7 pages, no figures.-- MSC2000 codes: Primary 42C05; Secondary 33C25.MR#: MR1882617 (2003b:42042)The concept of k-coherence of two positive measures μ1 and μ2 is useful in the study of the Sobolev orthogonal polynomials. If μ1 or μ2 are compactly supported on R then any 0-coherent pair or symmetrically 1-coherent
pair (μ1, μ2) must contain a Jacobi measure (up to affine transformation). Here examples of k-coherent pairs (k ≥ 1) when neither μ1 nor μ2 are Jacobi are constructed.Research of F. Marcellán supported by Dirección General de Investigación(Ministerio de Ciencia y Tecnología) of Spain under grant BFM2000-0206-C04-01. Research of A. Martínez-Finkelshtein partially supported by INTAS project 2000-272, a research grant of Dirección General de Enseñanza Superior (DGES) of Spain, project code PB95-1205, and by Junta de Andalucía, Grupo de Investigación FQM 0229. Research of J. J. Moreno-Balcázar partially supported by Junta de Andalucía, Grupo de Investigación FQM 0229, and INTAS project 2000-272.Publicad

Marcellán Español, Francisco José

Martínez-Finkelshtein, Andrei

Moreno Balcázar, Juan José

Universidad Carlos III de Madrid e-Archivo

MARGARITA MATHEMATICA EN MEMORIA DEJOSE´ JAVIER (CHICHO) GUADALUPE HERNA´NDEZ(Luis Espan˜ol y Juan L. Varona, editores),Servicio de Publicaciones, Universidad de La Rioja,Logron˜o, Spain, 2001.k-COHERENCE OF MEASURES WITH NON-CLASSICALWEIGHTSFRANCISCO MARCELLA´N, ANDREI MARTI´NEZ-FINKELSHTEINAND JUAN J. MORENO-BALCA´ZARDedicated to Prof. Dr. Jose´ Javier Guadalupe, Chicho.His memory will always remain with us.Abstract. The concept of k-coherence of two positive measures µ1 and µ2is useful in the study of the Sobolev orthogonal polynomials. If µ1 or µ2 arecompactly supported onRthen any 0-coherent pair or symmetrically 1-coherentpair (µ1, µ2) must contain a Jacobi measure (up to affine transformation). Hereexamples of k-coherent pairs (k ≥ 1) when neither µ1 nor µ2 are Jacobi areconstructed.1. Introduction and k-coherenceGiven two positive distribution functions (see, e.g., [10]), µ1 and µ2, supportedon R and with an infinite number of points of increase, we can define the Sobolevinner product(1) (p, q)S =∫pq dµ1 +∫p′q′ dµ2 ,on the space P of algebraic polynomials with real coefficients. The sequence Qn,degQn(x) = n, n = 0, 1, . . . of monic polynomials orthogonal with respect to (1)has been object of interest in the last years. Algebraic and analytic properties ofthese polynomials (known as Sobolev orthogonal polynomials) have been studiedand compared with the corresponding properties of the monic polynomial sequencesPn(µi, ·) such that for i = 1, 2,deg Pn(µi, ·) = n , 〈Pn(µi, x), xj〉i = 0 , j = 0, . . . , n− 1 ,with〈p, q〉i =∫pq dµi ,2000 Mathematics Subject Classification. Primary 42C05; Secondary 33C25.Key words and phrases. Coherent pairs of measures, Sobolev orthogonal polynomials.Research of F. Marcella´n supportedby Direccio´nGeneral de Investigacio´n (Ministerio de Cienciay Tecnolog´ıa) of Spain under grant BFM2000-0206-C04-01.Research of A. Mart´ınez-Finkelshtein partially supportedby INTAS project 2000-272, a researchgrant of Direccio´n General de Ensen˜anza Superior (DGES) of Spain, project code PB95-1205, andby Junta de Andaluc´ıa, Grupo de Investigacio´n FQM 0229.Research of J. J. Moreno-Balca´zar partially supported by Junta de Andaluc´ıa, Grupo de Inves-tigacio´n FQM 0229, and INTAS project 2000-272.7778 F. MARCELLA´N, A. MARTI´NEZ-FINKELSHTEIN AND J. J. MORENO-BALCA´ZAR(see [5]–[8]). This study is much simplified if we assume a link between the measuresµ1 and µ2, restrictive enough to follow the structural properties of the sequence{Qn}, but sufficiently loose to get some insight concerning the general situation.One of these links, appeared in [3], is the concept of the coherence of measures;it proved to be a fruitful tool in the study of the Sobolev orthogonal polynomials.All the coherent pairs of measures (µ1, µ2) have been obtained in [9]; in particular,it was shown that at least one of them must be a classical weight function (Jacobi,Laguerre or Hermite).Thus, in order to deal with pairs (µ1, µ2) with non-classical weights, we needto extend the concept of coherence. The main goal of this work is to show howto construct such pairs of measures, linked by the so-called k-coherence relation.These measures are in particular suitable for the study of structural and analyticproperties of the Sobolev polynomials {Qn} (see, e.g., [7]).Definition 1 (k-coherence). Let µi , i = 1, 2 , be two positive measures. Then(µ1, µ2) is a k-coherent pair of measures (k ≥ 0) if for every n ∈ N , n ≥ k + 1 ,there exist σ(n)n (µ1, µ2) , . . . , σ(n−k)n (µ1, µ2) ∈ R , with σ(n−k)n (µ1, µ2) 	= 0, such that(2) Pn(µ2, x) =P ′n+1(µ1, x)n+ 1−k∑j=0σ(n−j)n (µ1, µ2)P ′n−j(µ1, x)n − j .Remark: Relation (2) can be considered for k = −1, but this case is trivial andis only verified by the classical polynomials: Jacobi, Laguerre, Hermite and Bessel(see [10]). If k = 0 , we obtain the 0-coherence (or just coherence) introduced byIserles et al. in [3], and k = 1 yields the 1-coherence studied by de Bruin andMeijer in [2] as well as by Kwon et al in [4]. A particular case of 1-coherence is thesymmetric coherence, also introduced by Iserles et al. in [3], which takes place whenboth measures are symmetric with respect to the origin.When the measures µ1 and µ2 are linked by k-coherence, a connection betweenthe families {Qn} and {Pn(µ1, ·)} can be established. Indeed, denote kn(µi) =〈Pn(µi, ·), Pn(µi, ·)〉i , i = 1, 2 , and k˜n = (Qn, Qn)S . In order to simplify notationswe omit the explicit reference to the pair (µ1, µ2) in the coefficients σ(n−j)n , wheneverit cannot lead to confusion.We have the following relation between the polynomials Pn(µ1, x) and Qn(x):Proposition 1. Let (µ1, µ2) be a k-coherent pair of measures satisfying (2). Then,for n ≥ k + 1 , there exist α(n−j)n ∈ R , j = 0, . . . , k , with α(n−k)n 	= 0, such thatPn+1(µ1, x)−k∑j=0σ(n−j)nn+ 1n− j Pn−j(µ1, x) = Qn+1(x) +k∑j=0α(n−j)n Qn−j(x) .Proof. Consider the 1-parametric family of Sobolev inner products, given by(3) (p, q)S =∫pq dµ1 + λ∫p′q′ dµ2 , λ ≥ 0 .Standard arguments allow to express the polynomial Qn(x; λ), orthogonal with re-spect to (3), as a ratio of two determinants. It can be observed that Qn(x; 1) =k-COHERENCE OF MEASURES WITH NON-CLASSICAL WEIGHTS 79Qn(x), and that the coefficients of Qn(x; λ) are rational functions in λ such that foreach n a non-trivial limit polynomialRn(x) = limλ→∞Qn(x; λ)exists. This monic polynomial has degree n and satisfies (see [8]):〈Rn, 1〉1 = 0 , n ≥ 1 ,(4)〈R′n, xm〉2 = 0 , n ≥ 2 , 0 ≤m ≤ n− 2 .(5)From (5) we get(6) R′n+1(x) = (n + 1)Pn(µ2, x) , n ≥ 1 .For the time being, we have not used the concept of k-coherence. Now, by (2),relation (6) can be rewritten asR′n+1(x) = P′n+1(µ1, x)− (n+ 1)k∑j=0σ(n−j)n (µ1, µ2)P ′n−j(µ1, x)n− j , n ≥ k + 1 .Taking into account (4),(7) Rn+1(x) = Pn+1(µ1, x)−k∑j=0σ(n−j)nn+ 1n− j Pn−j(µ1, x) , n ≥ k + 1 .On the other hand, expanding Rn+1 in the basis {Q0, Q1, . . . , Qn+1}, and using (5)and (7) we get(8) Rn+1(x) = Qn+1(x) +k∑j=0α(n−j)n Qn−j(x) ,where, using (6) and (7),α(n−j)n =(Rn+1, Qn−j)Sk˜n−j=〈Rn+1, Qn−j〉1k˜n−j, j = 0, . . . , k .Moreover, we have, for j = k,α(n−k)n = −σ(n−k)nn+ 1n− kkn−k(µ1)k˜n−k	= 0 .It remains to gather (7) and (8). 2. k-coherent pairsGiven a positive and integrable function f on an open bounded interval ∆ ⊂R , f(x) dx stands for an absolutely continuous measure, supported on ∆ , whoseRadon-Nikodym derivative with respect to the Lebesgue measure dx on ∆ is f(x) .Furthermore, for ξ ∈ R , δ(ξ) will be the Dirac delta (mass point) at ξ.The complete classification of the coherent and symmetrically coherent pairs wasgiven by Meijer in [9]. In particular, he proved that if supp(µ1) = [−1, 1] then allthe 0-coherent pairs (µ1, µ2) are those described in the following table:80 F. MARCELLA´N, A. MARTI´NEZ-FINKELSHTEIN AND J. J. MORENO-BALCA´ZARTable 1Case µ1 µ21 (1− x)α−1(1 + x)β−1dx 1|x−ξ2|(1 − x)α(1 + x)βdx+Mδ(ξ2)2 |x− ξ1|(1− x)α−1(1 + x)β−1dx (1− x)α(1 + x)βdx3 (1 + x)β−1dx+Mδ(1) (1 + x)βdx4 (1− x)α−1dx+Mδ(−1) (1− x)αdxwith α, β > 0 , |ξ1| > 1 , |ξ2| ≥ 1 , and M ≥ 0 .In a similar way, all the symmetrically coherent pairs (µ1, µ2) were classified asfollows:Table 2Case µ1 µ25 (1− x2)α−1dx (1−x2)αx2+ξ22dx ξ2 	= 06 (1− x2)α−1dx (1−x2)αξ22−x2 dx+M [δ(−ξ2) + δ(ξ2)] |ξ2| ≥ 17 (x2 + ξ21)(1 − x2)α−1dx (1− x2)αdx ξ1 	= 08 (ξ21 − x2)(1 − x2)α−1dx (1− x2)αdx |ξ1| > 19 dx+M [δ(1) + δ(−1)] dxwith α > 0 and M ≥ 0 . Notice that in all the cases at least one of the measures isthe Jacobi weight.The main goal of this work is to give some non-trivial examples of k-coherentpairs (µ1, µ2) , k ≥ 1 , for which neither µ1 nor µ2 are Jacobi, combining coherentpairs from Tables 1 and 2:Proposition 2.(i) The pairCase 1 µ1 µ2|x− ξ1|(1− x)α−1(1 + x)β−1dx 1|x−ξ2| (1− x)α(1 + x)βdx+Mδ(ξ2)with α, β > 0 , |ξ1| > 1 , |ξ2| ≥ 1 and M ≥ 0 , is 1-coherent.(ii) PairsCase 2 µ1 µ22a (x2 + ξ21)(1− x2)α−1dx 1|x−ξ2 | (1− x2)αdx+Mδ(ξ2) ξ1 = 0, |ξ2| ≥ 12b (ξ21 − x2)(1− x2)α−1dx 1|x−ξ2 | (1− x2)αdx+Mδ(ξ2) |ξ1| > 1, |ξ2| ≥ 1k-COHERENCE OF MEASURES WITH NON-CLASSICAL WEIGHTS 81with α > 0 , and M ≥ 0 , andCase 3 µ1 µ23a |x− ξ1|(1− x2)α−1dx (1−x2)αx2+ξ22 dx |ξ1| > 1, ξ2 = 03b |x− ξ1|(1− x2)α−1dx (1−x2)αξ22−x2dx+M [δ(−ξ2) + δ(ξ2)] |ξ1| > 1, |ξ2| ≥ 1with α > 0 , are 2-coherent.(iii) PairsCase 4 µ1 µ24a (x2 + ξ21)(1− x2)α−1dx (1−x2)αx2+ξ22dx ξ1 = 0, ξ2 = 04b (x2 + ξ21)(1− x2)α−1dx (1−x2)αξ22−x2dx+M [δ(−ξ2) + δ(ξ2)] ξ1 = 0, |ξ2| ≥ 14c (ξ21 − x2)(1− x2)α−1dx (1−x2)αx2+ξ22dx |ξ1| > 1, ξ2 = 04d (ξ21 − x2)(1− x2)α−1dx (1−x2)αξ22−x2dx+M [δ(−ξ2) + δ(ξ2)] |ξ1| > 1, |ξ2| ≥ 1with α > 0 , and M ≥ 0 , are 3-coherent.In all the cases, the absolutely continuous parts of µ1 and µ2 are supported on[−1, 1] .Proof. Fix α, β > 0 corresponding to µ2 and denote the Jacobi measures(9) ν1 = (1− x)α−1(1 + x)β−1dx , ν2 = (1− x)α(1 + x)βdx ,with the following convention: if β does not appear explicitly in the expression ofµ2 (Cases 2–4), then assume(10) α = β .We use the fact that the derivative of a Jacobi polynomial is still a Jacobi poly-nomial (see [10]):(11) P ′n(ν1, x) = nPn−1(ν2, x) .Consider Case 1. Takeµ1 = |x− ξ1|(1− x)α−1(1 + x)β−1dx, µ2 = 1|x− ξ2|(1− x)α(1 + x)βdx+Mδ(ξ2) .Since µ2 is 0-coherent with ν1 (see Case 1 of Table 1), we havePn(µ2, x) =P ′n+1(ν1, x)n + 1− σ(n)n (ν1, µ2)P ′n(ν1, x)n,so that using (11),Pn(µ2, x) = Pn(ν2, x)− σ(n)n (ν1, µ2)Pn−1(ν2, x) .But ν2 is 0-coherent with µ1 (see Case 2, Table 1). Then,Pn(ν2, x) =P ′n+1(µ1, x)n+ 1− σ(n)n (µ1, ν2)P ′n(µ1, x)n,82 F. MARCELLA´N, A. MARTI´NEZ-FINKELSHTEIN AND J. J. MORENO-BALCA´ZARso that,Pn(µ2, x) =P ′n+1(µ1, x)n+ 1−(σ(n)n (µ1, ν2) + σ(n)n (ν1, µ2)) P ′n(µ1, x)n+ σ(n)n (ν1, µ2)σ(n−1)n−1 (µ1, ν2)P ′n−1(µ1, x)n− 1 , n ≥ 2 .This proves that (µ1, µ2) is 1-coherent with coherence parametersσ(n)n (µ1, µ2) = σ(n)n (µ1, ν2) + σ(n)n (ν1, µ2) ,σ(n−1)n (µ1, µ2) = −σ(n)n (ν1, µ2)σ(n−1)n−1 (µ1, ν2) .In the same way, if we combine coherent pairs of Table 1 or Table 2 we get allthe pairs that appear in Proposition 2. We omit the details and summarize thecoherence parameters σ(n−j)n (µ1, µ2) using the notation of (9) and (10):Case σ(n)n (µ1, µ2) σ(n−1)n (µ1, µ2)1 σ(n)n (µ1, ν2) + σ(n)n (ν1, µ2) −σ(n−1)n−1 (µ1, ν2)σ(n)n (ν1, µ2)2 σ(n)n (ν1, µ2) σ(n−1)n (µ1, ν2)3 σ(n)n (µ1, ν2) σ(n−1)n (ν1, µ2)4 0 σ(n−1)n (µ1, ν2) + σ(n−1)n (ν1, µ2)Case σ(n−2)n (µ1, µ2) σ(n−3)n (µ1, µ2)1 0 02 −σ(n−2)n−1 (µ1, ν2)σ(n)n (ν1, µ2) 03 −σ(n−2)n−2 (µ1, ν2)σ(n−1)n (ν1, µ2) 04 0 −σ(n−3)n−2 (µ1, ν2)σ(n−1)n (ν1, µ2)From the proof of Proposition 2 it can be seen that in all the cases the coherenceparameters σ(n−j)n (µ1, µ2) are sums or products of the parameters involved in thecoherence or symmetrically coherence relations, which are convergent as n → ∞(see [5]–[8]). Hence, σ(n−j)n (µ1, µ2) have finite limits when n → ∞, fact which canbe used in the study of the asymptotics of {Qn}. Actually, following the schemeproposed in [8] we can proveTheorem 1. Let (µ1, µ2) be a pair as in Proposition 2. Then,limn→∞Qn(x)Pn(µ2, x)=1Φ′(x)k-COHERENCE OF MEASURES WITH NON-CLASSICAL WEIGHTS 83uniformly on compact subsets of C \ [−1, 1] where Φ(x) = (x + √x2 − 1)/2 with√x2 − 1 > 0 when x > 1 .Open questions. In this paper some non-trivial k-coherent pairs of measures aregiven; still the problem of the complete classification of all k-coherent pairs (withk ≥ 1) remains open and has an independent interest. In this sense, in [4] theauthors have obtained all the 1-coherent pairs when one of the measures is classic(i.e., Jacobi, Laguerre or Hermite).Furthermore, the approach described in this work can be applied to obtain k-coherent pairs with unbounded support. Asymptotic properties for coherent andsymmetrically coherent pairs with unbounded support have been obtained in [6]and [1], respectively, but not much more in this direction is known.References[1] M. Alfaro, J. J. Moreno-Balca´zar, T. E. Pe´rez, M. A. Pin˜ar and M. L. Rezola, Asymptoticsof Sobolev orthogonal polynomials for Hermite coherent pairs, J. Comp. Appl. Math. 133(2001), 141–150.[2] M. G. de Bruin and H. G. Meijer, Zeros of orthogonal polynomials in a non-discrete Sobolevspace, Ann. Numer. Math. 2 (1995), 233–246.[3] A. Iserles, P. E. Koch, S. Nørsett and J. M. Sanz-Serna, On polynomials orthogonal withrespect to certain Sobolev inner products, J. Approx. Theory 65 (1991), 151–175.[4] K. H. Kwon, J. H. Lee and F. Marcella´n, Generalized coherent pairs, J. Math. Anal. Appl.253 (2001), 482–514.[5] F. Marcella´n, A. Mart´ınez-Finkelshtein and J. J. Moreno-Balca´zar, Asymptotics of Sobolevorthogonal polynomials for symmetrically coherent pairs of measures with compact support,J. Comp. Appl. Math. 81 (1997), 217–227.[6] F. Marcella´n and J. J. Moreno-Balca´zar, Strong and Plancherel-Rotach asymptotics of non-diagonal Laguerre-Sobolev, J. Approx. Theory 110 (2001), 54–73.[7] A. Mart´ınez-Finkelshtein, Asymptotics properties of Sobolev orthogonal polynomials, J.Comp. Appl. Math. 99 (1998), 491–510.[8] A. Mart´ınez-Finkelshtein, J. J. Moreno-Balca´zar, T. E. Pe´rez and M. A. Pin˜ar, Asymptotics ofSobolev orthogonal polynomials for coherent pairs of measures, J. Approx. Theory 92 (1998),280–293.[9] H. G. Meijer, Determination of all coherent pairs, J. Approx. Theory 89 (1997), 321–343.[10] G. Szego˝, Orthogonal polynomials, 4th edition, Amer. Math. Soc. Colloq. Publ. 23, Amer.Math. Soc., Providence, RI, 1975.F. Marcella´n: Departamento de Matema´ticas, Universidad Carlos III de Madrid,SpainE-mail address : pacomarc@ing.uc3m.esA. Mart´ınez-Finkelshtein and J. J. Moreno-Balca´zar: Departamento de Estad´ısticay Matema´tica Aplicada, Universidad de Almer´ıa, SpainE-mail address : andrei@ual.es, balcazar@ual.esURL: http://www.ual.es/~andrei, http://www.ual.es/~balcazar: Instituto Carlos I de F´ısica Teo´rica y Computacional, Universidad de Gra-nada, Spain

k-Coherence of measures with non-classical weights

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