Boundedness of Orthogonal Polynomials in Several Variables

Transcription

1 Iteratioal Joural of Mathematical Aalysis Vol. 10, 2016, o. 3, HIKARI Ltd, htt://dx.doi.org/ /ijma Boudedess of Orthogoal Polyomials i Several Variables Pavol Oršasý Deartmet of Alied Mathematics, Faculty of Mechaical Egieerig Uiversity of Žilia, Žilia, Slovaia Vladimír ulda Deartmet of Alied Mathematics, Faculty of Mechaical Egieerig Uiversity of Žilia, Žilia, Slovaia Helea Šamajová Deartmet of Alied Mathematics, Faculty of Mechaical Egieerig Uiversity of Žilia, Žilia, Slovaia Coyright c 2015 Pavol Oršasý, Vladimír ulda ad Helea Šamajová. This article is distributed uder the Creative Commos Attributio Licese, which ermits urestricted use, distributio, ad reroductio i ay medium, rovided the origial wor is roerly cited. Abstract I this aer we rove statemets of boudedess for orthogoal olyomials i several variables. We assume two systems of olyomials {P (x)}, {Q (x)} associated with a weight fuctio w (x) ad its geeralizatio w (x) h (x), where fuctio h (x) satisfies some coditios. Bouds of the system {Q (x)} whe the weight fuctio w (x) has searable variables are also give. Mathematics Subject Classificatio: 33C45, 33C50, 33D50 Keywords: weight fuctio, orthogoal olyomials i several variables, boudedess

2 118 Pavol Oršasý, Vladimír ulda ad Helea Šamajová 1 Itroductio Orthogoal olyomials i oe [1, 13, 15] or several variables [2, 9, 14] are very useful i may areas of the alied mathematics ad techical alicatios. We cosider two systems of orthogoal olyomials i oe real variable. Let { (x)} =0 be a system of orthoormal olyomials o I R with resect to the weight fuctio w(x) ad a system of orthoormal olyomials {q (x)} =0 o the same iterval I associated with a geeralized weight fuctio W (x) = w(x)h(x), where the factor h(x) satisfies some secific coditios. A tyical examle is the case whe the weight fuctio w(x) is associated with so-called classical orthogoal olyomials. I the aer [8] J. Korous dealt with the boudedess ad uiform boudedess of olyomials {q (x)} =0 orthoormal o a fiite iterval I. This cocet of the geeralizatio of the weight fuctio is aalyzed i [3, 4, 5, 10, 11, 12] ad for orthogoal olyomials i two variables i [6]. Some of techical alicatio of the olyomials i two variables may be foud i [7]. 2 Orthogoal olyomials i several variables I this aer we will use the otatio give i the moograh [2]. Let d-tule of o egative itegers be a stadart multi-idex, α = (, α 2,..., ) N d 0. The multi-idex α N d 0 ad x = (x 1, x 2,..., x d ) defie a moomial as x α = x 1 x α x d. The umber α = +α , is called the total degree of the moomial x α. Polyomial i d variables is a liear combiatio of moomials P (x) = α c α x α, where c α R. Total degree of this olyomial is the highest degree of its moomials. For N 0, we deote the set of olyomials i d variables as Π d. Polyomial i several variables is called a homogeeous olyomial, if each its moomial has the same total degree. Let us deote P d the set of such olyomials of the degree, i.e., P d = P (x) = c α x α, α =

3 Boudedess of orthogoal olyomials 119 where α = (, α 2,..., ) N d 0 ad x α = x 1 x α x d. Based o rior, each olyomial i d variables of total degree, ca be writte as a liear combiatio of relevat homogeeous olyomials P (x) = c α x α. The dimesioality of these sets i accordace with [2] are ( ) ( ) + d + d 1 dim Π d =, dim P d =. (3) It will be iterestig ad for further cosideratio also ecessary to clarify the relatioshi betwee the olyomials i d variable of total degree ad homogeeous olyomials of degree dim Π d = dim P d ( + d). d Let us tur the attetio to the orthogoal olyomials i several variables. Let us deote by, the scalar roduct o the set Π d. Usig the weight fuctio w(x), we have P, Q = P (x) Q (x) w (x) dx. The system of olyomials {Pα(x)} α =, = 0, 1,..., is orthogoal, if P α, P β = w(x)p α (x)p β (x)dx = 0, for ay α β. If also P α, P α = 1, the the system is orthoormal. Let the olyomial P (x) from the system {P α(x)} α = is give ad the total degree (deg P = ) of this olyomial is. The for ay olyomials of the lower degree Q (x) Π d 1, the coditio P, Q = 0 holds. I the followig, uless stated otherwise, we will cosider the olyomials i d variables. 3 Multilyig the weight fuctio by a ratioal fuctio Theorem 3.1. Let for two olyomials R r (x) ad S s (x) the followig iequalities 0 R r (x) K R, 0 < S s (x) K S

4 120 Pavol Oršasý, Vladimír ulda ad Helea Šamajová hold o the bouded domai R d (deg R = r, deg S = s). Let {Pα(x)} α =, {Q α(x)} α = be the systems of orthoormal olyomials, where Pα(x) be olyomials orthoormal o the bouded domai with resect to the weight fuctio h(x) ad Q α(x) be olyomials orthoormal o the same domai with resect to the geeralized weight fuctio ad let W (x) = R r (x) S s (x) h (x) (4) P α (x) H, (5) o, where H does ot deed either o ay variable x = (x 1, x 2,..., x d ), or o the degree of the olyomials Pα (x). The, for ay olyomial Q β (x), deg Q =, the followig iequalities hold: if s < if s Rr (x) Q β (x) [ ( ) + r + d + r ( ) s 1 + d ] KS KR H, (6) s 1 Rr (x) Q β (x) (d + + r)! KS KR H. (7) ( + r)!d! Proof. The roduct R r (x) Q β (x) will be exressed i the form R r (x) Q β (x) = +r c αpα (x), (8) where the coefficiets c α are give as c α = R r (x) Q β (x) Pα (x) h (x) dx. This relatioshi ca be easily writte i the followig form c R r (x) α = S s (x) Q β (x) S s (x) Pα (x) h (x) dx. If α + s <, these itegrals, based o the orthogoality of the olyomial Q β (x) o the domai with the weight fuctio (4), equal to zero. Thus for = α < s holds that c α = 0 ad we may write R r (x) Q β (x) = +r = s α = c αpα (x). (9)

5 Boudedess of orthogoal olyomials 121 Next, let o be alied 0 R r (x) K R, 0 < S s (x) K S, i.e., K R is the maximum of the olyomial R r (x) o the domai ad K S is the maximum of the olyomial S s (x) o the same domai. Usig the Schwarz iequality for o-zero coefficiets, we have c α R r (x) Q β (x) S s (x) P α (x) h (x) dx Ss (x) R r (x) K S Q β (x) P α (x) h (x) dx Ss (x) Rr (x) K S KR Q β (x) P α (x) h (x) dx Ss (x) ) 1 R r (x) S s (x) [Q β(x)] 2 2 h (x) dx ( [ P α (x) ] ) h (x) dx. K S KR ( From orthoormal roerties of olyomials Q β (x) ad P α (x) with the corresodig weight fuctios, we get K S KR. (10) c α The sum (9) has (accordig to (3)) at most ( ) + r + d dim Π d +r dim Π d s 1 = + r ( ) s 1 + d. s 1 terms. Thus the iequality (6) follows from the relatios (9), (10) ad (5). If s, the sum (8) has at most ( ) + r + d dim Π d (d + + r)! +r = = + r ( + r)!d! terms ad the iequality (7) follows from the relatios (8), (10) ad (5). 4 Reroducig erel The orthogoal olyomials i oe variable have may secific roerties. Oe of them is the three-term recurrece relatio ad the exressio of the

6 122 Pavol Oršasý, Vladimír ulda ad Helea Šamajová reroducig erel K (x, t) (the Christoffel-Darboux formula for olyomials (x) = a () x + a () 1 x ) K (x, t) = (x) (t) = =0 a() a (+1) (x) (t) (x) +1 (t), (11) x t which lays a fudametal role i may roblems i the theory of orthogoal olyomials. For orthogoal olyomials i d variables (x =(x 1,..., x d ) ad y =(y 1,..., y d )) the erel of the -th degree ca be, similarly to the case of the olyomials i oe or two variables, defied i the followig way K (x, y) = Pα (x) Pα (y). (12) 5 Weight fuctio with searated variables Let { Pα (x) } be a orthoormal system with the weight fuctio with α = searated variables w (x) = w (x 1, x 2,..., x d ) = w 1 (x 1 ) w 2 (x 2 )... w d (x d ), (13) where w 1 (x 1 ), w 2 (x{ 2 ),..., w} d (x { d ) are weight } { fuctios} associated with orthoormal systems (x 1 ), (x 2 ),..., (d) (x d ) i oe variables o corresodig itervals (a 1, b 1 ), (a 2, b 2 ),..., (a d, b d ), ( a i < b i, for i = 1, 2,..., d). Obviously, for the system of olyomials { P α (x) } orthoormal o = (a 1, b 1 ) (a 2, b 2 )... (a d, b d ), the followig formula holds P α (x) = (x 1 ) α 2 (x 2 )... (d) (x d ), (14) where α i are degrees of the olyomials αi (x i ) ad + α =, α =. I the revious sectio we discussed the reroducig erel. I this case, for the orthoormal system { Pα (x) } with the weight fuctio (13), we α = get accordig to (14) K (x, y) = = Pα (x) Pα (y) (x 1 )... (d) (x d ) (y 1 )... (d) (y d ),

7 Boudedess of orthogoal olyomials 123 where + α = ad it is obvious that K (x, y) (x 1 ) (y 1 ) α 2 (x 2 ) α 2 (y 2 )... =0 [ =0... ( (x 1 ) ) 2 =0 ( (d) (x d ) ) 2 =0 α 2 =0 ( (y 1 ) ) 2 α 2 =0 Thus, we obtai the estimatio [ K (x, y) K (x 1, x 1 ) K α 2 =0 ( (d) (y d ) ) 2 ] 1/2. ( α 2 (x 2 ) ) 2 =0 α 2 =0 (x 2, x 2 )... K (d) (x d, x d ) (d) (x d ) α (d) d (y d ) ( α 2 (y 2 ) ) /2, K (y 1, y 1 )... K (d) (y d, y d )] (15) where the erels K (i) (x i, x i ), i = 1, 2, {..., d are} accordig to the order of the systems of orthoormal olyomials (i) (x i ) with the weight fuctios w i (x i ). Theorem 5.1. Let { Pα (x) } be a system of orthoormal olyomials α = (14) with the weight fuctio w (x) = w 1 (x 1 ) w 2 (x 2 )... w d (x d ) o the domai = (a 1, b 1 ) (a 2, b 2 )... (a d, b d ). Let { Q α (x) } be a system of orthoormal olyomials with the weight α = fuctio W (x) = w (x) h (x) o. Let the fuctio h (x) be bouded o ad The, the followig estimatio 0 < M h (x). (16) Q β (x) [ K holds o. (x 1, x 1 ) K (x 2, x 2 )... K (d) (x d, x d ) ] 1/2 Proof. The olyomials Q β (x) ca be rereseted i the form ( + 1) d M (17) Q β (x) = c α Pα (x), (18)

8 124 Pavol Oršasý, Vladimír ulda ad Helea Šamajová where the coefficiets are determied by the equalities c α = w (y) Q β (y) Pα (y) dy. (19) Substitutig (19) ito the exasio (18), we obtai Q β (x) = w (y) Q β (y) Pα (y) Pα (x) dy. The sum i the itegrad is the erel of the -th degree give by (12) Q β (x) = w (y) Q β (y) K (x, y) dy. Accordig to (15) we get Q β (x) [ K (x 1, x 1 )... K (d) (x d, x d ) ] 1/2 w (y) Q β (y) ( K (y 1, y 1 )... K (d) (y d, y d ) ) 1/2 dy. The, usig the coditio (16), we get Q β (x) [ K 1 M ( ( (x 1, x 1 )... K (d) (x d, x d ) ] 1/2 w (y) h (y) ( Q β (y) ) 2 dy ) 1/2 w (y) [ K (y 1, y 1 )... K (d) (y d, y d ) ] 1/2 dy). From the roerties of orthoormality of the system { Pα (y) } with the α = weight fuctio w (y) h (y), Q β (x) b1 1 [ K M ( (x 1, x 1 )... K (d) (x d, x d ) ] 1/2 w (y) [ K (y 1, y 1 )... K (d) (y d, y d ) ] ) 1/2 dy follows. Sice erels K (i) (x i {, x i ) are associated } i the order with the systems of orthoormal olyomials (i) (x i ) with the weight fuctios w i (x i ), for i = 1, 2,..., d, we have w (y) [ K (y 1, y 1 )... K (d) (y d, y d ) ] dy = a 1 w 1 (y 1 ) K (y 1, y 1 ) dy 1... bd a d w d (y d ) K (d) (y d, y d ) dy d = ( + 1) d.

9 Boudedess of orthogoal olyomials 125 Fially, we get the statemet of the theorem Q β (x) [ K (x 1, x 1 ) K (x 2, x 2 )... K (d) (x d, x d ) ] 1/2 ( + 1) d M. Acowledgemets. The authors gratefully acowledge the Scietific rat Agecy VEA of the Miistry of Educatio of Slova Reublic ad the Slova Academy of Scieces for suortig this wor uder the rat No. 1/0234/13. Refereces [1] J. Duham, Fourier Series ad Orthogoal Polyomials, Dover Publicatios, Dover Ed editio, [2] C. F. Dul, Y. Xu, Orthogoal Polyomials of Several Variables, Ecycloedia of Mathematics ad its Alicatios 81, Cambridge Uiversity Press, htt://dx.doi.org/ /cbo [3] B. Ftore, Some roerties of geeralized Hermite olyomials, Studies of the Uiversity of Žilia, Mathematical Series, 21 (2007), 1-6. [4] B. Ftore, M. Marčoová, Marov tye olyomial iequality for geeralized Hermite weight, Tatra Mt. Math. Publ., 49 (2011), htt://dx.doi.org/ /v [5] B. Ftore, Differetial recurrece formula for the secod derivatives of orthogoal olyomials, Studies of the Uiversity of Žilia, Mathematical Series, 13 (2001), [6] B. Ftore, P. Oršasý, Korous tye iequalities for orthogoal olyomials i two variables, Tatra Mt. Math. Publ., 58 (2014), htt://dx.doi.org/ /tmm [7] A. Kajor, J. Huzvar, B. Ftore, M. Vatuch, Criterio equatios of heat trasfer for horizotal ies oe above aother at atural covectio i liear method of aroximatio, Commuicatios: Scietific Letters of the Uiversity of Žilia, 16 (2014), o. 3A, [8] J. Korous, The develomet of fuctios of oe real variable i the certai series of orthogoal olyomials, Rozravy II. třídy Česé aademie věd v Praze, č. 1 (1938), (I Czech)

10 126 Pavol Oršasý, Vladimír ulda ad Helea Šamajová [9] H. L. Krall, I. M. Sheffer, Orthogoal Polyomials i Two Variables, A. Mat. Pura Al., Series 4, 76 (1967), htt://dx.doi.org/ /bf [10] E. Levi, D. S. Lubisy, Orthogoal Polyomials with Exoetial Weights, i Caadia Mathematical Society Boos i Math., Vol. 4, Sriger, New Yor, htt://dx.doi.org/ / [11] P. Nevai, éza Freud, Orthogoal olyomials ad Christoffel fuctios. A case study, J. Arox. Theory, 48 (1986), o. 1, htt://dx.doi.org/ / (86)90016-x [12] P. Oršasý, Iequalities for some systems of orthogoal olyomials usig reroducig erel fuctios, Alied Mathematical Scieces, 9 (2015), o. 126, htt://dx.doi.org/ /ams [13] P. K. Sujeti, Classical Orthogoal Polyomials, Naua, Moscow, (I Russia) [14] P. K. Sujeti, Orthogoal Polyomials i Two Variables, ordo ad Breach Sciece Publishers, Amsterdam, [15]. Szegő, Orthogoal Polyomials, Naua, Moscow, (i Russia) Received: November 2, 2015; Published: Jauary 29, 2016