Orthogonal polynomials for exponential weights x2α(1 – x2)2ρe–2Q(x) on [0, 1)

and Rong Liu
  • Corresponding author
  • College of Computer and Information Sciences, Fujian Agriculture and Forestry University, Fuzhou, Fujian 350002, P. R. China
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Abstract

Let Wα,ρ = xα(1 – x2)ρeQ(x), where α > –12 and Q is continuous and increasing on [0, 1), with limit ∞ at 1. This paper deals with orthogonal polynomials for the weights Wα,ρ2 and gives bounds on orthogonal polynomials, zeros, Christoffel functions and Markov inequalities. In addition, estimates of fundamental polynomials of Lagrange interpolation at the zeros of the orthogonal polynomial and restricted range inequalities are obtained.

1 Introduction and results

In this paper, for α > –12, we set

Wα,ρ(x)=xα(1x2)ρW(x),x[0,1),

for which the moment problem possesses a unique solution, and discuss the orthogonal polynomials for the weight Wα,ρ2 on [0, 1). The main results tell us that adding an even factor (1 – x2)2ρ to the weight x2αe–2Q(x), α > –12, under sufficient conditions for ρ and Q(x), its properties will be invariant. It is an important and meaningful extension to the case ρ = 0 (we can see [1, 2]).

Assume that

I=[0,d),0<d

and

W=eQ,

where Q : I → [0, ∞) is continuous. All power moments for W exist. Such W is called an exponential weight on I. In the paper, for 0 < p ≤ ∞, ∥⋅∥Lp(I) is the usual Lp (quasi) norm on the interval I.

Levin and Lubinsky [3, 4] discussed orthogonal polynomials for exponential weights W2 on [–1, 1] and (c, d), c < 0 < d, respectively. Kasuga and Sakai [5] dealt with generalized Freud weights |x|2αW(x)2 in (–∞, ∞). Liu and Shi [6] considered generalized Jacobi-Exponential weights UW, where U(x) is generalized Jacobi weights on (c, d), c < 0 < d, and gave the estimates of the zeros of orthogonal polynomials for UW. Meanwhile, Shi [7] gave the estimates of the Lp Christoffel functions for UW on (c, d). In [8], Liu and Shi got further estimations of the Lp Christoffel functions for UW on [–1, 1]. In [9], Notarangelo stated analogues of the Mhaskar-Saff inequality for doubling-exponential weights on (–1, 1). The above references dealt with exponential weights on a real interval (c, d) containing 0 in its interior. In [1, 2], Levin and Lubinsky dealt with exponential weights x2αW(x)2, α > –1/2, in [0, d), since the results of [3, 4] cannot be applied through such one-sided weights. All the results on one-sided case and two-sided case are useful in polynomial approximation. Mastroianni and Notarangelo [10, 11] considered Lagrange interpolation processes based on the zeros for exponential weight on (–1, 1) and the real semiaxis, respectively.

Levin and Lubinsky [1, 2] defined an even weight corresponding to the one-sided weight. The weight is denoted that

I=(d,d)

and for xI*,

Q(x):=Q(x2),

W(x):=exp(Q(x)).

Throughout, c, C0, C1, … stand for positive constants independent of variables and indices, unless otherwise indicated and their values may be different at different occurrences, even in subsequent formulas. Moreover, CnDn means that there are two constants c1 and c2 such that c1Cn/Dnc2 for the relevant range of n. We write c = c(λ) or cc(λ) to indicate dependence on or independence of a parameter λ. Pn stands for the set of polynomials of degree at most n.

A function f : [0, d) → (0, ∞) is said to be quasi-increasing if there exists C > 0 such that

f(x)Cf(y),0<xy<d.

Definition 1.1

(see [1, Definition 1.1]). LetI = [0, d). Assume thatW = eQwhere Q : I → [0, ∞) satisfies the following properties:

  • xQ′(x) ∈ C(I) with limit 0 at 0 andQ(0) = 0.
  • Qexists in (0, d), whileQ*is positive in (0, d).
  • limxdQ(x)=.
  • The function
    T(x):=xQ(x)Q(x),x(0,d)
    is quasi-increasing in (0, d), with
    T(x)Λ>12,x(0,d).
  • There existsC1 > 0 such that
    |Q(x)|Q(x)C1Q(x)Q(x),a.e.x(0,d).
    Then we writeW ∈ 𝓛(C2). If also there exists a compact subintervalJofI*, andC2 > 0 such that
    Q(x)|Q(x)|C2|Q(x)|Q(x),a.e.xI{J},
    then we writeW ∈ 𝓛(C2+).

For W ∈ 𝓛(C2) and t > 0, the Mhaskar-Rahmanov-Saff number 0 < at := at(Q) is defined by the equation

t=1π01atxQ(atx)[x(1x)]1/2dx.

Put for t > 0,

Δt:=Δt(Q):=[0,at),ηt:=ηt(Q):=tT(at)2/3,

φt(x):=φt(Q;x):=x+att2(a2tx)tatx+atηt,x[0,at],φt(at),x>at,φt(0),x<0.

We also need a modification of φt, namely

φt(x):=xx+att2φt(x)=x(a2tx)tatx+atηt,x[0,at],φt(at),x>at.

The orthogonal polynomial of degree n for Wα,ρ2 is denoted by pn(Wα,ρ2, x) or just pn(x). Thus

Ipn(x)pm(x)Wα,ρ2dx=δnm

and

pn(x)=γnxn+,

where γn = γn(Wα,ρ2) > 0.

The zeros of pn(x) are denoted by

xnn<xn1,n<<x2n<x1n,

and the corresponding fundamental polynomials of Lagrange interpolation are polynomials jnPn–1. The classical Christoffel function is

λn(Wα,ρ2;x)=infPPn(PWα,ρL2(I)/|P(z)|)2.

Considering the factor (1 – x2)ρ, we introduce the following weight

Q^(x):=Q(x)+ρq(x),q(x):=ln(1x2),W^(x):=eQ^(x).

Before stating our results, we need some corresponding notations,

Δ^t:=Δt(Q^):=[0,a^t),a^t:=at(Q^),η^t:=ηt(Q^),T^(x):=T(Q^;x),φ^t(x):=φt(Q^;x).

The following theorems are similar in spirit with their analogues for weights (1 – x2)ρeQ(x) on two-sided intervals [12]. while the results of [12] cannot be applied to one-sided case. Furthermore, the formulation of the results are different, just as there are between the Laguerre and Hermite weights.

Theorem 1.1

LetI = [0, 1) andW ∈ 𝓛(C2) (orW ∈ 𝓛(C2+)). Suppose that there existsλsuch that forxI ∖ {0},

Q(x)λ1+x2(1x2)2,

where

λ>2|ρ|ΛΛ12,ρ<0,0,ρ0,

andΛis defined by (1.3). Then we have ∈ 𝓛(C2) (or 𝓛(C2+)).

According to the above Theorem and applying Theorem 1.5 in [1] and Theorem 1.4 and Theorem 1.5 in [2], we gain the following Theorem 1.2. We also get the following Theorem 1.3, by using Theorem 1.2 and Theorem 1.3 in [1].

Theorem 1.2

Letα > –12, W ∈ 𝓛(C2) and the other assumptions of Theorem 1.1 be valid. Assume that 0 < p ≤ ∞.

  • LetL, ζ ≥ 0. Letβ > –1pifp < ∞ andβ ≥ 0 ifp = ∞. There existC1, n0 > 0 such that for nn0andPPn,
    PW^(x)xβLp(I)C1PW^(x)xβLp[La^nn2,a^n(1ζη^n)].
    Moreover, givenr > 1, there existC2, n0, ν > 0 such that for nn0andPPn,
    PW^(x)xβLp(a^rn,1)exp(C2nν)PW^(x)xβLp(Δ^n).
  • Letβ > –1pifp < ∞ andβ ≥ 0 ifp = ∞. Given 0 < r < 1. Then for n ≥ 1, PPnand for someC,
    PW^(x)φ^n(x)xβLp(I)CPW^(x)xβLp(I),PW^(x)xβLp(I)Cn2a^nPW^(x)xβLp(I)
    and
    PW^(x)xβLp[a^rn,1)Cna^nT^(a^n)PW^(x)xβLp(I).
  • LetL > 0. Then uniformly for n ≥ 1 andx ∈ [0, ân(1 + Lη̂n)],
    λn(Wα,ρ2;x)φ^n(x)W^2(x)x+a^nn22α.
    Moreover, there existsC > 0 such that uniformly for n ≥ 1 andxI,
    λn(Wα,ρ2;x)Cφ^n(x)W^2(x)x+a^nn22α.
  • Then uniformly for n ≥ 1,
    supxIpn(Wα,ρ2;x)W^(x)x+a^nn2αx+a^nn2(a^nx)141.
  • There existC3, C4 > 0 such that for n ≤ 1 and 1 ≤ jn – 1,
    xjnxj+1,nC3φ^n(xjn),xnna^nn2
    and
    a^n(1C4η^n)x1n<a^n+α+14.
    Furthermore, for each fixedjandn, xjnis a non-decreasing function ofα.

Theorem 1.3

Letα > –12, W ∈ 𝓛(C2+) and the other assumptions of Theorem 1.1 be valid.

  • Let 0 < β < 1, then uniformly for n ≥ 1,
    supxI|pn(x)|W^(x)x+a^nn2αna^n12,supx[a^βn,1)|pn(x)|W^(x)x+a^nn2αa^n12(nT^(a^n))16
    and
    xjnxj+1,nφ^n(xjn),1j<n.
    IfW ∈ 𝓛(C2), these estimates hold withreplaced byC.
  • There existsn0such that uniformly forn > n0, 1 ≤ jn,
    pnWα,ρ(xjn)φ^n(xjn)1xjn(a^nxjn)14,
    pn1Wα,ρ(xjn)a^n1xjn(a^nxjn)14,maxxIjn(x)W^(x)x+a^nn2α(Wα,ρ)1(xjn)1
    and
    1x1na^nη^n.
    If we assume instead thatW ∈ 𝓛(C2), then (1.8) holds withreplaced byCand (1.9) holds withreplaced byC.
  • For jn – 1 andx ∈ [xj+1,n, xjn],
    pnWα,ρ(x)min|xxjn|,|xxj+1,n|φ^n(xjn)1xjn(a^nxjn)14.

Here we give the following two theorems as examples of Theorem 1.1.

Theorem 1.4

LetI = [0, 1), τ > 0, and

Q(x)=(1x)τ1,x[0,1).

  • Then
    T(x)1,x(0,1).
  • If
    τ>4|ρ|,ρ<0,0,ρ0,
    then we have ∈ 𝓛(C2+).

Theorem 1.5

LetI = [0, 1), k ≥ 1, τ > 0, and

Q(x)=expk(1x)τexpk(1),x[0,1).

  • Then the relation of (1.10) holds.
  • If
    τ>4|ρ|j=1kexpj(1),ρ<0,0,ρ0,
    then we have ∈ 𝓛(C2+).

Remark 1.1

For Theorem 1.4 and Theorem 1.5, Levin and Lubinsky in [1, 2] discussed the case whenρ = 0.

We shall give some technical lemmas in Section 2 and the proofs of Theorem 1.1, Theorem 1.4 and Theorem 1.5 in Section 3.

2 Auxiliary lemmas

Lemma 2.1

  • Let the assumptions of Theorems 1.1 be valid. Then there existμ > 0 and
    λ=2|ρ|μ,ρ0
    such that
    μQ(x)|ρ|q(x),xI{0}
    and
    μQ(x)|ρ|q(x),xI{0}.
  • LetI = [0, 1) andW ∈ 𝓛(C2). Assume that there existsμ > 0 such that (2.2) is valid. Then (2.1) holds. Moreover, there existsλsuch that forxI ∖ {0}, (1.6) and (1.7) hold.

Proof

The case when ρ = 0 is trivial. Let ρ ≠ 0.

(a) Given any ε > 0, choose

μ=Λ(12+ε)Λ,ρ<0,2ρε,ρ>0.

Then

λ=2|ρ|μ>2|ρ|ΛΛ12,ρ<0,0,ρ>0.

Clearly,

q(x)=2x1x2,q(x)=2(1+x2)(1x2)2.

Then inequality (1.6) with the above relations may be written as

Q(x)|ρ|μq(x),x(0,1).

so the relation (2.1) follows from (2.5) directly.

If we introduce the notation

Q¯(x):=Q(x)|ρ|μq(x),xI,

then according to (2.1),

Q¯(x)0,xI{0}.

With the help of (2.7) for x ∈ (0, 1),

Q¯(x)=0xQ¯(z)dz0,

and

Q¯(x)=0xQ¯(z)dz0.

Inequalities of (2.7), (2.8) and (2.9) give the estimates

μQ(i)(x)|ρ|q(i)(x),x(0,1),i=0,1,2.

Here according to (1.1) and (1.2) we introduce

Q¯(s):=Q(s)|ρ|μq(s),sI,

and then for sI* ∖ {0} = (–1, 1) ∖ {0},

Q¯(s)=Q(s)|ρ|μq(s)=4s2[Q(s2)|ρ|μq(s2)]+2[Q(s2)|ρ|μq(s2)].

Since s ∈ (–1, 1) ∖ {0} is equivalent to s2 ∈ (0, 1), so using (2.12) and (2.10) we obtain

Q¯(s)0,s(1,1){0}.

This proves the relation (2.2).

(b) Given (2.2), using (2.12) it is shown that one of the following cases will hols:

Case 1:

Q(x2)|ρ|μq(x2)0

and

Q(x2)|ρ|μq(x2)0,x(1,1){0};

Case 2:

Q(x2)|ρ|μq(x2)0,x(1,1){0},Q(x2)|ρ|μq(x2)0,x(1,1){0}

and

Q(x2)|ρ|μq(x2)2x2Q(x2)|ρ|μq(x2),x(1,1){0}.

According to (2.7) and (2.8), it is clear that Case 1 gives (2.1).

Here Case 2 means that

Q¯(x)0,x(0,1)

and

Q¯(x)0,x(0,1).

Using the same arguments as (2.7) and (2.8), we see that Case 2 is contradictory. So we get (2.1).

Further, set λ=2|ρ|μ, where μ is defined by (2.3), then coupling with (2.1) we obtain (1.6) and (1.7).□

Remark 2.1

According to the above Lemma (b), the result implies Q″(x) > 0, x ∈ (0, 1). Moreover, we see that the assumptions of Theorems 1.1 and Lemma 2.1(b) are equivalent.

Lemma 2.2

Let

h(x)=(1x)τ1(or(1x)τ),τ>0x[0,1).

Then

h(x)=τ(1x)τ1

and

h(x)=τ(1+τ)(1x)τ2τ(1+x2)(1x2)2.

Proof

By a short calculation we gain

h(x)=τ(1+τ)(1+x)τ+2(1x2)τ+2τ(1+τ)(1+x2)τ+2(1x2)τ+2τ(1+x2)(1x2)2,x[0,1).

3 Proof of theorems

Proof of Theorem 1.1

Here let ρ ≠ 0. The theorem for the case when ρ = 0 is trivial. In deed, the authors in [1, 2] discussed the case when ρ = 0. We use the idea of Theorem 1.1 in [12] with modification, and according to Lemma 2.1 it is more easier to get μ since we only require ε > 0 in formula (2.3).

We set

Q^(x)=Q(x)+ρq(x),q(x):=ln(1x2),xI.

With Definition 1.1 (a) for Q we have

Q^(0)=0

and by (2.4)

xQ^(x)=xQ(x)+2ρxx1x2,xI,

which shows that x′(x) is continuous in I, with limit 0 at 0.

This proves the properties listed in Definition 1.1 (a) with Q replaced by .

By (2.4) and Definition 1.1 (b) for Q,

Q^(x)=Q(x)+ρ2(1+x2)(1x2)2,

which means that ″(x) exists in (0, 1).

In the following parts, the notations μ, Q and Q* are defined by (2.3), (2.6) and (2.11), respectively.

By the notation Q(x) in (2.6)

Q^(x)=Q¯(x)+(|ρ|μ+ρ)q(x),xI,

then coupling with (2.9) and noticing that 0 < μ < 1 for ρ < 0, we see

Q^(x)>0,x(0,1)

and

limx1Q^(x)=.

Meanwhile, by (3.1), (2.10), (2.4) and the fact (|ρ|μ+ρ)>0

Q^(x)>0,x(0,1).

Here according to the notation Q* in (2.11)

Q^(s)=Q¯(s)+(|ρ|μ+ρ)q(s),s(1,1){0}.

By (2.4) and the fact (|ρ|μ+ρ)>0,

(|ρ|μ+ρ)q(s)=(|ρ|μ+ρ)(4s2q(s2)+2q(s2))>0,

provided s ∈ (–1, 1) ∖ {0}, which is equivalent to s2 ∈ (0, 1).

Hence, combining (3.3), (2.2) and (3.4), we get

Q^(s)>0,s(1,1){0}.

This proves the properties listed in Definition 1.1 (b) and (c) with Q replaced by .

In what follows we separate two cases. First, we give an inequality analogous to (2.10).

Using (2.2) and the same arguments as (2.8) and (2.9) with replaced Q(x) by Q*(s) for s ∈ (0, 1), we have Q*′(s) ≥ 0 and Q*(s) ≥ 0.

Similarly, by (2.2) for s ∈ (–1, 0),

Q¯(s)=s0Q¯(z)dz0

and

Q¯(s)=s0Q¯(z)dz0.

Hence, we have the estimates

μ|Q(i)(s)||ρq(i)(s)|,s(1,1){0},i=0,1,2.

  1. Case 1ρ < 0.(d) In this case by (3.2) and (2.10) for x ∈ (0, 1),
    Q^(x)Q^(x)Q(x)+|ρ|q(x)Q(x)+ρq(x)1+μ1μQ(x)Q(x)
    and
    Q^(x)Q^(x)Q(x)|ρ|q(x)Q(x)(1μ)Q(x)Q(x).
    Thus, for the function T^(x)=xQ^(x)Q^(x)
    (1μ)T(x)T^(x)1+μ1μT(x),x(0,1).
    According to Definition 1.1(d) for T(x) and (3.8) for 0 < xy < 1,
    T^(x)1+μ1μT(x)C1+μ1μT(y)C1+μ(1μ)2T^(y).
    We see that (x) is quasi-increasing in (0, 1) and by (3.8), (1.3) and (2.3) for x ∈ (0, 1),
    T^(x)(1μ)T(x)Λ(1μ)=Λ^>12.
    This proves Definition 1.1(d) with Q replaced by .(e) By (2.10), (1.4) and (3.7) for a.e. x ∈ (0, 1),
    |Q^(x)|Q^(x)|Q(x)|+|ρq(x)|Q(x)|ρq(x)|1+μ1μ|Q(x)|Q(x)C11+μ1μQ(x)Q(x)C11+μ(1μ)2Q^(x)Q^(x).
    This proves ∈ 𝓛(C2).Replacing x ∈ (0, 1) with s2, s ∈ (–1, 1) ∖ {0} and multiplying by 2s in the equality of (3.6) we get
    |Q^(s)|Q^(s)1+μ1μ|Q(s)|Q(s),s(1,1){0}.
    By (3.5), (1.5) and (3.9) for a.e. tI* ∖ {J},
    Q^(s)|Q^(s)||Q(s)||ρq(s)||Q(s)|(1μ)Q(s)|Q(s)|C2(1μ)|Q(s)|Q(s)C2(1μ)21+μ|Q^(s)|Q^(s).
    This proves ∈ 𝓛(C2+).
  2. Case 2ρ > 0.(d) By (3.2) and (2.10) for x ∈ (0, 1),
    Q^(x)Q^(x)Q(x)+ρq(x)Q(x)(1+μ)Q(x)Q(x)
    and
    Q^(x)Q^(x)Q(x)Q(x)+ρq(x)Q(x)(1+μ)Q(x).
    Thus for the function T^(x)=xQ^(x)Q^(x),
    11+μT(x)T^(x)(1+μ)T(x),x(0,1).
    According to Definition 1.1(d) for T(x) and (3.12) for 0 < xy < 1,
    T^(x)(1+μ)T(x)C(1+μ)T(y)C(1+μ)2T^(y),
    which shows that (x) is quasi-increasing in (0, 1).Now, we set a function K(x) = xq′(x) – q(x). By (2.4)
    K(x)0,x[0,1)
    and hence K(x) ≥ K(0) = 0, which means
    xρq(x)ρq(x),x[0,1).
    By (3.13) and (1.3)
    xQ^(x)xQ(x)+ρxq(x)ΛQ(x)+ρq(x)min{Λ,1}[Q(x)+ρq(x)]=min{Λ,1}Q^(x),
    which gives (x) ≥ min{Λ, 1} > 1/2, x ∈ (0, 1).This proves Definition 1.1(d) with Q replaced by .(e) By (2.10), (1.4) and (3.11) for a.e. x ∈ (0, 1),
    |Q^(x)|Q^(x)Q(x)+ρq(x)Q(x)(1+μ)Q(x)Q(x)C1(1+μ)Q(x)Q(x)C1(1+μ)2Q^(x)Q^(x).
    Now we obtain ∈ 𝓛(C2).By (3.10) and with the same argument as (3.9) we have
    |Q^(s)|Q^(s)(1+μ)|Q(s)|Q(s),s(1,1){0}.
    By (3.5), (1.5) and (3.14) for a.e. tI* ∖ {J},
    Q^(s)|Q^(s)|Q(s)|Q(s)+ρq(s)|11+μQ(s)|Q(s)|C21+μ|Q(s)|Q(s)C2(1+μ)2|Q^(s)|Q^(s).
    So we conclude that ∈ 𝓛(C2+).□

Proof of Theorem 1.4

(a) Set f(x) = xQ′(x) – Q(x), and then applying Lemma 2.2 we have

f(x)=xQ(x)=τ(1+τ)x(1x)τ20,x[0,1),

which meas that f(x) ≥ f(0) = 0. It is equivalent to

T(x)1,x(0,1).

(b) Using (2.14) we have

Q(x)τ(1+x2)(1x2)2,x[0,1).

Then by (1.11)

λ=τ>4|ρ|,ρ<0,0,ρ>0.

By (3.15) we see Λ = 1 and hence coupling with the above relation we obtain that (1.6) and (1.7) are valid. Thus applying Theorem 1.1 we get W ∈ 𝓛(C2+).□

Proof of Theorem 1.5

(a) Put

h(x)=(1x)τ,

and

gj(x)=expj(h(x)),x[0,1).

By calculation

Q(x)=gk(x)=h(x)j=1kgj(x)=h(x)j=1kexpj(h(x)),

Q(x)=h(x)j=1kgj(x)+h(x)j=1kgj(x)i=1ijkgi(x)=h(x)j=1kgj(x)+h(x)2j=1ki=1j1gi(x)j=1kgj(x)=j=1kexpj(h(x))h(x)+h(x)2j=1ki=1j1expi(h(x)).

By (3.16), (2.13) and (2.14) for x ∈ [0, 1),

f(x)=(xQ(x)Q(x))=xQ(x)=xj=1kexpj(h(x))τ(1+τ)(1x)τ2+h(x)2j=1ki=1j1expi(h(x))0,

and hence by the same argument as Theorem 1.4 (a) inequality (1.10) holds.

(b) By (3.16) and (2.14)

Q(x)j=1kexpj(h(x))h(x)j=1kexpj(1)τ(1+τ)(1x)τ2j=1kexpj(1)τ1+x2(1x2)2.

Then by (1.12)

λ=τj=1kexpj(1)>4|ρ|,ρ<0,0,ρ>0.

By the statements of (a), we see Λ = 1 and hence coupling with the above relation (1.6) and (1.7) are valid. Thus applying Theorem 1.1 we get W ∈ 𝓛(C2+).□

Acknowledgement

The research is supported in part by the National Natural Science Foundation of China (No. 11626060) and by Scientific Research Fund of Fujian Provincial Education Department (No. JAT160172).

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    R. Liu and Y.G. Shi, Generalized Christoffel functions for Jacobi-exponential weights on [−1, 1], Acta Math. Hungar. 148 (2016), no. 1, 17–42, .

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    I. Notarangelo, Polynomial inequalities and embedding theorems with exponential weights in (−1, 1), Acta Math. Hungar. 134 (2012), no. 3, 286–306, .

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    G. Mastroianni and I. Notarangelo, Lagrange interpolation with exponential weights on (−1, 1), J. Approx. Theory 167 (2013), 65–93, .

    • Crossref
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    G. Mastroianni and I. Notarangelo, Lagrange interpolation at Pollaczek-Laguerre zeros on the real semiaxis, J. Approx. Theory 245 (2019), 83–100, .

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    Y.G. Shi, Orthogonal polynomials for Jacobi-exponential weights (1− x 2)ρ e Q(x) on (−1, 1), Acta Math. Hungar. 140 (2013), no. 4, 363–376, .

    • Crossref
    • Export Citation

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    A.L. Levin and D.S. Lubinsky, Orthogonal polynomials for exponential weights x 2ρ e −2Q(x) on [0, d) II, J. Approx. Theory 139 (2006), 107–143, .

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    A.L. Levin and D.S. Lubinsky, Christoffel functions and orthogonal polynomials for exponential weights on [−1, 1], Memoirs Amer. Math. Soc. 111 (1994), no. 535, .

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    A.L. Levin and D.S. Lubinsky, Orthogonal Polynomials for Exponential Weights, Springer, New York, 2001.

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    T. Kasuga and R. Sakai, Orthogonal polynomials with generalized Freud type weights, J. Approx. Theory 121 (2003), no.1, 13–53, .

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    R. Liu and Y.G. Shi, The zeros of orthogonal for Jacobi-exponential weights, Abstr. Appl. Anal. 2012 (2012), Article ID 386359, .

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    Y.G. Shi, Generalized Christoffel functions for Jacobi-exponential weights, Acta Math. Hungar. 140 (2013), 71–89, .

    • Crossref
    • Export Citation
  • [8]

    R. Liu and Y.G. Shi, Generalized Christoffel functions for Jacobi-exponential weights on [−1, 1], Acta Math. Hungar. 148 (2016), no. 1, 17–42, .

    • Crossref
    • Export Citation
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    I. Notarangelo, Polynomial inequalities and embedding theorems with exponential weights in (−1, 1), Acta Math. Hungar. 134 (2012), no. 3, 286–306, .

    • Crossref
    • Export Citation
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    G. Mastroianni and I. Notarangelo, Lagrange interpolation with exponential weights on (−1, 1), J. Approx. Theory 167 (2013), 65–93, .

    • Crossref
    • Export Citation
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    G. Mastroianni and I. Notarangelo, Lagrange interpolation at Pollaczek-Laguerre zeros on the real semiaxis, J. Approx. Theory 245 (2019), 83–100, .

    • Crossref
    • Export Citation
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    Y.G. Shi, Orthogonal polynomials for Jacobi-exponential weights (1− x 2)ρ e Q(x) on (−1, 1), Acta Math. Hungar. 140 (2013), no. 4, 363–376, .

    • Crossref
    • Export Citation
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