Quasi-definiteness of generalized Uvarov transforms of moment 2019. 8. 1.¢  70 Generalized...

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  • QUASI-DEFINITENESS OF GENERALIZED UVAROV TRANSFORMS OF MOMENT FUNCTIONALS

    D. H. KIM AND K. H. KWON

    Received 11 March 2001

    When σ is a quasi-definite moment functional with the monic orthogonal polynomial system {Pn(x)}∞n=0, we consider a point masses perturbation τ of σ given by τ := σ + λ

    ∑m l=1

    ∑ml k=0((−1)

    kulk/k!)δ (k)(x − cl), where λ,

    ulk, and cl are constants with ci �= cj for i �= j. That is, τ is a gen- eralized Uvarov transform of σ satisfying A(x)τ = A(x)σ, where A(x) =∏m

    l=1(x− cl) ml+1. We find necessary and sufficient conditions for τ to be

    quasi-definite. We also discuss various properties of monic orthogonal poly- nomial system {Rn(x)}∞n=0 relative to τ including two examples.

    1. Introduction

    In the study of Padé approximation (see [5, 10, 21]) of Stieltjes type mero- morphic functions

    ∫b a

    dµ(x)

    z−x +

    m∑ l=1

    ml∑ k=0

    Clk k!(

    z−cl )k+1 , (1.1)

    where −∞ ≤ a < b ≤ ∞, Clk are constants, and dµ(x) is a positive Stieltjes measure, the denominators Rn(x) of the main diagonal sequence of Padé approximants satisfy the orthogonality

    ∫b a

    Rn(x)π(x)dµ(x)+

    m∑ l=1

    ml∑ k=0

    Clk∂ k [ πRn

    ]( cl

    ) = 0, π ∈ Pn−1, (1.2)

    Copyright c© 2001 Hindawi Publishing Corporation Journal of Applied Mathematics 1:2 (2001) 69–90 2000 Mathematics Subject Classification: 33C45 URL: http://jam.hindawi.com/volume-1/S1110757X01000225.html

    http://jam.hindawi.com/volume-1/S1110757X01000225.html

  • 70 Generalized Uvarov transforms

    where Pn is the space of polynomials of degree ≤ n with P−1 = {0}. That is, Rn(x) (n ≥ 0) are orthogonal with respect to the measure

    dµ+

    m∑ l=1

    ml∑ k=0

    (−1)kClkδ (k)

    ( x−cl

    ) , (1.3)

    which is a point masses perturbation of dµ(x). Orthogonality to a positive or signed measure perturbed by one or two point masses arises naturally also in orthogonal polynomial eigenfunctions of higher order (≥ 4) ordi- nary differential equations (see [14, 15, 16, 19]), which generalize Bochner’s classification of classical orthogonal polynomials (see [6, 18]). On the other hand, many authors have studied various aspects of orthogonal polynomials with respect to various point masses perturbations of positive-definite (see [1, 2, 8, 14, 27, 28]) and quasi-definite (see [3, 4, 9, 11, 12, 20, 23]) moment functionals. In this work, we consider the most general such situation. That is, we consider a moment functional τ given by

    τ := σ+λ

    m∑ l=1

    ml∑ k=0

    (−1)kulk

    k! δ(k)

    ( x−cl

    ) , (1.4)

    where σ is a given quasi-definite moment functional, λ, ulk, and cl are complex numbers with ul,ml �= 0 and ci �= cj for i �= j, that is, τ is obtained from σ by adding a distribution with finite support. We give necessary and sufficient conditions for τ to be quasi-definite. When τ is also quasi-definite, we discuss various properties of orthogonal polynomials {Rn(x)}∞n=0 relative to τ in connection with orthogonal polynomials {Pn(x)}∞n=0 relative to σ. These generalize many previous works in [4, 9, 11, 12, 20, 23].

    2. Necessary and sufficient conditions

    For any integer n ≥ 0, let Pn be the space of polynomials of degree ≤ n and P =

    ⋃ n≥0 Pn. For any π(x) in P, let deg(π) be the degree of π(x) with the

    convention that deg(0) = −1. For the moment functionals σ, τ (i.e., linear functionals on P) (see [7]), c in C, and a polynomial φ(x) =

    ∑n k=0 akx

    k, let

    〈 σ′,π

    〉 := −

    〈 σ,π′

    〉 ; 〈φσ,π〉 := 〈σ,φπ〉;〈

    (x−c)−1σ,π 〉

    := 〈 σ,θcπ

    〉 ;

    ( θcπ

    ) (x) :=

    π(x)−π(c)

    x−c ; (2.1)

    (σφ)(x) :=

    n∑ k=0

    ( n∑

    j=k

    ajσjk

    ) xk; 〈στ,π〉 = 〈σ,τπ〉, π ∈ P.

  • D. H. Kim and K. H. Kwon 71

    We also let

    F(σ)(z) :=

    ∞∑ n=0

    σn

    zn+1 (2.2)

    be the (formal) Stieltjes function of σ, where σn := 〈σ,xn〉 (n ≥ 0) are the moments of σ. Following Zhedanov [29], for any polynomials A(z), B(z), C(z), D(z) with no common zero and |C|+ |D| �≡ 0, let

    S(A,B,C,D)F(σ)(z) := AF(σ)+B

    CF(σ)+D . (2.3)

    If S(A,B,C,D)F(σ) = F(τ) for some moment functional τ, then we call τ a rational (resp., linear) spectral transform of σ (resp., when C(z) = 0). Then S(A,B,C,D)F(σ) = F(τ) if and only if

    xA(x)σ = C(x)(στ)+xD(x)τ,

    〈σ,A〉+x(σθ0A)(x)+xB(x) = (στ)(θ0C)(x)+〈τ,D〉+x(τθ0D)(x). (2.4) In particular, for any c and β in C, let

    U(c,β)F(σ) := (z−c)F(σ)+β

    z−c (2.5)

    be the Uvarov transform (see [28, 29]) of F(σ). Then for any {ci}ki=1 and {βi}

    k i=1 in C,

    F(τ) := U ( ck,βk

    ) · · ·U(c1,β1)F(σ) = A(z)F(σ)+B(z) A(z)

    , (2.6)

    where A(z) = ∏k

    i=1(z−ci), B(z) = ∑k

    i=1 βi ∑k

    j=1 j �=i

    (z−cj), and by (2.4)

    A(x)τ = A(x)σ. (2.7)

    In this case, we say that τ is a generalized Uvarov transform of σ. Conversely, if (2.7) holds for some polynomial A(x) (�≡ 0), then

    F(τ) = A(z)F(σ)+

    ( τθ0A

    ) (z)−

    ( σθ0A

    ) (z)

    A(z) (2.8)

    and F(τ) is obtained from F(σ) by deg(A) successive Uvarov transforms (see [29]), that is, τ is a generalized Uvarov transform of σ.

    In the following, we always assume that τ is a moment functional given by (1.4), where σ is a quasi-definite moment functional. Let {Pn(x)}∞n=0 be the monic orthogonal polynomial system (MOPS) relative to σ satisfying the

  • 72 Generalized Uvarov transforms

    three term recurrence relation

    Pn+1(x) = ( x−bn

    ) Pn(x)−cnPn−1(x), n ≥ 0,

    ( P−1(x) = 0

    ) . (2.9)

    Since (1.4) implies (2.7) with A(x) = ∏m

    l=1(x−cl) ml+1, τ is a generalized

    Uvarov transform of σ. Then our main concern is to find conditions under which a generalized Uvarov transform τ, given by (1.4), of σ is also quasi- definite. In other words, we are to solve the division problem (2.7) of the moment functionals.

    Let

    Kn(x,y) :=

    n∑ j=0

    Pj(x)Pj(y)〈 σ,P2j

    〉 , n ≥ 0 (2.10) be the nth kernel polynomial for {Pn(x)}∞n=0 and K(i,j)n (x,y)=∂xi∂yjKn(x,y). We need the following lemma which is easy to prove.

    Lemma 2.1. Let V = (x1,x2, . . . ,xn)t and W = (y1,y2, . . . ,yn)t be two vec- tors in Cn. Then

    det ( In +VW

    t )

    = 1+

    n∑ j=1

    xjyj, n ≥ 1, (2.11)

    where In is the n×n identity matrix.

    Theorem 2.2. The moment functional τ is quasi-definite if and only if dn �= 0, n ≥ 0, where dn is the determinant of (

    ∑m l=1(ml + 1)) ×

    ( ∑m

    l=1(ml +1)) matrix Dn:

    Dn := [ Atl(n)

    ]m t,l=1

    , n ≥ 0, (2.12)

    where

    Atl(n) =

    [ δtlδki +λ

    ml−i∑ j=0

    ul,i+j

    i!j! K(k,j)n

    ( ct,cl

    )]mt ml k=0, i=0

    . (2.13)

    If τ is quasi-definite, then the MOPS {Rn(x)}∞n=0 relative to τ is given by

    Rn(x) = Pn(x)−λ

    m∑ l=1

    ml∑ i=0

    ml−i∑ j=0

    ul,i+j

    i!j! K

    (0,j) n−1

    ( x,cl

    ) R(i)n

    ( cl

    ) , (2.14)

  • D. H. Kim and K. H. Kwon 73

    where {R(i)n (cl)} m ml l=1,i=0 are given by

    Dn−1

     

    Rn ( c1

    ) R′n

    ( c1

    ) ...

    R (m1) n

    ( c1

    ) Rn

    ( c2

    ) ...

    R (mm) n

    ( cm

    )

     

    =

     

    Pn ( c1

    ) P′n

    ( c1

    ) ...

    P (m1) n

    ( c1

    ) Pn

    ( c2

    ) ...

    P (mm) n

    ( cm

    )

     

    , n ≥ 0 (D−1 = I). (2.15)

    Moreover, 〈 τ,R2n

    〉 =

    dn

    dn−1

    〈 σ,P2n

    〉 , n ≥ 0 (d−1 = 1). (2.16)

    Proof. (⇒). Assume that τ is quasi-definite and expand Rn(x) as Rn(x) = Pn(x)+

    n−1∑ j=0

    CnjPj(x), n ≥ 1, (2.17)

    where Cnj = 〈σ,RnPj〉/〈σ,P2j 〉, with 0 ≤ j ≤ n−1. Here, 〈 σ,RnPj

    〉 =

    〈 τ−λ

    m∑ l=1

    ml∑ k=0

    (−1)kulk

    k! δ(k)

    ( x−cl

    ) ,RnPj

    = −λ

    m∑ l=1

    ml∑ k=0

    ulk

    k!

    k∑ i=0

    ( k

    i

    ) R(i)n

    ( cl

    ) P

    (k−i) j

    ( cl

    ) (2.18)

    so that

    Rn(x) = Pn(x)−λ

    n−1∑ j=0

    Pj(x)〈 σ,P2j

    〉 m∑ l=1

    ml∑ k=0

    ulk

    k!

    k∑ i=0

    ( k

    i

    ) R(i)n

    ( cl

    ) P

    (k−i) j

    ( cl

    )

    = Pn(x)−λ

    m∑ l=1

    ml∑ k=0

    ulk

    k!

    k∑ i=0

    ( k

    i

    ) R(i)n

    ( cl

    ) K

    (0,k−i) n−1

    ( x,cl

    )

    = Pn(x)−λ

    m∑ l=1

    ml∑ i=0

    ml−i∑ j=0

    ul,i+j

    i!j! K

    (0,j) n−1

    ( x,cl

    ) R(i)n

    ( cl

    ) .

    (2.19)