 2 F x F (2α + 1) + y2 x F x (2β + 1) F + y y =, α > β > 1 2 are called geeralized biaxisymmetric potetials. I this paper, the characterizatio of lower der ad lower type of etire GBASP F i term of their approximatio err E p (F ) i L p -m, p 1 have bee obtaied. The aalysis utilizes the Bergma ad Gilbert Itegral Operat Method to exted results from classical fuctio they o the best polyomial approximatio of aalytic fuctios of oe complex variable. 1. Itroductio Let F = F (x, y) be a real valued regular solutio to the geeralized biaxially symmetric potetial equatio 2 x (2α + 1) (2β + 1) ] + + F =, α > β > 1 y2 x x y y 2, (1) (α, β) fixed i a eighbourhood of the igi where the aalytic Cauchy data F x (, y) = F y (x, ) = is satisfied alog the sigular lies i Σ (α,β), the ope uit hypersphere. These solutios called the geeralized biaxially symmetric potetials (GBASP) ca be expaded i Σ (α,β) uiquely as i terms of the complete set R (α,β) F (x, y) = (x, y) = (x 2 + y 2 ) P (α,β) = a R (α,β) (x, y) (2) (x 2 y 2 )r 2] /P (α,β) (1), =, 1, 2,... (3) of biaxisymmetric harmoic potetials, where P (α,β) are Jacobi polyomials (1],14]), r 2 = x 2 + y 2. After quadratic trasfmatio 1] R (α,β) gives various special fuctios f suitable limits of α ad β. F example α = β = gives the zoal harmoics, so that F iterprets as a axisymmetric potetial o E 3 ad α = β = 1 2 gives the eve circular harmoics o E 2 where the iterpretatio i F = R e f, f is real aalytic. The Euler-Poisso-Darboux equatio, arisig i gas dyamics, is viewed i terms of equatio (1) after a trasfmatio (see Gilbert ad Roger 5] pp.223]). The GBASP 21 Mathematics Subject Classificatio. 3B1, 41A1. Key wds ad phrases. L p -Nm, lower der, lower type, Best polyomial approximatio err, etire fuctio ad geeralized biaxisymmetric potetials. 13

2 14 HUZOOR H. KHAN AND RIFAQAT ALI F (x, y) the suggest geeralizatios of aalytic fuctios ad have a variety of Physical iterpolatios (2, 3, 5, 7]). Some examples also ca be foud i Askey 1]. Mccoy 1] developed a operat ad its iverse, which associates with each GBASP, a uique eve aalytic fuctio. Thus, let f(z) = a z 2, z = x + iy C (4) = be the uique associated eve aalytic fuctio. The operat mappig f oto GBASP F (x, y) = a R (α,β) (x, y), α > β > 1 2, uiquely is give by = F (x, y) = H (α,β) (f) = ξ 2 = x 2 y 2 t 2 + 2i(xyt cos s) µ (α,β) (t, s) = γ (α,β) (1 t 2 ) α β 1 t 2β+1 (si s) 2α f(ξ) µ α,β (t, s)dsdt (5) γ (α,β) = 2Γ(α + 1)/Γ( 1 )Γ(α β)γ(β + 1/2). 2 The iverse operat H 1 (α,β) is give by f(z) = H 1 (α,β) (F ) = 1 the keral S (α,β) is give by 1 F rξ, r(1 ξ) 1/2 ]S(α,β) (z/r, ξ)dξ (6) S (α,,β,) (τ, ξ) = S (α,β)(τ, ξ)(1 ξ) α (1 + ξ) β, (1 r) ( α + β + +2 S (α,β) (τ, ξ) = η α,β (1 + τ)α + β F 1 ; α + β + 3 2τ(1 + ξ) ) ; β + 1; 2 2 (1 + τ) 2 η (α,β) = Γ(α + β + 2)/2 α+β,+1 Γ(α + 1)Γ(β + 1). The malizatios H (α,β) (1) = H 1 (α,β) (1) = 1 are take. The keral S (α,β)(τ, ξ) ia aalytic o τ < 1 f 1 ξ 1. The local fuctio elemets F ad f are harmoically/aalytically, cotiued by cotour defmatio by the Evelope Method 3]. It was proved 3, 4], that GBASP F is regular i the hypersphere (α,β) : x 2 + y 2 2 if ad oly if its associate f is aalytic i the disk D : x 2 + y 2 2. Further we have which ca be aalytically cotiued as f(x + i) = F (x, ), x < f(z) = F (z, ), z <. The der, type, lower der ad lower type of a etire fuctio GBASP are defied from the maximum modulus M r (F ) = { F (x, y) : x 2 + y 2 r 2 as i fuctio they 8], respectively. ρ(f ) λ(f ) T (F ) t(f ) if if log log M r (F ) log r log M r (F ) r ρ(f ).

3 ON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC We ow have Remark 1. The ders, types,lower ders ad lower types of the GBASP ad the associate are respectively equal 1]. Hece, the otatios (i) ρ = ρ(f ) = ρ(f) (ii) π = T (F ) = T (f) (iii) λ = λ(f ) = λ(f) (iv) t = t(f ) = t(f). Further we defie the approximatio errs i L p -ms. Let A p ( (α,β) ) ad a p (D ) deote the spaces of GBASP ad its associate that remai regular ad respectively aalytic o (α,β) : x 2 + y 2 2 ad D : x 2 + y 2 2 with fiite ms ( 1/q F = F dxdy) p (7) (α,β) ( 1/q, f = F p dx dy) (8) D f fixed p 1. F each iteger, defie the sets of polyomials { Φ = a k z 2k : a k real, G = {H (α,β) (P ) : P Φ k= with Φ, = {p, (z) = p (z/) : p Φ ad G, = {H (α,β) (p, ) : p, Φ,. Now the best polyomial approximates to the GBASP ad associate are defied as e (p) (f) = if { f p, : p, Φ, E (p) (F ) = if { F P, : P, G,. It is kow 15] that f each there is a extremal polyomial p, Φ, f which e (p) (f) = { f p,. I the followig it will be evidet that f each, E (p) (F ) = F P,, P, G,. It is obvious that whe = 1 i the above otatios, the subscripts are dropped. Mccoy 1] studied the growth parameters of F i terms of the errs E (F ) obtaiig global existece criterio f GBASP of Sato idex k usig the results obtaied by Reddy 12]. Also i a subsequet paper 11] he studied der ad type of F i terms of E (p) (F ). Srivastava 13], studied the growth ad polyomial approximatio of GBASP i sub m. However they did ot cosider other imptat growth parameters such as lower der, lower type i terms of E (p) (F ), which play very sigificat role i the study of growth of a etire GBASP. I this paper we have tried to fill this gap. 2. Prelimiaries Let w = φ(z) be the uivalet fuctio mappig the complemet of D oto w > 1 such that φ( ) = ad φ ( ) >. Set D r = {z : φ(z) = r, r > 1, the we have

4 16 HUZOOR H. KHAN AND RIFAQAT ALI Lemma 1. Let the F A p ( ), p 1, be a etire fuctio GBASP of der ρ, type T, lower der λ, lower type t, the ρ λ = ρ(f ) λ(f ) T ρ t ρ if if log log M r (F ) log r log M r (F ) r ρ(f ) if if log log M r (f), log r log log M r (f) r ρ(f), where M r (F ) = z Dr { F (z),, r > 1 ad M r (f) = z =r f(z) Proof. Takig the defiitio of der type, lower der ad lower type of etire GBASP ito accout with Remark 1 the proof proceeds o the lies of Lemma ]. Lemma 2. Let the f a p (D), p 1, be a etire fuctio ad r ( 1) be a fixed umber, the e (p) (f ) K M r (f)(r /r), r > 2r, (9) F all sufficietly large values of. Here K is a costat depedig o D, ad p but idepedet of ad r. Proof. Sice f is etire 11], it follows that 9] p.114], there exists a sequece of polyomials {p, p beig of degree almost, such that f(z) p (z) 3 2 M r (f) (r /r) +1 1 (r /r), z D, (1) f all sufficietly large values of ad all r > r. It is well kow 9] Chapter XII] that there exist polyomials p, (p) Φ, such that e (p) (f ) = f p (),, =, 1, 2,... (11) Lemma ow follows from (8), (11) ad (1). 3. Mai results I this sectio we shall prove our mai results. Theem 1. The etire fuctio GBASP F A p ( α,β ), p 1, is of lower der λ if ad oly if k log k 1 λ = max lim if { k log E (p) k (F ). (12) Here maximum is take over all icreasig sequeces { k of positive itegers. Proof. Let F A p ( α,β ) be a etire fuctio, the f < < 1, F A p ( α,β ). Hece, the f is etire as is F = H (α,β) (f) usig Holder s iequality to get F = F p γ p (α,β) H (α,β) f dsdt, f p dsdt. Now Applyig Fubii s theem, we get ] F p γ p (α,β) f p dxdy dsdt, x 2 +y 2 2

5 ON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC γ p (α,β) f p ds dt = πγ p (α,β) f p F π 1/p γ (α,β) f p it gives E(F p ) π 1/p γ (α,β) e p (f ). (13) Let f ay icreasig sequece { k, lim if k log k 1 log E p k (F ) = θ( k) = θ. First, let θ >. The f arbitrary ε, θ > ε >, we have ] E p k /(θ ε) k (F ) > k 1, k > k (ε). Defie the sequece ] 1/(θ ε) r k = e k 1, k = 1, 2,... f r k r r k+1, k > k, we have from (9), (13) ad above iequality. log M r (f) log E p k (F ) log(π 1/p γ(α, β) log K + k log(r/r ) = ( rk+1 e Hece f k > k > k log k+1 + k log r k k log r k O(1) θ ε ) θ ε ( rk+1 ) θ ε ( rk+1 ) θ ε log(rk /e) + log rk log r O(1) e e ( rk+1 ) θ ε ( > 1 log r ) O(1). e log log M r (f) > (θ ε) log(r k+1 ) + O(1) > (θ ε) log r + O(1) log log λ if M r (f) θ. log r The iequality obviously holds if θ =. Sice { k was ay icreasig sequece, we obtai k log k 1 λ = λ(f ) max θ( k ) = max lim if { k log E p k (F ). (14) To obtai the reverse iequality, we have f f = H 1 (α,β)(f ) 1 1 S(α, β) (F ) dξ. Applyig Holder s iequality i the above estimate, we get f p ω p α,β 1 1 (F ) p dξ

6 18 HUZOOR H. KHAN AND RIFAQAT ALI where Iview of Fubiis theem, { ω α,β = S α,β (τ, ξ) : τ, ξ 1. f p ω p (α,β) So that f a p (D ). Thus we get Thus f ay sequece{ k, x 2 +y 2 2 2ω p (α,β) F p f p 2 1/p ω (α,β) F. e p (f ) 2 1/p ω (α,β) E p (F ), f ay. lim if k log k 1 log e p lim if k (f) ] F p dx dy dξ, k log k 1 log E p (F ). By usig theem 1 of Jueja 6] f q = 2 i view of (11), we have λ = max lim if { k k log k 1 log e p k (f) max { k lim if Combiig (14) ad (15) we get (12). Hece the proof is complete. k log k 1 log E p k (F ). (15) We ca prove the above theem by imposig, a weaker coditio o the sequece {e p (f). Hece we have Theem 2. Let F A p ( α,β ), p 1 be a etire fuctio GBASP of lower der λ ad pose that {e p (f)/e p +1 (f) fm a o-decreasig fuctio of f >. The λ if log log E p (F ). (16) Proof. Proceedig as the proof of first part of Theem 1, we ca easily show that λ lim if log log E p (F ). We ow apply theem 2A ad 2B of Reddy 12] p ] with (11) f etire fuctio f. The log λ if log e p (f). Sice e p (f ) 2 1/p w (α,β) E p (F ), we get λ lim if Hece the proof is complete. log log E(F p lim if ) Fially we obtai characterizatio f the lower type of F log log e p (f) = λ. Theem 3. Let F A p ( α,β ), p 1 be a etire fuctio GBASP of lower der ρ ad lower type t. If {e p (f)/e p +1 (f) fms a o-decreasig fuctio of The ρ/. E(F p )] (17) t ρ if

7 ON THE BEST POLYNOMIAL APPROXIMATION OF GENERALIZED BIAXISYMMETRIC ρ/ Proof. Let lim if E(F p )] = η, < η <. F arbitrary ε, < ε < η ad all sufficietly large > = (ε), we have ( (η ε) ) /ρ, E(F p ) > from (9), (13) ad above iequality, we get log M r (f) ( (η ε) ) ρ log + log(r/r ) log(π 1/p γ α,β ) log K. Choose = ρ(η ε)(r/r ) ρ. The f large values of r, >, i above, we get log M r (f) (η ε)(r/r ) ρ O(1). Dividig by r ρ ad proceedig to limits as r, we get t ρ lim if lim if E p (F )] ρ/. (18) The above iequality holds if η =. To prove reverse iequality i (17), uder the give coditio o e p (f ), we have by a result of Reddy 12]Theem 4A] i of (11). ρ/ e p (f )] ρ t t ρ if Hece the proof is complete. e p (f )] ρ/ lim if Refereces ρ/. E p (F )] 1] Askey, R., Orthogoal polyomials ad sphericalfuctio, Regioal coferece Series i Applied Math. SIAM, Philadelphia, Pa, ] Colto, D.L., Partial differetial equatio i the complex domais, Research Notes i Mathematics 4, Petma, Sa Fracisco, Calif., ] Gilbert, R.P., Fuctio theetic methods i partial differetial equatio, Math. i Sci. ad Egieerig 54, Academic Press, New Yk, ] Gilbert, R.P., Costrctive methods f elliptic equatio, Lecture Notes i Math. 365, Spriger Verlag, Berli ad New Yk, ] Gilbert, R.P., Roger Newto, Eds., Aalytic methods i mathematical physics, Gdaad Breach Sciece Publ, New Yk, ] Jueja, O.P., Approximatio of a etire fuctio, J. Approx. They 11, (1974), ] Kellogg, O.D., Foudatios of potetial they, Frederic, Ugar, New Yk, ] Levi, B.Ja., Distributio of zero of etire fuctios, Trasl. Math Moographs, Vol. 5, Amer. Math. Soc., Providece, R.I, ] Markushevich, A.I., They of fuctios of complex variable 3, Pretice-Hall Ic. Eglewood Cliffs, N.J., ] Mccoy, P.A., Polyomial approximatio of geeralized biaxisymmetric potetials, J. Approx. They 25, (1979), ] Mccoy, P.A., Best L p approximatio of geeralized biaxisymmetric potetials, Proc. Amer. Math. Soc. No. 7 79, (198), ] Reddy, A.R., Approximatio of etire fuctio, J. Approx. They 3, (197), ] Srivastava, G.S., O the growth ad polyomial approximatiom of geeralized biaxisymmetric potetials, Soochow J. Math No. 4 23, (1997), ] Szego, G., Orthogoal polyomials, Amer. Math. Soc.collog. Publ.No. 22, Amer. Math. Soc., Providece, R.I., ] Walsh, J.L., Iterpolatio ad approximatio by ratioal fuctios i the complex domai, Amer. Math. Soc. R.I, ] Wiiarski, T.N., Approximatio ad iterpolatio of etire fuctios, A. Polo. Math. 23, (197),

8 11 HUZOOR H. KHAN AND RIFAQAT ALI Departmet of Mathematics Aligarh Muslim Uiversity Aligarh - 222, Idia address: huzokha@yahoo.com Departmet of Applied Mathematics Aligarh Muslim Uiversity Aligarh - 222, Idia address: rifaqat.ali1@gmail.com