Generalized Jacobi Koornwinder s-type Bernstein polynomials bases transformations

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1 Iteratoal Joural of Mathematcs Vol. 27, No. 11 ( (13 pages c World Scetfc Publshg Compay DOI: /S129167X Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos Mohammad A. AlQudah School of Basc Sceces ad Humates Germa Jordaa Uversty, Amma 1118, Jorda mohammad.qudah@gju.edu.jo Maalee N. AlMhedat Departmet of Basc Sceces, Uversty of Petra Amma 11196, Jorda malmhedat@uop.edu.jo Receved 16 March 215 Revsed 2 February 216 Accepted 3 August 216 Publshed 14 October 216 Ths paper provdes a explct closed form of geeralzed Jacob Koorwder s polyomals of degree r terms of the Berste bass of fxed degree. Moreover, explct forms of geeralzed Jacob Koorwder s type ad Berste polyomals bases trasformatos are cosdered. Keywords: Jacob Koorwder s polyomals; Berste polyomals; trasformato; bass. Mathematcs Subject Classfcato 21: 33C45, 33C5, 42C5, 5A1, 33B15 1. Itroducto Approxmato s mportat to may umercal methods, as may arbtrary cotuous fuctos ca be approxmated by polyomals. O the other had, polyomals ca be characterzed may dfferet bases. Every type of polyomal bass, such as the moomal power, Jacob, Berste ad Hermte bass form, has ts stregth. By sutable choce of the bass, umerous problems ca be solved ad may complcatos ca be removed. Rababah [13] provded a explct form of classcal Jacob ad Berste polyomals bases trasformatos. I addto, a explct form for geeralzed Tschebyscheff polyomals of Koorwder s type ad Berste polyomals chage of bases was dscussed [1]. I ths paper a geeralzato of the results Correspodg author

2 M. A. AlQudah & M. N. AlMhedat [1] s studed. A explct closed form of geeralzed Jacob Koorwder s polyomals of degree r terms of the Berste bass of fxed degree ad the explct forms of geeralzed Jacob Koorwder s polyomals ad Berste polyomals bases trasformatos are provded Berste polyomals The Berste polyomals have bee studed thoroughly, there exst may extesve treatmets of these polyomals [4]. They are ow for ther aalytc ad geometrc propertes [3, 7, 16], partcular ths bass s ow to be optmally stable. Defto 1.1. The + 1 Berste polyomals Bv (x ofdegree, x [, 1], are defed by ( Bv (x = x v (1 x v, v =, 1,...,, v (1.1 else, where ( v are bomal coeffcets. Farou [4] surveyed Berste bass propertes, he descrbed some of the ey propertes ad algorthms assocated wth the Berste bass. Here, we brefly summarzed some of the ey propertes of these polyomals. Berste polyomals are all oegatve, Bv (x, x [, 1], satsfy symmetry relato Bv (x = B v (1 x, have a sgle uque maxmum of ( ( at x =, =,...,, ther roots are x =, 1 wth multplctes ad they form a partto of uty B (x =1. The Berste polyomals of degree ca be defed by combg two Berste polyomals of degree 1. The th-th-degree Berste polyomal s gve by B (x =(1 xb 1 (x+xb 1 1 (x, =,...,; 1, where B(x = ad B (x = for < or >. For more detals, see Farou [4]. Moreover, t s possble to wrte Berste polyomals of degree r where r terms of the Berste polyomals of degree usg the followg degree elevato defed by [5] ( r+ B r (x = r r ( ( B (x, =, 1,...,r. (1.2 = These remarable propertes ad others [4] mae Berste polyomals sgfcat for the developmet of Bézer curves ad surfaces computer-aded geometrc desg (CAGD. The Berste polyomals are the stadard bass for the Bézer

3 Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos represetatos of curves ad surfaces CAGD. However, the Berste polyomals are ot orthogoal ad could ot be used effectvely the least-squares approxmato [14], thus the calculatos performed fdg the least-squares approxmato polyomal of degree m do ot decrease the calculatos to obta the least-squares approxmato polyomal of degree m+1. Sce the, a theory of approxmato has bee developed ad may approxmato techques have bee preseted ad examed. The method of least-squares approxmato accompaed by orthogoal polyomals s oe of these methods Least-squares approxmato I the followg, we defe the cotuous least-squares approxmatos of a fucto f(x by usg polyomals wth stadard power bass, {1,x,x 2,...,x }. Defto 1.2. For a cotuous fucto f(x defedo[, 1], the least-squares approxmato requres fdg a least-squares polyomal p (x = a φ (x that mmzes the error E(a,a 1,...,a = [f(x p (x] 2 dx. A ecessary codto for E(a,a 1,...,a to have a mmum over all values a,a 1,...,a s = E = 2 [f(x p a (x]φ (xdx, =,...,. Thus, for =, 1,...,, a that mmze f(x a φ (x 2 satsfy f(xφ (xdx = a φ (xφ (xdx, whch gves a system of ( + 1 equatos, called ormal equatos, ( +1 uows: a,=,...,.those (+1 uows of the least-squares polyomal p (x ca be foud by solvg the ormal equatos. By choosg φ (x =x, as a bass, the f(xx dx = a x + dx = a + +1, ad the matrx of coeffcets of the ormal equatos are gve as

4 M. A. AlQudah & M. N. AlMhedat The coeffcet matrx of the ormal equatos s the Hlbert matrx whch has roud-off error dffcultes ad s otorously ll-codtoed for eve modest values of. However, such computatos ca be made computatoally effectve by usg orthogoal polyomals. Thus, choosg {φ (x,φ 1 (x,...,φ (x} to be orthogoal smplfes the least-squares approxmato problem. The coeffcet matrx of the ormal equatos wll be dagoal, whch smplfes calculatos ad gves a closed form for a, =, 1,...,. Moreover, oce p (x s ow, t s oly eeded to compute a +1 to get p +1 (x, whch turs out to be computatoally effcet. See [14] for more detals o the least-squares approxmatos Classcal Jacob polyomals The classcal Jacob polyomals of degree, P (α,β (x, are a set of orthogoal polyomals defed as solutos to the dfferetal equato (1 x 2 y +[β α (α + β +2x]y + ( + α + β +1y =. These polyomals are orthogoal, except for a costat factor, wth respect to the weght fucto W (x =(1 x α (1 + x β, α,β > 1. The followg orthogoalty relato [6, 11] wll be used the proof of the ma result: (1 x α x β P (α,β (xp m (α,β (xdx f m, Γ(α +1Γ(β +1 f m = =, = Γ(α + β +2 (1.3 Γ( + α +1Γ( + β +1 (2 + α + β +1Γ( +1Γ( + α + β +1 f m = N. Uvarate classcal Jacob orthogoal polyomals are tradtoally defed o [ 1, 1]. However, t s more coveet to use [, 1]. Usg Pochhammer symbol s more approprate, but the combatoral otato gves more compact ad readable formulas, these have also bee used by Szegö [15]. For coveece we recall the followg explct expresso for Jacob polyomals of degree x [9]: P (α,β (x := ( ( + α + β (x ( +1 x 1, 2 2 whch ca be trasformed terms of the Berste bass o x [, 1] as ( +α ( +β P (α,β (2x 1 := ( 1 +1 ( B (x

5 Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos 1.3. Geeralzed Jacob Koorwder s polyomals Koorwder [1] troduced orthogoal polyomals for whch the weght fucto s a lear combato of the Jacob weght fucto ad two delta fuctos at 1 ad 1. He descrbed a geeral class of Jacob-type polyomals {P (α,β,m,n (x} = of degree, wth weght fucto (1 x α (1 + x β + lear combato of δ(x +1adδ(x 1. For α, β > 1 adm,n, geeralzed Jacob Koorwder s-type polyomals, also referred to as the Jacob-type polyomals or geeralzed Jacob polyomals, are defed [8, 1] by P (α,β,m,n (x =P (α,β (x+mq (α,β (x+nr (α,β (x+mns (α,β (x, where Q (α,β (x =R (α,β (x =S (α,β (x =adfor =1, 2, 3,..., ad Q (α,β R (α,β S (α,β (x = (β +2 1(α + β +2 1 (α +1! [( + α + β +1P (α,β (x (β +1(x 1DP (α,β (x], (1.4 (x = (α +2 1(α + β +2 1 (β +1! [( + α + β +1P (α,β (x (α +1(x +1DP (α,β (x] (1.5 (x = (α + β +2 (α + β +2 1 (α +1(β +1!( 1! [( + α + β +1P (α,β (x {(β +1(x 1 + (α +1(x +1}DP (α,β (x]. (1.6 Usg the symmetry relato [1] of geeralzed Jacob polyomals P (α,β,m,n (x =( 1 P (β,α,n,m ( x, =, 1, 2,..., ad the recurrece relato [2, 1, 15] P (α+1,β+1 1 (x =P (α+1,β (x P (α,β+1 (x, =1, 2, 3,..., (1.4, we obta the followg represetato: Q (α,β (x = (β +2 1(α + β +2 1 (α +1 1! Smlarly, usg the symmetry relato (x +1DP (α 1,β+1 (x, =1, 2, 3,... Q (α,β (x =( 1 R (β,α ( x, =, 1, 2,..., (

6 M. A. AlQudah & M. N. AlMhedat ad (x 2 1D 2 P (α,β (x =( + α + β +1P (α,β (x [(β +1(x 1 +(α +1(x +1]DP (α,β (x, =, 1,..., (1.5 ad(1.6, respectvely, we get the followg represetatos: R (α,β (x = (α +2 1(α + β +2 1 (β +1 1! (x 1DP (α+1,β 1 (x, =1, 2, 3,..., (1.8 S (α,β (x = (α + β +2 (α + β +2 1 (α +1(β +1!( 1! The represetatos (1.7 ad(1.8 ca be wrtte as ad Q (α,β (x = R (α,β (x = (x 2 1D 2 P (α,β (x, =1, 2, 3,... q (α,β P (α,β (x wth q (α,β = (β +2 1(α + β +2 1 (α +1 1 ( 1! r (α,β P (α,β (x wth r (α,β = (α +2 1(α + β (β +1 1 ( 1! (1.9 Usg P (α,β (x (+αp (α,β+1 1 (x =(x 1DP (α,β (x, =1, 2, 3,..., (1.9 we obta S (α,β (x = s (α,β P (α,β (x wth s (α,β = (α + β +2 (α + β +2 1 (α +1(β +1( 1!( 2!. Therefore, for α, β > 1 adm, N, geeralzed Jacob Koorwder s polyomals {P (α,β,m,n (x} = are orthogoal o the terval [ 1, 1] wth respect to the weght fucto Γ(α + β +2 2 α+β+1 Γ(α +1Γ(β +1 (1 xα (1 + x β + Mδ(x +1+Nδ(x 1, (1.1 ad ca be wrtte as where P (α,β,m,n (x =(1+λ P (α,β λ = Mq (α,β + Nr (α,β (x+ λ P (α,β (x, ( MNs (α,β. (1.12

7 Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos For more propertes of geeralzed Jacob polyomals, the reader s referred to [8, 1, 15]. 2. Ma Results I ths secto, we show how geeralzed Jacob Koorwder s polyomals P r (α,β,m,n (x ofdegreer ca be wrtte explctly as a lear combato of the Berste polyomal bass of degree r. The we provde a closed form for the trasformato matrx of geeralzed Jacob Koorwder s polyomals bass to Berste polyomals bass ad for Berste polyomals bass to geeralzed Jacob Koorwder s polyomals bass Geeralzed Jacob Koorwder s polyomals usg Berste bass The ext theorem shows how geeralzed Jacob Koorwder s polyomals P r (α,β,m,n (x ca be wrtte as a lear combato of the Berste polyomal bass. Theorem 2.1. For α, β > 1 ad M, N, geeralzed Jacob Koorwder s polyomals P r (α,β,m,n (x of degree r have the followg Berste represetato: P (α,β,m,n r (x = where λ ( r+β r = ( 1 r η (α,β,r s defed (1.12, η (α,β,r (α,β. Moreover, the coeffcets η,r η (α,β,r = (r + α +1 ( + β B r (x+ = (r+α ( r+β r λ = ( 1 η (α,β, ( r satsfy the recurrece relato B (x, (2.1, =, 1,...,r, ad η (α,β,r = η (α,β 1,r, =1,...,r. (2.2 Proof. To wrte a geeralzed Jacob polyomal P r (α,β,m,n (x ofdegreer as a lear combato of the Berste polyomal bass B r (x, =, 1,...,r,of degree r explct form, we beg wth substtutg (1.1 to(1.11 toget P (α,β,m,n r (x = = = ( r+α ( r+β r ( r r B r r (x+ ( r+α ( r+β ( 1 r r ( r B r (x = + λ λ j= ( +α ( +β ( 1 j j j ( Bj j (x j= ( +α ( +β j j ( j B j (x

8 M. A. AlQudah & M. N. AlMhedat By defg η (α,β,r = ( r+α ( r+β r ( r, =, 1,...,r, t s clear that η (α,β,r = ( r+β r. Usg smple combatoral dettes smplfcatos, we have the recurrece relato defed ( Bases trasformatos Rababah [12] provded some results cocerg the uvarate Chebyshev case, the followg theorem we use smlar approach to geeralze the results for geeralzed Jacob Koorwder s polyomals case. The followg theorem combes the superor performace of the least-squares of geeralzed Jacob Koorwder s polyomals wth the geometrc sghts of the Berste polyomals bass. Theorem 2.2. The etres M,r,,r =, 1,...,, of the trasformato matrx of the geeralzed Jacob polyomal bass to Berste polyomal bass of degree are gve by M,r = ( 1 m(,r r ( r ( 1 r =max(,+r + ( 1 m(, λ j=max(,+ ( r + α ( r + β r ( ( + α ( 1 j j j ( + β j. (2.3 Proof. Ay polyomal p (x, x [, 1], of degree ca be wrtte uquely as a lear combato of certa elemetary bass. Wrte p (x as a lear combato of the Berste polyomal bass p (x = c r Br (x r= ad as a lear combato of geeralzed Jacob Koorwder s polyomals p (x = d P (α,β,m,n (x. = We are terested the trasformato matrx M, wherec = M d that trasforms the geeralzed Jacob coeffcets {d } = to the Berste coeffcets {c r} r= : c M, M,1 M,2... M, d c 1.. = M 1, M1,1 M1,2... M1, d (2.4 c M, M,1 M,2...M, d

9 Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos Usg Theorem 2.1 we wrte the geeralzed Jacob polyomals (1.11 ofdegree r terms of Berste polyomal bass of degree as P r (α,β,m,n (x = Nr, B (x, r =, 1,...,, (2.5 = where Nr, are the etres of the ( +1 ( + 1 bass coverso matrx N. Thus the elemets of c cabewrttetheform: c = d r Nr,. (2.6 r= Comparg Eqs. (2.4 ad(2.6 showsthatm = N T. Now, we eed to wrte each Berste polyomal of degree r where r terms of Berste polyomals of degree. By substtutg the degree elevato (1.2 defed by [5] to P (α,β,m,n r (x = = ( 1 r η (α,β,r B r (x+ λ = ( 1 η (α,β, B (x ad rearragg the order of summatos, we fd that the etres of the matrx N are gve by ( 1 m(,r r ( ( ( r r + α r + β Nr, = ( 1 r r + =max(,+r ( 1 m(, λ j=max(,+ ( ( + α ( 1 j j j ( + β j Thus, the matrx M ca be obtaed by trasposg the etres of the matrx N.. I Corollary 2.1, we express geeralzed Jacob Koorwder s polyomals P r (α,β,m,n (x ofdegreer terms of the Berste bass of fxed degree. Corollary 2.1. The geeralzed Jacob polyomals P (α,β,m,n (x,..., P (α,β,m,n (x of degree less tha or equal to ca be expressed the Berste bass of fxed degree by the followg formula: P r (α,β,m,n (x = Nr,B (x, r =, 1,...,, where, for =,...,, ( r Nr, = + 1 m(,r =max(,+r = ( 1 m(, λ ( ( ( r r + α r + β ( 1 r r j=max(,+ ( ( + α ( 1 j j j ( + β j. (2.7

10 M. A. AlQudah & M. N. AlMhedat Proof. Usg (2.5 from the proof of Theorem 2.2, ay geeralzed Jacob Koorwder s polyomal P r (α,β,m,n (x of degree r cabeexpressed terms of Berste bass of fxed degree as P r (α,β,m,n (x = = N r, B (x,r =, 1,...,, where the matrx N ca be obtaed by trasposg the etres of the matrx M defed (2.3. We dscussed earler some aalytc ad geometrc propertes for Berste polyomals, t s worth metog that Berste polyomals ca be tegrated as B (xdx = 1 +1, =, 1,...,. Moreover, the product of two Berste polyomals s a Berste polyomal ad s gve by ( +m +j B (xbj m(x =( m ( j B +m +j (x. The followg terestg theorem wll be used to smplfy Theorem 2.4. Theorem 2.3. Let Br (x be the Berste polyomal of degree ad P (α,β,m,n (x be the geeralzed Jacob polyomal of degree, the, for, r =, 1,...,, we have ( 1 1 (1 x α x β Br (xp (α,β,m,n (xdx =Ψ,r r where λ d s defed by (1.12, Ψ,r, =, + d= λ d Ψ,r j,d, ( ( + α + β ( 1 B(β + r + +1,+ + α r +1 (2.8 ad B(x, y s the Beta fucto. Proof. Usg (2.1, the tegral ca be smplfed to I = + I = (1 x α x β ( r d= (1 x α x β B r (xp(α,β,m,n x r (1 x r (xdx ( +α ( +β ( 1 ( B (x ( λ d (1 x α x β x r (1 x r r ( d d+α ( d+β ( 1 d j j d j ( d Bj j d (xdx j=

11 = Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos ( ( ( + α + β 1 ( 1 x β+r+ (1 x ++α r dx r ( d ( ( d + α d + β + λ d ( 1 d j r j d j d= j= x β+r+j (1 x +d+α r j dx. (2.9 The tegrals the last equato are Beta fuctos B(x,y wthx 1 = β + r + +1,y 1 = + + α r +1,x 2 = β + r + j +1 ad y 2 = + d + α r j +1. Theorem 2.4. The etres a,r,,r =, 1,...,, of the matrx trasformato of the Berste polyomal bass to the geeralzed Jacob polyomal bass of degree are gve by a,r = ( r where λ d s defed by (1.12, Ψ,r, h (α,β = [ (α,β h (1 + λ 2 Ψ,r, + d= λ d Ψ,r j,d s defed by (2.8 ad (2 + α + β +1Γ( +1Γ( + α + β +1. Γ( + α +1Γ( + β +1 ], (2.1 Proof. To wrte the Berste polyomal bass to geeralzed Jacob polyomal bass of degree, we vert the trasformato formula (2.4 to get d = M 1 c. (2.11 We wrte as a,r, b,r,,r=,...,, for the etres of M 1 ad N 1, respectvely. The trasformato of Berste polyomal to geeralzed Jacob polyomal bass of degree cathebewrtteas B r (x = = b r,p (α,β,m,n (x. (2.12 To fd the etres b r,,,r =, 1,...,, we multply (2.12 by (1 x α x β P (α,β,m,n (x ad tegrate over [, 1] to get Br (x(1 xα x β P (α,β,m,n (xdx = = b r, (1 x α x β P (α,β,m,n (xp (α,β,m,n (xdx. (

12 M. A. AlQudah & M. N. AlMhedat By the orthogoalty relato (1.3 weget Br (x(1 x α x β P (α,β,m,n (xdx = b r, Usg Theorem 2.3, weobta ( [ (α,β b r, = r h (1 + λ 2 Ψ,r, + where λ d s defed by (1.12, Ψ,r, h (α,β = d= s defed by (2.8 ad (1 + λ 2. (2.14 h (α,β ] λ d Ψ,r j,d, (2.15 (2 + α + β +1Γ( +1Γ( + α + β +1. Γ( + α +1Γ( + β +1 The etres of M 1 are obtaed by trasposto of N 1. Acowledgmets The authors would le to tha the edtor ad aoymous revewers for ther valuable commets ad suggestos, whch were helpful mprovg the paper. Refereces [1] M. A. AlQudah, Geeralzed Tschebyscheff of the secod d ad Berste polyomals chage of bases, Eur. J. Pure Appl. Math. 8(3 ( [2] H. Bavc ad R. Koeoe, O a dfferece equato for geeralzatos of Charler polyomals, J. Approx. Theory 81 ( [3] G. Far, Curves ad Surface for Computer Aded Geometrc Desg: APractcal Gude, Computer Scece ad Scetfc Computg Seres (Academc Press, MA, [4] R. T. Farou, The Berste polyomal bass: A ceteal retrospectve, Comput.- Aded Geom. Des. 29(6 ( [5] R. T. Farou ad V. T. Raja, Algorthms for polyomals Berste form, Comput.-Aded Geom. Des. 5(1 ( [6] I. Gradshte ad I. Ryzh, Tables of Itegrals, Seres, ad Products (Academc Press, New Yor, 198. [7] J. Hosche, D. Lasser ad L. L. Schumaer, Fudametals of Computer Aded Geometrc Desg (A. K. Peters, [8] J. Koeoe ad R. Koeoe, Dfferetal equatos for geeralzed Jacob polyomals, J. Comput. Appl. Math. 126(1 ( [9] R. Koeoe, T. H. Koorwder, P. Lesy ad R. Swarttouw, Hypergeometrc Orthogoal Polyomals ad Ther q-aalogues (Sprger-Verlag, Berl, 21. [1] T. H. Koorwder, Orthogoal polyomals wth weght fucto (1 x α (1 + x β + Mδ(x +1+Nδ(x 1, Ca. Math. Bull. 27(2 ( [11] F. W. J. Olver, D. W. Lozer, R. F. Bosvert ad C. W. Clar (eds., NIST Hadboo of Mathematcal Fuctos (Cambrdge Uversty Press, Cambrdge, 21. [12] A. Rababah, Trasformato of Chebyshev Berste polyomal bass, Comput. Methods Appl. Math. 3(4 ( [13] A. Rababah, Jacob Berste bass trasformato, Comput. Methods Appl. Math. 4(2 (

13 Geeralzed Jacob Koorwder s-type Berste polyomals bases trasformatos [14] J. Rce, The Approxmato of Fuctos: Lear Theory (Addso-Wesley, Readg, [15] G. Szegö, Orthogoal Polyomals, 4th ed., Amerca Mathematcal Socety Colloquum Seres, Vol. 23 (Amerca Mathematcal Socety, Provdece, [16] F. Yamaguch ad F. Yamaguch, Curves ad Surfaces Computer Aded Geometrc Desg (Sprger-Verlag, Berl,