Orthonormal polynomials with exponentialtype

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1 Joural of Approxiatio Theory ) Orthooral polyoials with expoetialtype weights H.S. Jug a,, R. Sakai b a Departet of Matheatics Educatio, Sugkyukwa Uiversity, Seoul , Republic of Korea b Fukai 675, Ichiba-cho, Toyota-city, Aichi , Japa Received 28 Deceber 2006; received i revised for 11 Deceber 2007; accepted 30 Deceber 2007 Couicated by Doro S Lubisky Available olie 5 Jauary 2008 Abstract Let R=, ) ad let w ρ x) := x ρ exp Qx)), where ρ > 2 1 ad Qx) C2 : R R + =[0, ) is a eve fuctio. I this paper we cosider the properties of the orthooral polyoials with respect to the weight w 2 ρ x), obtaiig bouds o the orthooral polyoials ad spacig o their zeros. Moreover, we estiate A x) ad B x) defied i Sectio 4, which are used i represetig the derivative of the orthooral polyoials with respect to the weight w 2 ρ x) Elsevier Ic. All rights reserved. MSC: 41A10; 41A17 Keywords: Expoetial weight; Orthooral polyoials; Zeros 1. Itroductio ad preliiaries Let R =, ). Let Qx) C 2 : R R + =[0, ) be a eve fuctio ad wx) = exp Qx)) be such that 0 x w 2 x) dx for all = 0, 1, 2,...Forρ > 2 1,weset w ρ x) := x ρ wx), x R. The we ca costruct the orthooral polyoials p,ρ x) = p wρ 2 ; x) of degree with respect to wρ 2 x). That is, p,ρ x)p,ρ x)w 2 ρ x) dx = δ Kroecker s delta) Correspodig author. E-ail addresses: hsu90@skku.edu H.S. Jug), ryozi@crest.oc.e.jp R. Sakai) /$ - see frot atter 2008 Elsevier Ic. All rights reserved. doi: /j.jat

2 216 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) ad p,ρ x) = γ x +, γ = γ,ρ > 0. Moreover, we deote the zeros of p,ρ x) by x,,ρ x 1,,ρ x 2,,ρ x 1,,ρ. A fuctio f : R + R + is said to be quasi-icreasig if there exists C > 0 such that fx) Cf y) for 0 xy. For ay two sequeces {b } =1 ad {c } =1 of o-zero real ubers or fuctios), we write b c if there exists a costat C>0idepedet of or x) such that b Cc for large eough. We write b c if b c ad c b. We deote the class of polyoials of degree at ost by P. Throughout C, C 1,C 2,... deote positive costats idepedet of, x, t, ad polyoials of degree at ost. The sae sybol does ot ecessarily deote the sae costat i differet occurreces. We shall be iterested i the followig subclass of weights fro [3]. Defiitio 1.1. Let Qx) : R R + be eve cotiuous fuctio ad satisfies the followig properties: a) Q x) is cotiuous i R, with Q0) = 0. b) Q x) exists ad is positive i R\{0}. c) li Qx) =. x d) The fuctio Tx):= xq x) Qx), x = 0 is quasi-icreasig i 0, ) with Tx) Λ > 1, x R + \{0}. e) There exists C 1 > 0 such that Q x) Q x) C Q x) 1 Qx) a.e.x R\{0}. The we write wx) FC 2 ). If there also exist a copact subiterval J 0) of R, ad C 2 > 0 such that Q x) Q x) C Q x) 2 a.e.x R\J, Qx) the we write wx) FC 2 +). Here are soe typical exaples of FC 2 +). Defie for α + >1, 0, ad α 0 Q l,α, x) := x exp l x α ) α exp l 0)),

3 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) where α = 0ifα = 0, otherwise α = 1 ad exp l x) := expexp...expx))...)) deotes the lth iterated expoetial. We also defie Q α x) := 1 + x ) x α 1, α > 1. The the expoets exp Q l,α, x)) ad exp Q α x)) belog to FC 2 +). I R +, we cosider aother expoetial weights. Defiitio 1.2. Let wt) = e Rt) where R : R + R +. Let Qx) = Rt), t = x 2 ad satisfies the followig properties: a) t 1/2 R t) is cotiuous i R + with liit 0 at 0 ad R0) = 0. b) R t) exists i 0, ) while Q x) is positive 0, ). c) li Rt) =. t d) The fuctio Tt):= tr t), t 0, ) Rt) is quasi-icreasig i 0, ) with Tt) Λ > 1 2, t 0, ). e) There exists C 1 > 0 such that R t) R t) C R t) 1 Rt) a.e.t 0, ). The we write w LC 2 ). If there also exist a copact subiterval J 0) of R ad C 2 > 0 such that Q x) Q x) C Q x) 2 Qx) a.e.x R\J, the we write w LC 2 +). Siilarly for ρ > 1 2,weset w ρ t) := t ρ wt), t R +. The the orthooral polyoial of degree with respect to w 2 ρ t) is deoted by p,ρt) = p w 2 ρ ; t). More precisely, p,ρt) satisfies that ad 0 p,ρ t) p,ρ t) w 2 ρ t) dt = δ p,ρ t) = γ t +, γ = γ,ρ > 0.

4 218 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) The zeros of p,ρ t) are deoted by 0 t,,ρ t 1,,ρ t 2,,ρ t 1,,ρ. Let 0 p. The L p Christoffel fuctios λ,p w ρ ; x) with a weight w ρ x) are defied by λ,p w ρ ; x) = if P P 1 Especially if p = 2, we have λ wρ 2 ; x) := if P P 1 Pw ρ p u) du/ P p x). P w ρ ) 2 u) du/p 2 x) = 1 1 j=0 p2 j,ρ x). The correspodig Christoffel fuctios λ,p w ρ ; t) with a weight w ρ are defied siilarly. The ubers λ,j = λ x j,,ρ ) ad λ,j = λ t j,,ρ ), j = 1, 2,..., are called the Christoffel ubers. Levi ad Lubisky [3] ivestigated the weight wx) FC 2 ) ad the orthooral polyoials with respect to w 2 x). IR +, they [4,5] cosidered the weight fuctios w ρ t) = t ρ wt) where wt) = e Rt) LC 2 ), Rt) = Qx), t = x 2 ad estiate the orthooral polyoials p,ρ t) with respect to the weights w ρ t). Furtherore, they proved w FC 2 ), wx) = e Qx) w LC 2 ), wt) = e Rt), Rt) = Qx), t = x 2 i [4, Defiitio 1.1 ad Lea 2.2]. If Qx) satisfies 1 Λ 1 xq x)) Q Λ 2, x) where Λ i, i = 1, 2 are costats, the we call exp Qx)) the Freud-type weight. The the class FC 2 ) cotais the Freud-type weights. For certai geeralized Freud-type weight wx), Kasuga ad Sakai [2] ivestigated the orthooral polyoials associated with w ρ x) = x ρ wx), obtaiig bouds o the orthooral polyoials, zeros, Christoffel fuctios, ad the restricted rage iequalities. I [1], we ivestigated the ifiite fiite rage iequality, a estiate for the Christoffel fuctio ad the Markov Berstei iequality with respect to the weights w ρ x) = x ρ wx), wx) FC 2 ). I this paper we cosider the properties of the orthooral polyoials with respect to the weight w ρ x) = x ρ wx) o R, wx) FC 2 ). First, we prove the relatios betwee p,ρ x) ad p,ρ t), which are siilar to the relatio betwee Herite polyoials ad Laguerre polyoials. Fro this relatios, we ivestigate the bouds o the orthooral polyoials p,ρ x) ad spacig o their zeros. Moreover, we estiate A x) ad B x) defied i Sectio 4, which are used i represetig the derivative of the orthooral polyoials with respect to the weight wρ 2x). I the followig we itroduce useful otatios. a) Mhaskar Rahaov Saff MRS) ubers a x ad ã t are defied as the positive roots of the followig equatios: x = 2 π 1 0 a x uq a x u) 1 u 2 ) 1/2 du, x > 0, t = 1 π 1 0 ã t vr ã t v) dv, t > 0. {v1 v)} 1/2

5 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) b) Let η x = xt a x )) 2/3, x > 0, η t = t Tã t )) 2/3, t > 0. c) The fuctios φ u x) ad φ u t) are defied as the followig: a2u 2 x2 φ u x) = u[a u + x + a u η u )a u x + a u η u )] 1/2, x a u, φ u a u ), a u x, t +ã u u 2 ) 1/2 ã 2u t) φ u t) = uã u t +ã u η u ) 1/2, t ã u, φ u ã u ), ã u t, φ u 0), t 0. This paper is orgaized as follows. I Sectio 2, we state the relatios betwee p,ρ x) ad p,ρ t) ad the bouds o the orthooral polyoials ad spacig o their zeros. I Sectio 3, we prove the results of Sectio 2. I Sectio 4, we estiate A x) ad B x) defied i Sectio 4, which are used i represetig the derivative of the orthooral polyoials with respect to the weight wρ 2 x). Fially Sectio 5 is a appedix cotaiig various estiates ad well kow theores fro [1,3 5]. 2. Theores I the followig theore, we state the relatios betwee p,ρ x) ad p,ρ t). Theore 2.1. Let wx) = wt) with t = x 2. The the orthooral polyoials p,ρ x) with ρ > 1 2 o R ca be etirely reduced to the orthooral polyoials i R+ as follows: For = 0, 1, 2,..., p 2,ρ x) = p,1/2ρ 1/2) t) ad p 2+1,ρ x) = x p,1/2ρ+1/2) t). These forulas are i soe respects the aalogues of [6, 5.6.1) ad 4.1.5)]. Especially, [6, 5.6.1)] shows that Herite polyoials ca be reduced to Laguerre polyoials with the paraeter α =± 2 1. Moreover, these forulas are used alost everywhere i provig the other results of Sectio 2. For zeros, we prove: Theore 2.2. Let ρ > 2 1 ad wx) FC2 +). The a) For the iiu positive zero x [/2],,ρ, x [/2],,ρ a 1 ad for the axiu zero x 1,,ρ, 1 x 1,,ρ η a. b) For 1 ad 1 j 1, x j,,ρ x j+1,,ρ φ x j,,ρ ). 2.1)

6 220 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) If we assue that wx) FC 2 ) istead, the i a), for soe costat C>0 x [/2],,ρ a 1, a 1 Cη ) x 1,,ρ a +ρ/2, ad b) holds with replaced by. I the followig theores, we ivestigate the bouds o the orthooral polyoials p,ρ x). Theore 2.3 cf. Levi ad Lubisky [3, Theore 1.17]). Let ρ > 2 1 ad let wx) FC2 ). The uiforly for 1, sup p,ρ x)wx) x + a ) ρ x 2 a 2 x R 1/4 1. Theore 2.4 cf. Levi ad Lubisky [3, Theore 1.18]). Let ρ > 2 1 ad let wx) FC2 +). The uiforly for 1 we have the followig: sup p,ρ x)wx) x + a ) ρ 1/2 a T a )) 1/6. x R If wx) FC 2 ), this estiate holds with replaced by. Recall that the Lagrage fudaetal polyoials at the zeros of p,ρ x) are polyoials l j,,ρ x) P 1, give by l j,,ρ x) = p,ρ x) x x j,,ρ )p,ρ x j,,ρ). Theore 2.5 cf. Levi ad Lubisky [3, Theore 1.19 a) ad b)]). Let wx) FC 2 +) ad ρ > 1 2. The there exists 0 such that uiforly for 0 ad 1 j, a) b) c) p,ρ w x j,,ρ) x j,,ρ + a ) ρ φ x j,,ρ ) 1 [a 2 x2 j,,ρ ] 1/4. p 1,ρ x j,,ρ ) wx j,,ρ ) ax x R l j,,ρ x)wx) x + a x j,,ρ + a ) ρ w 1 x j,,ρ ) ) ρ a 1 [a2 x2 j,,ρ ]1/4. x j,,ρ + a ) ρ 1. d) For j 1 ad x [x j+1,,ρ,x j,,ρ ], p,ρ x) wx) x + a ) ρ i{ x x j,,ρ, x x j+1,,ρ }φ x j,,ρ ) 1 [a 2 x2 j,,ρ ] 1/4. 2.2) If we assue that wx) FC 2 ) istead, the a) holds with replaced by C ad b) holds with replaced by >.

7 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Theore 2.6 cf. Levi ad Lubisky [3, Theore 13.6]). Let wx) FC 2 ). Assue pρ > 1/2 if 0 p ad ρ 0 if p =. The we have for 1, p,ρ x)wx) x + a ) ρ 1, p 4, a LpR) 1/p 1/2 log1 + T a )) 1/4, p = 4, T a )) p 1 ), p > 4. Let { Φ x) := ax η, 1 x } a ad x + := { x, x > 0, 0, x 0. Theore 2.7. Let wx) FC 2 +) ad 0 s t. Assue pρ > 2 1 if 0 p ad ρ 0 if p =. The we have for 2, Φt/4 1/p)+ x) p,ρ w)x) x + a ) ρ s L p R) Φt/4 1/p)+ x) p,ρ w)x) x + a ) ρ s L p a /2 x a 1 η )) a 1/p t/2 log if s = t ad 4 pt, a 1/p s/2 if st ad 4 pt or if pt 4, if p =. a s/2 3. Proofs of theores To prove our theores we use the results of [3 5]. Proof of Theore 2.1. For k = 0, 1, 2,..., 1, p,1/2ρ 1/2) x 2 )x 2k wρ 2 x) dx = 2 0 = 0 p,1/2ρ 1/2) x 2 )x 2k wρ 2 x) dx p,1/2ρ 1/2) t)t k w 1/2ρ 1/2) 2 t) dt = 0. I the above equatio, we have the first equality by the itegratio of eve fuctio, the secod equality by the substitutio t = x 2, ad the fial equality by the orthogoality for the polyoials of degree at ost 1. For k = 0, 1, 2,..., 1, we have p,1/2ρ 1/2) x 2 )x 2k+1 wρ 2 x) dx = 0 sice the itegrad is odd by the defiitios. O the other had, we have p 2,1/2ρ 1/2) x2 )wρ 2 x) dx = 2 0 = 0 p 2,1/2ρ 1/2) x2 )wρ 2 x) dx p 2,1/2ρ 1/2) t) w2 1/2ρ 1/2) t) dt = 1.

8 222 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Siilarly, we have for k = 0, 1, 2,...,2 ad x p,1/2ρ+1/2) x 2 )x k wρ 2 x) dx = 0 x p,1/2ρ+1/2) x 2 )) 2 w 2 ρ x) dx = 1. Therefore, the result is proved. Proof of Theore 2.2. Fro Theore 2.1 we have the followig: p 2,ρ x) = p,1/2ρ 1/2) t), p 2+1,ρ x) = xp 2,ρ+1 x) = t 1/2 p,1/2ρ+1/2) t). We oly give the proof of the case of p 2,ρ x), because for the case of p 2+1,ρ x) we see that 1 2 ρ 2 1 ) ay be replaced with 2 1 ρ ). a) Sice x,2,ρ 2 = t,,1/2ρ 1/2) ad x1,2,ρ 2 = t 1,,1/2ρ 1/2), we have the results by A.1) ad a) of Theore A.3. b) By A.9), A.2), ad A.1), we have for j = 1, 2,..., 1, x 2 j,2,ρ x2 j+1,2,ρ = t j,,1/2ρ 1/2) t j+1,,1/2ρ 1/2) φ tj,,1/2ρ 1/2) ) φ2 x j,2,ρ ) Sice for j = 1, 2,..., 1bya) x j,2,ρ + a ) x j,2,ρ ) x j,2,ρ + x j+1,2,ρ ad x,2,ρ x +1,2,ρ = 2x,2,ρ a φ 2x,2,ρ ), we have for j = 1, 2,...,, ) 1/2 xj,2,ρ 2 + a2 2. x j,2,ρ x j+1,2,ρ φ 2 x j,2,ρ ). 3.1) For j = + 1,+ 2,...,2 we also obtai 3.1) by the syetry of x j,2,ρ. For the case wx) FC 2 ), the proof is the sae as the above. Proof of Theore 2.3. First we prove the result for the eve case. Let ρ > 2 1 ad γ = 2 1 ρ 2 1 ). The γ > 2 1 ad we have by A.1) p 2,ρ x) wx) x + a ) ρ x 2 a2 2 1/4 ) ρ/2 p 2,ρ x) wx) x 2 + a2 2 x 2 a2 2 1/4 2 ) γ ) 1/4 = p,γ t) wt) t + ã 2 t + ã 2 t ã 1/4.

9 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Therefore, we have by Theore A.4 sup p 2,ρ x) wx) x + a ) ρ x 2 a2 2 x R 1/4 1. Now, we show the result for the odd case. Let γ = 2 1 ρ ). The sice p 2+1,ρ x) = xp 2,ρ+1 x) ad x + a ) 2+1 x + a ) 2, we obtai p 2+1,ρ x) wx) p 2,ρ+1 x) wx) x + a p,γ t) wt) t + ã 2 ) ρ x 2 a /4 By Theores A.4 ad A.5 ) γ+1/4 p,γ t) wt) t + ã 2 t ã 1/4 1 ad ) γ+1/4 p,γ t) wt) t + ã ã +1 ã 1/4 1, 2 x + a ) 2 ρ+1 x 2 a /4 + a2+2 2 a2 2 1/4) ) γ+1/4 t ã 1/4 + ã +1 ã 1/4). { ã t 1/4 ã +1 ã 1/4, 0 t ã /2,t ã 2, ã 1/2 Tã )) 1/6 ã 1/4 ã +1 ã 1/4, ã /2 tã 2, because ã +1 ã ã / Tã )) by A.8). O the other had, fro A.11) we have for x j,2+2,ρ+1 with ε 1 a 2 x j,2+2,ρ+1 ε 2 a 2,0ε 1 ε 2 1, p 2+1,ρ w)x j,2+2,ρ+1 ) x j,2+2,ρ+1 + a ) ρ 2+1 xj,2+2,ρ a /4 a 3/2+ρ p 2,ρ+1 w)x j,2+2,ρ+1 ) a 3/2+ρ p,γ wt j,+1,γ ) > a 3/2+ρ ã 1/2 γ 1, because ε1ã 2 t j,+1,γ ε2ã 2 by b) of Theore A.6 ad A.1). Therefore, we have sup p 2+1,ρ x) wx) x + a ) ρ 2+1 x 2 a2+1 2 x R /4 1. Proof of Theore 2.4. Let γ = 2 1 ρ 2 1 ). The A := sup p 2,ρ x) wx) x R x + a 2 2 ) ) ρ γ+1/4 sup p,γ t) wt) t + ã t R + 2.

10 224 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Now we estiate A. By Theore A.2, Theore A.4, A.1), ad A.2), there exists soe costat L>0such that ) γ+1/4 A sup p,γ t) wt) t + ã t ã 1 L η ) 2 ax ã t) 1/4 ã η ) 1/4 ) ã 1/4 1/6 Tã ) t ã 1 L η ) a 1/2 2 2T a 2 )) 1/6. For the lower bouds, we use the Berstei iequality [5, Theore )]. The ) t 1/4 1,,γ p,γ w γ t 1,,γ ) φ t 1,,γ ) = p,γ w ) t1,,γ ) φ t 1,,γ )t γ+1/4 1,,γ sup p,γ w ) ) γ+1/4 t) t + ã t R + 2 A. O the other had, we have by A.10) ad a) of Theore A.3 t 1/4 1,,γ p,γ t 1,,γ) w γ t 1,,γ ) φ t 1,,γ ) ã t 1,,γ ) 1/4 ã η ) 1/4 ã 1/4 { Tã )} 1/6 a 1/2 {T a )} 1/6. Cosequetly, we have A a 1/2 {T a )} 1/6. The case of p 2+1,ρ x) is siilar. Lea 3.1. Let w LC 2 +) ad ρ > 2 1. The uiforly for 1 2 ) ρ ) 1/2 p,ρ 0) 3.2) ã ad for t [0,t,,ρ ], ã 2 ) ρ+1 ) 1/2 p,ρ t) t t,,ρ. 3.3) ã Proof. By Theore A.5, we kow that p,ρ 0) 2 ) ρ ) 1/2. ã ã ã O the other had, sice p,ρ t) is decreasig o [0,t,,ρ] we have by the ea value property, A.10), ad a) of Theore A.3, p,ρ 0) p,ρ 2 ) ρ ) 1/2 t,,ρ) t,,ρ. ã Therefore, 3.2) is proved. Siilarly for t [0,t,,ρ ], we have by the ea value property p,ρ t) t t,,ρ p,ρ 2 ) ρ+1 t,,ρ) t t,,ρ ã ã ã ) 1/2

11 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) ad we have by A.12), p,ρ t) t t,,ρ p,ρ 2 ) ρ+1 ) 1/2 t,,ρ) t t,,ρ. ã ã Therefore, 3.3) is also proved. Proof of Theore 2.5. a) To prove a), we use A.10). Let γ = 2 1 ρ 2 1 ). The for j = 1, 2,...,2 p 2,ρ x j,2,ρ) wx j,2,ρ ) x j,2,ρ + a ) 2 ρ 2 p 2,ρ x j,2,ρ) wx j,2,ρ ) x j,2,ρ ρ = 2 p,γ w γ t j,,γ )t 3/4 j,,γ φ t j,,γ ) 1 [t j,,γ ã t j,,γ )] 1/4 t 3/4 j,γ φ 2 x j,2,ρ ) 1 xj,2,ρ 1 [x2 j,2,ρ a2 2 x2 j,2,ρ )] 1/4 x 3/2 j,2,ρ φ 2 x j,2,ρ ) 1 [a 2 2 x2 j,2,ρ ] 1/4. Here, we used a) of Theores 2.2, 2.1, A.10), A.1) A.3), ad so o. Now, let γ = 1 2 ρ ). The sice x j,2+1,ρ = x j,2,ρ+1 for j = 1, 2,...,ad p 2+1,ρ x) = xp 2,ρ+1 x) + p 2,ρ+1x), we have for j = 1, 2,...,,+ 2,...,2 + 1 by the eve case p 2+1,ρ w)x j,2+1,ρ) x j,2+1,ρ + a ρ ) p 2,ρ+1 w)x j,2,ρ+1) x j,2,ρ+1 + a ) ρ φ 2+1 x j,2+1,ρ ) 1 a x2 j,2+1,ρ ) 1/4. For x +1,2+1,ρ = 0, we obtai by Lea 3.1 ) ρ p 2+1,ρ w)0) a2+1 p 2,ρ+10) a φ 2+1 0) 1 a2+1 2 ) 1/4. a 3/2 Therefore, a) is proved. b) By the defiitio of λ x) = λ wρ 2 ; x) we have ) ρ = p,γ 0) a ) ρ λ 1 x j,,ρ) = γ 1,ρ γ,ρ p,ρ x j,,ρ)p 1,ρ x j,,ρ ).

12 226 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) So, we have p 1,ρ x j,,ρ ) = γ,ρ γ 1,ρ λ 1 x j,,ρ)p,ρ x j,,ρ)) 1. The we have by a) ad Theore A.9 p 1,ρ w)x j,,ρ ) x j,,ρ + a ) ρ = γ,ρ λ 1 γ x j,,ρ)w 2 x j,,ρ ) x j,,ρ + a ) 2ρ 1,ρ p,ρ w)x j,,ρ) x j,,ρ + a ) ρ 1 γ,ρ 1/4 a 2 γ j) x2. 1,ρ Here γ 1,ρ /γ,ρ a see Lea 4.7 i Sectio 4, ad ote that its proof is idepedet of Theore 2.5). Therefore, b) is proved. To prove c) ad d), let 1 ρ 1 ) γ := ρ + 1 ) 2 2 if is eve if is odd ad := [ 2 ]. c) First, we easily have ax l j,,ρ w)x) x + a ) ρ w 1 x j,,ρ ) x j,,ρ + a ) ρ x R l j,,ρw)x j,,ρ ) x j,,ρ + a ) ρ w 1 x j,,ρ ) x j,,ρ + a ) ρ = 1. Therefore, it reais to obtai the upper bouds. By Theore A.8, it suffices to prove the upper bouds for x a. First, suppose 1 j. The we have by Theore 2.1 B := l j,,ρ x) wx) x + a ) ρ w 1 x j,,ρ ) x j,,ρ + a ) ρ l j,,γ t) wt) t + ã 2 ) γ w 1 t j,,γ) t j,,γ + ã 2 x + x j,,ρ ) x +a /) 1/2 x 3/2 j,,ρ if = 2, x x + x j,,ρ ) x +a /) 1/2 ) γ x 5/2 j,,ρ if = For x a if x 2 x j,,ρ or x j,,ρ a /2, the x / x j,,ρ 1. Therefore, for these cases we have B 1 by A.12). Now, suppose 2 x j,,ρ x a ad x j,,ρ a /2. The

13 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) sice x x j,,ρ x j,,ρ ad x x j,,ρ x, we also have by a) ad A.4) p,ρ x) B = x x j,,ρ )p,ρ x j,,ρ) wx) x + a ) ρ w 1 x j,,ρ ) x j,,ρ + a ) ρ p,ρ x)wx) x + a ) ρ x x j,,ρ φ x j,,ρ )a 2 x2 j,,ρ )1/4 a3/2 p,ρ x)wx) x + a ) ρ x x j,,ρ. The for x a /2, we have by Theore 2.3 ad a) of Theore 2.2 B a3/2 a 2 x2 ) 1/4 1 x j,,ρ ad for x a /2, we have by Theore 2.4 ad A.5), B a3/2 x a a 1/2 T a )) 1/6 x 1. Therefore, we also have B 1 for these case. Thus, we proved for 1 j sup l j,,ρ w)x) x + a ) ρ w 1 x j,,ρ ) x j,,ρ + a ) ρ 1. Moreover, whe = 2 + 1, we have by b), A.6), Theores 2.3 ad 2.4 l +1,2+1,ρ x) wx) x + a ) ρ w 1 x +1,2+1,ρ ) x +1,2+1,ρ + a ) ρ = p 2,ρ+1x)wx) x + a ) ρ 2+1 p 2,ρ+1w) 1 a ) ρ 0) a3/2 a 3/2, x a /2 1. a 3/2 T a )) 1/6, x a /2 If we use the syetry of the zeros, the c) ca be proved for all j1 j ). d) To prove d), we use A.13), Leas A.1 ad 3.1. Suppose 1 j 1 ad let x [x j+1,,ρ,x j,,ρ ]. The we have by A.13) p,ρ x) wx) x + a ) ρ p,γ t) w γ t)t 1/4 i{ t t j,,γ, t t j+1,,γ } φ 1 t j,,γ)ã t j,,γ ) 1/4 i{ x 2 xj,,ρ 2, x2 xj+1,,ρ 2 }φ 1 x j,,ρ) x j,,ρ 1 a 2 x2 j,,ρ ) 1/4 i{ x x j,,ρ, x x j+1,,ρ }φ 1 x j,,ρ) a 2 x2 j,,ρ ) 1/4.

14 228 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Moreover, whe = 2 + 1, fro 3.3) we have for x [x +1,2+1,ρ,x,2+1,ρ ], p 2+1,ρ x) wx) x + a ) ρ = xp 2,ρ+1 x) wx) x + a ) ρ ) ρ/2 t 1/2 p,γ t) wt) t + ã t 1/2 t t 3,,γ a 3/2 2 i{ x, x x,2+1,ρ } a 3/2 ã 7/4 i{ x x +1,2+1,ρ, x x,2+1,ρ }. Whe = 2, sice fro 3.3) we have for x [0,x,2,ρ ], p 2,ρ x) wx) x + a ) 2 ρ p,1/2ρ 1/2) t) wt) t + ã ) 3/4 t t,,1/2ρ 1/2) a 3/2 ã x x,2+1,ρ, ) ρ/2 ã ) 1/2 we obtai i this case that for x [x +1,2,ρ,x,2,ρ ], p 2,ρ x) wx) x + a ) 2 ρ i{ x x +1,2,ρ, x x,2,ρ }. 2 a 3/2 If we use the syetric property d) ca be proved for + 1 j. Therefore, d) is proved copletely. Proof of Theore 2.6. Fro Theores A.8 ad 2.3 we have p,ρ w)x) x + a ) ρ p,ρ w)x) x + LpR) a ) ρ LpLa/ x a1 Lη)) a 1/2 1 x ) 1/4 a Lp La / x a 1 Lη )) 1, p 4, a 1/p 1/2 log1 + T a )) 1/4, p = 4, T a )) p 1 ), p > 4. Now, we estiate the lower bouds of p,ρ w)x) x + a ) ρ Lp. Sice by 2.2), 2.1), ad R) Lea A.1 xj,,ρ ) ρ p dx x j+1,,ρ xj,,ρ p,ρ w)x) x + a x j+1,,ρ i{ x x j,,ρ, x x j+1,,ρ } p dxφ x j,,ρ ) p x j,,ρ x j+1,,ρ p+1 φ x j,,ρ ) p a 2 x2 j,,ρ p/4 x j,,ρ x j+1,,ρ a j,,ρ) 2 x2, ) p/4 ) p/4 a 2 x2 j,,ρ

15 we have fro a) of Theore 2.2 p,ρ w)x) H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) x + a Therefore, we have the result. ) ρ LpR) > ) 1/p x1,,ρ a 2 x2) p/4 dx x,,ρ a 1 η ) a 1/p 1/2 a 1 η ) a 2 x2) p/4 dx ) 1/p 1, p 4, log1 + T a )) 1/4, p = 4, T a )) p 1 ), p > 4. Proof of Theore 2.7. First, assue 4 pt. Sice by Theore A.8 Φt/4 1/p) x) p,ρ w)x) x + a ) ρ s L p a 1 η ) x ) η t/4 1/p) p,ρ w)x) x + a ) ρ s L ps a 1 η ) x ) η t/4 1/p) p,ρ w)x) x + a ) ρ s L ps x a 1 η ))) Φt/4 1/p) x) p,ρ w)x) x + a ) ρ s ad fro Theore 2.3 for x a 1 η ), we have Φ t/4 1/p) Φt/4 1/p) x) p,ρ w)x) x) p,ρ w)x) x + a x + a ) ρ s a s/2 ) ρ s L p R) L p x a 1 η ))) 1 x ) t s)/4 1/p, a { a 1/p t/2 a 1/p s/2 1 u t s)/4 1/p log, s = t, Lp u 1 η ) a 1/p s/2, s t. Now, we estiate the lower bouds. Siilarly to the proof of Theore 2.6, sice by 2.2), 2.1), ad Lea A.1, xj,,ρ Φt/4 1/p) x) p,ρ w)x) x + a ) ρ s p dx x j+1,,ρ x j,,ρ x j+1,,ρ a sp/2 1 x ) t s)p/4 1 j,,ρ, a we have fro a) of Theore 2.2, Φt/4 1/p) x) p,ρ w)x) x + a ) ρ s L p a /2 x a 1 η )) a > a s/2 1 η ) 1 x ) ) t s)p/4 1 1/p { a 1/p t/2 log, s = t, dx a /2 a a 1/p s/2, s t.

16 230 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) For the case 0 pt 4, the result ca be proved siilarly. Especially, whe p =, we kow easily the result fro Theore 2.3. Cosequetly, we proved the result. 4. Further properties of p,ρ x) I the rest of this paper we let p x) = p,ρ x) siply ad we assue that ρ > 1 2. Theore 4.1. We have a represetatio: Here where p x) = A p x) x)p 1 x) B x)p x) 2ρ. 4.1) x A x) = 2b p 2 u)qx, u)w2 ρ u) du, B x) = 2b p u)p 1 u)qx, u)wρ 2 u) du, Qx, t) = Q x) Q u), b = γ { 1 ρ, is odd, ad ρ x u γ = 0, is eve. Proof. Usig the reproducig kerel K x, u) = b {p x)p 1 u) p u)p 1 x)}/x u) we have easily the results by the sae ethod as [2, Theore 1.6]. Theore 4.2 cf. Levi ad Lubisky [3, Theore 13.7]). Let wx) FC 2 ), L>0, ad γ>0. The there exist C, 0 > 0 such that for 0 ad γ a x a 1 + Lη ), A x) φ 2b x) 1 {a Lη ) 2 x 2 } 1/2, B x) A x), 4.2) ad for γ a x εa with 0 ε1sall eough, there exists 0 λε) 1 such that B x) λε)a x). 4.3) Reark 4.3. Let wx) FC 2 ).Ifρ 0orΛ 2the 4.2) ad 4.3) holds for x a 1+Lη ). For 1 Λ 2, if Q x) is bouded o a certai iterval 0,c], or if there exist 0 δ 2 ad a positive costat C such that Q x) Cx δ o a certai iterval 0,c] ad ρ δ 1 2 the 4.2) ad 4.3) holds for x a 1 + Lη ). Moreover, for 1 Λ 2, if Q x) is o-icreasig a certai iterval 0,c] ad Λ + 2ρ 1 0, the 4.2) ad 4.3) holds for x a 1 + Lη ). Proof of Theore 4.2. First we ca show 4.2) by repeatig the ethods of the proof of [3, Theore 13.7; 5, Theore 4.1]. The secod iequality i 4.2) is show by the Cauchy Schwarz iequality. Forula 4.2) is proved by dividig ito two parts, that is, the upper bouds part ad the lower bouds part. Let us defie Θ,ρ x) = A x) 2b φ x) a 2 x2 1/2.

17 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) To prove Theore 4.2 we eed soe leas. Lea 4.4. Assue that wx) FC 2 ). The a) b) c) d) 0 0 p w ρ ) 2 u)uq u) du = + ρ p w ρ ) 2 u) Q u) du = p w ρ ) 2 u)q u) du. 0 a p w) 2 u) u + a ) 2ρ uq u) du. p w) 2 u) u + a ) 2ρ Q u) du. a Proof. Sice a) ad b) are siilar results to [3, Lea 12.7], we ca prove the by repeatig the ethods of the proof of [3, Lea 12.7; 5, Theore 4.2]. Moreover, c) ad d) ca be obtaied easily, usig a), b), ad their proofs. Lea 4.5 cf. Levi ad Lubisky [3, Theore 12.11]). Let wx) FC 2 ). For γ a x Γ we have Θ,ρ x) 1, where Γ = a 1 Mη ) ad M>0 is chose such that x 1,,ρ >a 1 M/2)η ). Proof. We split Θ,ρ x) ito two parts as the followig; [ ] p ) 2 Θ,ρ x) = w ρ u)qx, u) duφ x) a 2 x2 1/2 + u γa /2 u γa /2 =: A + B. Here, if we use the ethod of [3, Chapter 12] with Lea 4.4 ad Theore 2.3, the we ca show B 1 by replacig ψ x) with p w) 2 x) x + a ) 2ρ a 2 x 2 1/2. O the other had, we kow by Theore 2.3 that A 1 u 2ρ a u +a /) 2ρ Qx, u) duφ x) a 2 x2 1/2. u γa /2 The sice for γ a x a /2 ad u γa 2 Qx, u) a 4.4) ad for a /2 x Γ ad u γa 2 Qx, u) Q a 2 )/a Ta ) a 2, we have A 1. So, the lea is proved. Now we prove 4.2).

18 232 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Proof of the upper bouds of A x). Let γ a x a 1+Lη ). If we distiguish three rages of x; i) γ a x Γ, ii) Γ x a 1 + Lη ), ad iii) a 1 + Lη ) x Γ, the the upper bouds for A x) ca be proved easily by repeatig ethods of proofs of [3, Theore 13.7; 5, Theore 4.1] for upper bouds. Proof of the lower bouds of A x). Siilarly to the ethods of [3, Lea 13.8; 5, Theore 4.2], we ca choose the ubers θ 0, 1) ad α > 1 satisfyig that uiforly for r [0, 2], p w ρ ) 2 u)q u) du. [a θ,a 2 ]\[a r/α,a αr ] a If we use these ubers θ 0, 1) ad α > 1, the the lower bouds for A x) ca be proved easily by repeatig ethods of [3, Theore 13.7; 5, Theore 4.1] for lower bouds. I the followig, we will prove 4.3). Let γ a x εa for 0 ε 2 1 sall eough. By 4.4) we obtai u γa /2 u 2ρ u +a /) 2ρ Qx, u) a 2 u2 ) 1/2 du a 2 O a u γa /2 ) a 2. Choose 0 θ 1 satisfyig that Qx, u) u a θ aθ 2 u2 1/2 du C σ θ x) Cθ aθ 2 x2 1/2 a2θ 2 Cθ x2 a 2. Here, u 2ρ du u +a /) 2ρ σ u x) = 1 au π 2 a2 u x2 ) 1/2 Qx, s) a u au 2 s2 ) 1/2 ds, x [ a u,a u ] is the desity of the equilibriu easure of total ass for the field Q. It is show i [3, Theore 5.3] that for L>1 σ t x) ta2 t x 2 ) 1/2 alt 2 x2 ), x a t. The usig Theore 2.3, p w ρ ) 2 u)qx, u) du u a θ + Sice for x [0,a /2], Q x) C a x a ) Λ 1 O u γa /2 a γa /2 u a θ u 2ρ Qx, u) u +a /) 2ρ a 2 u2 ) u 2ρ 1/2 du Qx, u) u +a /) 2ρ a 2 u2 ) 1/2 du ) a 2 + Cθ a )

19 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) see [3, Lea )]), we have for a θ u a 2, Qx, u) Qx, u) = 2 uq x) xq u) x 2 u 2 Q εa ) +ε Q u) ε Λ 1 a a 2 + ε Q u). 4.6) a The usig Cauchy Schwartz iequality, 4.6), ad b) of Lea 4.4, we have p u)p 1 u)w 2 ρ u)qx, u) du a θ u a 2 p u)p 1 u)wρ 2 u) Qx, u) Qx, u) du ε a 0 a θ u a 2 p u)p 1 u) w 2 ρ u) Q u) du +ε Λ 1 a 2 p u)p 1 u) wρ 2 u) du 0 ε a 2 + ε Λ 1 a ) O the other had, sice Qx, u) Q u) for u a 2, usig Cauchy Schwartz iequality ad Theore A.7 we have for soe costat C>0 p u)p 1 u)w 2 ρ u)qx, u) du u a 2 Here we ote that if γ a x 1 2 a, the A x) b a 2. Oe C ). Therefore, if we take θ > 0i4.5) ad ε>0i 4.7) sall eough the by 4.5), 4.7), ad 4.8), we kow that there exists 0 λε) 1 such that B x) λε) A x). b b Sice b > 0, 4.3) follows. Therefore, Theore 4.2 is proved copletely. To prove Reark 4.3, we restate it i the followig: Reark 4.6. Let wx) FC 2 ). The 4.2) ad 4.3) holds for x a 1 + Lη ) if oe of the followig coditios is satisfied: a) ρ 0; b) Λ 2; c) 1 Λ 2, Q x) is bouded o a certai iterval 0,c]; d) 1 Λ 2, there exist 0 δ 2 ad a positive costat C such that Q x) Cx δ o a certai iterval 0,c] ad ρ δ 1 2 ; e) 1 Λ 2, Q x) is o-icreasig a certai iterval 0,c] ad Λ + 2ρ 1 0.

20 234 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Proof. It is sufficiet to prove Lea 4.5 for x γ a. a) Sice u 2ρ u + a ) 2ρ, we have siilarly to the estiate of B i Lea 4.5 Θ,ρ x) p w) 2 u) u + a ) 2ρ Qx, u) duφ x) a 2 x2 1/2 1. For the other cases we will prove that for x γ a u γa /2 ) 2 p w ρ u)qx, u) du a 2. This suffices for provig Lea 4.5. b) For x γ a ad u γa 2 there exists a certai ξ betwee x ad u such that we have by d), e) of Defiitio 1.1 ad [3, Lea )] Qx, u) = Q ξ) C Q ξ)) 2 Tξ) Q ξ) Qξ) ξ Tξ) ξ a Λ ξ Λ 1 Tξ) a Λ ξ Λ 2 a Λ a 2. Therefore, we have ) 2 ) 2 p w ρ u)qx, u) du p u γa /2 a 2 w ρ u) du a 2. c) The we have by Theore 2.3 ) 2 p w ρ u)qx, u) du p w ρ ) 2 u) du u γa /2 1 a u γa /2 u γa /2 u 2ρ u +a /) 2ρ du 1 a 2. d) Let x,u 0. First, we obtai Qx, u) = 1 x x u u Q s) ds C x u The by Theore 2.3, ) 2 p w ρ u)qx, u) du u γa /2 x u 1 a s δ ds = C x1 δ u 1 δ x u p w ρ ) 2 u) 1 u δ du C 1 u δ. u γa /2 u 2ρ δ u γa /2 u +a /) 2ρ du 1 a ) 1 δ a = a ) 2 δ a 2 a 2.

21 e) Sice Q u) u H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) ) Q u) u = Q u) Q u)/u u Q x) Q u) x u = Q u) Q ξ) u 0 for a certai ξ 0,u),wehave = x Q ) x) Q u) 0. u x x u The we have by Theore 2.3 ad [3, Lea )] ) 2 p w ρ u)qx, u) du u γa /2 1 a u γa /2 u γa /2 1 ) 2ρ a a 2 Λ a 2 a 2. Therefore, Lea 4.5 for x γ a is proved. ) 2 Q u) p w ρ u) du u u 2ρ Q u) u +a /) 2ρ du 1 ) 2ρ u 2ρ Q u) du u a a u γa /2 u u Λ+2ρ 2 du 1 ) 2ρ ) Λ+2ρ 1 u γa /2 a a a a Λ The followig lea is useful. Lea 4.7 cf. Levi ad Lubisky [5, Lea 5.2b)]). Let wx) FC 2 +). The b a. Proof. This is siilar to the proof of [5, Lea 5.2]. By Cauchy Schwartz iequality, Theores A.8 ad 2.3 b := γ 1,ρ γ,ρ = xp x)p 1 x)w 2 ρ x) dx a. For x j = x j,,ρ = 0, we kow that by 4.1) ad the Christoffel Darboux forula b 1 = λ j {p 1 x j )} 2 A x j ). Sice we kow fro 2.1) ad A.4) that the uber of zeros of p x) lyig i [a /4,a /2] is at least >, we have by 4.2) ad Gauss-quadrature forula, b 2 λ j {p 1 x j )} 2 A x j ) b x j [a /4,a /2] a 2 λ j {p 1 x j )} 2 a 2 x j [a /4,a /2] p 1 2 x)w2 ρ x) dx a 2. Therefore, we obtai the lea. a Λ

22 236 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Fro Lea 4.7 we obtai the followig. Corollary 4.8. Let wx) FC 2 +), L>0ad γ > 0. The there exist C, 0 > 0 such that for 0 ad γ a x a 1 + Lη ), A x) a φ x) 1 {a Lη ) 2 x 2 } 1/2. 4.8) Moreover, if either of the coditios i Reark 4.6 is satisfied, the 4.8) holds for x a 1 + Lη ). Ackowledgets This paper was supported by Faculty Research Fud, Sugkyukwa Uiversity, Authors thak the referee for ay valuable coets ad correctios. Appedix A. Lea A.1. a) For u>0, we have au 2 =ã u/2, Ta u ) = 2 Tã u/2 ) ad η u = 2 1/3 η u/2. A.1) b) Uiforly for t [0, ã ] ad u>0, we have φ 2u φ t) u t) t +ã u u 2. A.2) ) 1/2 c) Uiforly for ad x φ +M φ x), M > 0, φ x j ) φ x j+1, ), 1 j 1, A.3) ad uiforly for x εa, 0 ε1, we have φ 2 x) φ x) a. d) For soe ε>0ad for large eough u, Ta u ) Cu 2 ε. e) There exists L 0 such that for ay fixed L>L 0 ad uiforly for u>0, 1 a u 1 a Lu Ta u ) ad Ta Lu ) Ta u ). f) For t>0ad 2 s t 2s, uiforly for u>0, 1 a s 1 s 1 t Ta s ). a t A.4) A.5) A.6) A.7) A.8) Proof. a) They are [4, 2.6), 2.5), 2.9)]. b) This is [5, 2.13)]. c) This is easily proved by the defiitio of φ x). d) This is [3, 3, 3.38)]. e) This is [3, Lea ), Lea )]. f) This is [3, 3.50)].

23 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Theore A.2 Levi ad Lubisky [4, Theore 1.5]). Let w LC 2 ). Let 0 p ad L, λ > 0. Let β > p 1 if p, ad β 0 if p =. a) There exist C, 0 > 0 such that for 0 ad P P, P w)t)t β Lp R + ) P w)t)t β Lp [Lã 2,ã 1 λ η ]). b) Give r>1, there exist C, 0 > 0 ad α > 0 such that for 0 ad p P, P w)t)t β Lp ã r, ) exp C α ) P w)t)t β Lp [0,ã ]). Theore A.3 Levi ad Lubisky [4, Theore 1.4; 5, Theore 1.4). Let ρ > 2 1 ad let wx) LC 2 +). The a) For the iiu zero t,,ρ we have t,,ρ ã 2 ad the axiu zero t 1,,ρ, we have for soe C>0 1 t 1,,ρ η ã. b) For 1 ad 1 j 1, t j,,ρ t j+1,,ρ φ t j,,ρ ). A.9) If we assue that wx) LC 2 ) istead, the i a) for soe costat C>0 t,,ρ ã 2, ã 1 C η ) t 1,,ρ ã +ρ+1/4, ad b) holds with replaced by. Theore A.4 Levi ad Lubisky [4, Theore 1.2]). Let ρ > 2 1 ad let w LC2 ). Let p,ρ t) be the th orthooral polyoial for the weight w 2 ρ. The uiforly for 1, [ sup p ) ρ,ρt) wt) t + ã t R + 2 t +ã 2) ã t) 1/4] 1. Theore A.5 Levi ad Lubisky [5, Theore 1.2]). Let ρ > 2 1 ad let w LC2 +). Let p,ρ t) be the th orthooral polyoial for the weight w 2 ρ. The uiforly for 1, ad sup p,ρ t) ) ρ ) wt) t + ã 1/2 t R + 2 ã sup p,ρ t) ) ρ wt) t + ã t ã β 2 ã 1/2 T ã )) 1/6. If w LC 2 ), these estiates hold with replaced by.

24 238 H.S. Jug, R. Sakai / Joural of Approxiatio Theory ) Theore A.6 Levi ad Lubisky [5, Theore 1.3]). Let w LC 2 +) ad ρ > 1 2. There exists 0 such that uiforly for 0, 1 j, a) p,ρ w ρ t j,,ρ ) φt j,,ρ ) 1 [t j,,ρ ã t j,,ρ )] 1/4. A.10) b) p 1,ρ w ρ t j,,ρ ) ã 1 [t j,,ρã t j,,ρ )] 1/4. A.11) ) ρ c) ax l j,ρ t) wt) t + ã t R + 2 w ρ 1 t j,,ρ) 1. A.12) d) For j 1 ad t [t j+1,,ρ,t j,,ρ ], p,ρ w ρ t) i{ t t j,,ρ, t t j+1,,ρ } φ t j,,ρ ) 1 [t j,,ρ ã t j,,ρ )] 1/4. A.13) If we assue istead that w LC 2 ), the a) holds with replaced by ad b) holds with replaced by. > Theore A.7 Jug ad Sakai [1, Theore 2.4]). Let wx) FC 2 ),0p ad L 0. Let β R. The give r > 1, there exists a positive costat C 2 such that we have for ay polyoial P P P w β )x) Lp a r x ) exp C 2 α ) P w β )x) Lp L a x a 1 Lη )). Theore A.8 Jug ad Sakai [1, Theore 2.5]). Let wx) FC 2 ),0p, β R, ad L 0. The we have for ay polyoial P P, P w)x) x + a ) β P w)x) x + a ) β. L p R) L p La / x a 1 Lη )) Theore A.9 Jug ad Sakai [1, Theore 2.7]). Let ρ > 1/p, 0 p ad let wx) FC 2 ). a) Let L>0. The uiforly for 1 ad x a 1 + Lη ), we have λ p w ρ ; x) φ x)w p x) x + a ) ρp. b) Moreover, uiforly for 1 ad x R, λ p w ρ ; x) > φ x)w p x) x + a ) ρp. Refereces [1] H.S. Jug, R. Sakai, Iequalities with expoetial weights, J. Coput. Appl. Math ) 2008) [2] T. Kasuga, R. Sakai, Orthooral polyoials for geeralized Freud-type weights, J. Approx. Theory ) [3] A.L. Levi, D.S. Lubisky, Orthogoal Polyoials for Expoetial Weights, Spriger, New York, [4] A.L. Levi, D.S. Lubisky, Orthogoal polyoials for expoetial weights x 2ρ e 2Qx) o [0,d), J. Approx. Theory ) [5] A.L. Levi, D.S. Lubisky, Orthogoal polyoials for expoetial weights x 2ρ e 2Qx) o [0,d),II, J. Approx. Theory ) [6] G. Szegő, Orthogoal Polyoials, Aerica Matheatical Society Colloquiu Publicatios, vol. 23, Aerica Matheatical Society, Providece, RI, 1975.